Daily Papers

Stochastic optimizers are central to deep learning, yet widely used methods such as Adam and Adan can degrade in non-stationary or noisy environments, partly due to their reliance on momentum-based magnitude estimates. We introduce Ano, a novel optimizer that decouples direction and magnitude: momentum is used for directional smoothing, while instantaneous gradient magnitudes determine step size. This design improves robustness to gradient noise while retaining the simplicity and efficiency of first-order methods. We further propose Anolog, which removes sensitivity to the momentum coefficient by expanding its window over time via a logarithmic schedule. We establish non-convex convergence guarantees with a convergence rate similar to other sign-based methods, and empirically show that Ano provides substantial gains in noisy and non-stationary regimes such as reinforcement learning, while remaining competitive on low-noise tasks such as standard computer vision benchmarks.

We demonstrate the possibility of what we call sparse learning: accelerated training of deep neural networks that maintain sparse weights throughout training while achieving dense performance levels. We accomplish this by developing sparse momentum, an algorithm which uses exponentially smoothed gradients (momentum) to identify layers and weights which reduce the error efficiently. Sparse momentum redistributes pruned weights across layers according to the mean momentum magnitude of each layer. Within a layer, sparse momentum grows weights according to the momentum magnitude of zero-valued weights. We demonstrate state-of-the-art sparse performance on MNIST, CIFAR-10, and ImageNet, decreasing the mean error by a relative 8%, 15%, and 6% compared to other sparse algorithms. Furthermore, we show that sparse momentum reliably reproduces dense performance levels while providing up to 5.61x faster training. In our analysis, ablations show that the benefits of momentum redistribution and growth increase with the depth and size of the network. Additionally, we find that sparse momentum is insensitive to the choice of its hyperparameters suggesting that sparse momentum is robust and easy to use.

The magnitude of a finite metric space has recently emerged as a novel invariant quantity, allowing to measure the effective size of a metric space. Despite encouraging first results demonstrating the descriptive abilities of the magnitude, such as being able to detect the boundary of a metric space, the potential use cases of magnitude remain under-explored. In this work, we investigate the properties of the magnitude on images, an important data modality in many machine learning applications. By endowing each individual images with its own metric space, we are able to define the concept of magnitude on images and analyse the individual contribution of each pixel with the magnitude vector. In particular, we theoretically show that the previously known properties of boundary detection translate to edge detection abilities in images. Furthermore, we demonstrate practical use cases of magnitude for machine learning applications and propose a novel magnitude model that consists of a computationally efficient magnitude computation and a learnable metric. By doing so, we address the computational hurdle that used to make magnitude impractical for many applications and open the way for the adoption of magnitude in machine learning research.

The Mechanistic Interpretability (MI) program has mapped the Transformer as a precise computational graph. We extend this graph with a conservation law and time-varying AC dynamics, viewing it as a physical circuit. We introduce Momentum Attention, a symplectic augmentation embedding physical priors via the kinematic difference operator p_t = q_t - q_{t-1}, implementing the symplectic shear q_t = q_t + γp_t on queries and keys. We identify a fundamental Symplectic-Filter Duality: the physical shear is mathematically equivalent to a High-Pass Filter. This duality is our cornerstone contribution -- by injecting kinematic momentum, we sidestep the topological depth constraint (L geq 2) for induction head formation. While standard architectures require two layers for induction from static positions, our extension grants direct access to velocity, enabling Single-Layer Induction and Spectral Forensics via Bode Plots. We formalize an Orthogonality Theorem proving that DC (semantic) and AC (mechanistic) signals segregate into orthogonal frequency bands when Low-Pass RoPE interacts with High-Pass Momentum. Validated through 5,100+ controlled experiments (documented in Supplementary Appendices A--R and 27 Jupyter notebooks), our 125M Momentum model exceeds expectations on induction-heavy tasks while tracking a 350M baseline within sim2.9% validation loss. Dedicated associative recall experiments reveal a scaling law γ^* = 4.17 times N^{-0.74} establishing momentum-depth fungibility. We offer this framework as a complementary analytical toolkit connecting Generative AI, Hamiltonian Physics, and Signal Processing.

Vector space embedding models like word2vec, GloVe, fastText, and ELMo are extremely popular representations in natural language processing (NLP) applications. We present Magnitude, a fast, lightweight tool for utilizing and processing embeddings. Magnitude is an open source Python package with a compact vector storage file format that allows for efficient manipulation of huge numbers of embeddings. Magnitude performs common operations up to 60 to 6,000 times faster than Gensim. Magnitude introduces several novel features for improved robustness like out-of-vocabulary lookups.

Metric space magnitude, an active field of research in algebraic topology, is a scalar quantity that summarizes the effective number of distinct points that live in a general metric space. The {\em weighting vector} is a closely-related concept that captures, in a nontrivial way, much of the underlying geometry of the original metric space. Recent work has demonstrated that when the metric space is Euclidean, the weighting vector serves as an effective tool for boundary detection. We recast this result and show the weighting vector may be viewed as a solution to a kernelized SVM. As one consequence, we apply this new insight to the task of outlier detection, and we demonstrate performance that is competitive or exceeds performance of state-of-the-art techniques on benchmark data sets. Under mild assumptions, we show the weighting vector, which has computational cost of matrix inversion, can be efficiently approximated in linear time. We show how nearest neighbor methods can approximate solutions to the minimization problems defined by SVMs.

Metric space magnitude, an active subject of research in algebraic topology, originally arose in the context of biology, where it was used to represent the effective number of distinct species in an environment. In a more general setting, the magnitude of a metric space is a real number that aims to quantify the effective number of distinct points in the space. The contribution of each point to a metric space's global magnitude, which is encoded by the {\em weighting vector}, captures much of the underlying geometry of the original metric space. Surprisingly, when the metric space is Euclidean, the weighting vector also serves as an effective tool for boundary detection. This allows the weighting vector to serve as the foundation of novel algorithms for classic machine learning tasks such as classification, outlier detection and active learning. We demonstrate, using experiments and comparisons on classic benchmark datasets, the promise of the proposed magnitude and weighting vector-based approaches.

Efficiently exploring complex loss landscapes is key to the performance of deep neural networks. While momentum-based optimizers are widely used in state-of-the-art setups, classical momentum can still struggle with large, misaligned gradients, leading to oscillations. To address this, we propose Torque-Aware Momentum (TAM), which introduces a damping factor based on the angle between the new gradients and previous momentum, stabilizing the update direction during training. Empirical results show that TAM, which can be combined with both SGD and Adam, enhances exploration, handles distribution shifts more effectively, and improves generalization performance across various tasks, including image classification and large language model fine-tuning, when compared to classical momentum-based optimizers.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Normalization techniques are a boon for modern deep learning. They let weights converge more quickly with often better generalization performances. It has been argued that the normalization-induced scale invariance among the weights provides an advantageous ground for gradient descent (GD) optimizers: the effective step sizes are automatically reduced over time, stabilizing the overall training procedure. It is often overlooked, however, that the additional introduction of momentum in GD optimizers results in a far more rapid reduction in effective step sizes for scale-invariant weights, a phenomenon that has not yet been studied and may have caused unwanted side effects in the current practice. This is a crucial issue because arguably the vast majority of modern deep neural networks consist of (1) momentum-based GD (e.g. SGD or Adam) and (2) scale-invariant parameters. In this paper, we verify that the widely-adopted combination of the two ingredients lead to the premature decay of effective step sizes and sub-optimal model performances. We propose a simple and effective remedy, SGDP and AdamP: get rid of the radial component, or the norm-increasing direction, at each optimizer step. Because of the scale invariance, this modification only alters the effective step sizes without changing the effective update directions, thus enjoying the original convergence properties of GD optimizers. Given the ubiquity of momentum GD and scale invariance in machine learning, we have evaluated our methods against the baselines on 13 benchmarks. They range from vision tasks like classification (e.g. ImageNet), retrieval (e.g. CUB and SOP), and detection (e.g. COCO) to language modelling (e.g. WikiText) and audio classification (e.g. DCASE) tasks. We verify that our solution brings about uniform gains in those benchmarks. Source code is available at https://github.com/clovaai/AdamP.

We introduce Leap+Verify, a framework that applies speculative execution -- predicting future model weights and validating predictions before acceptance -- to accelerate neural network training. Inspired by speculative decoding in language model inference and by the Automatically Scalable Computation (ASC) architecture for program execution, Leap+Verify decomposes training into three dynamically detected regimes (chaotic, transition, stable) using activation-space cosine similarity as a real-time Lyapunov proxy signal. Within each regime, analytic weight predictors (momentum, linear, quadratic extrapolation) attempt to forecast model parameters K training steps ahead; predictions are accepted only when validated against a held-out loss criterion. We evaluate Leap+Verify on GPT-2 124M and Qwen 2.5-1.5B trained on WikiText-103 across five random seeds, sweeping prediction depth K in {5, 10, 25, 50, 75, 100}. Momentum-based prediction (Adam moment extrapolation) fails catastrophically at both scales, with predicted losses exceeding actuals by 100-10,000x -- a universal norm explosion in optimizer-state extrapolation. Finite-difference predictors (linear, quadratic) succeed where momentum fails: at 124M, they achieve 24% strict acceptance at K=5 in stable regimes; at 1.5B, they achieve 37% strict acceptance in transition regimes. The scale-dependent finding is in regime distribution: GPT-2 124M spends 34% of training in stable regime, while Qwen 1.5B spends 64% in chaotic regime and reaches stable in only 0-2 of 40 checkpoints. Larger models are more predictable when predictable, but less often predictable -- the practical bottleneck shifts from predictor accuracy to regime availability. Cross-seed results are highly consistent (less than 1% validation loss variance), and the three-regime framework produces identical phase boundaries (plus or minus 50 steps) across seeds.

Adaptive gradient methods, e.g. Adam, have achieved tremendous success in machine learning. Scaling the learning rate element-wisely by a certain form of second moment estimate of gradients, such methods are able to attain rapid training of modern deep neural networks. Nevertheless, they are observed to suffer from compromised generalization ability compared with stochastic gradient descent (SGD) and tend to be trapped in local minima at an early stage during training. Intriguingly, we discover that substituting the gradient in the second raw moment estimate term with its momentumized version in Adam can resolve the issue. The intuition is that gradient with momentum contains more accurate directional information and therefore its second moment estimation is a more favorable option for learning rate scaling than that of the raw gradient. Thereby we propose AdaMomentum as a new optimizer reaching the goal of training fast while generalizing much better. We further develop a theory to back up the improvement in generalization and provide convergence guarantees under both convex and nonconvex settings. Extensive experiments on a wide range of tasks and models demonstrate that AdaMomentum exhibits state-of-the-art performance and superior training stability consistently.

The vast majority of modern deep learning models are trained with momentum-based first-order optimizers. The momentum term governs the optimizer's memory by determining how much each past gradient contributes to the current convergence direction. Fundamental momentum methods, such as Nesterov Accelerated Gradient and the Heavy Ball method, as well as more recent optimizers such as AdamW and Lion, all rely on the momentum coefficient that is customarily set to β= 0.9 and kept constant during model training, a strategy widely used by practitioners, yet suboptimal. In this paper, we introduce an adaptive memory mechanism that replaces constant momentum with a dynamic momentum coefficient that is adjusted online during optimization. We derive our method by approximating the objective function using two planes: one derived from the gradient at the current iterate and the other obtained from the accumulated memory of the past gradients. To the best of our knowledge, such a proximal framework was never used for momentum-based optimization. Our proposed approach is novel, extremely simple to use, and does not rely on extra assumptions or hyperparameter tuning. We implement adaptive memory variants of both SGD and AdamW across a wide range of learning tasks, from simple convex problems to large-scale deep learning scenarios, demonstrating that our approach can outperform standard SGD and Adam with hand-tuned momentum coefficients. Finally, our work opens doors for new ways of inducing adaptivity in optimization.

Earthquake prediction and seismic hazard assessment remain fundamental challenges in geophysics, with existing machine learning approaches often operating as black boxes that ignore established physical laws. We introduce POSEIDON (Physics-Optimized Seismic Energy Inference and Detection Operating Network), a physics-informed energy-based model for unified multi-task seismic event prediction, alongside the Poseidon dataset -- the largest open-source global earthquake catalog comprising 2.8 million events spanning 30 years. POSEIDON embeds fundamental seismological principles, including the Gutenberg-Richter magnitude-frequency relationship and Omori-Utsu aftershock decay law, as learnable constraints within an energy-based modeling framework. The architecture simultaneously addresses three interconnected prediction tasks: aftershock sequence identification, tsunami generation potential, and foreshock detection. Extensive experiments demonstrate that POSEIDON achieves state-of-the-art performance across all tasks, outperforming gradient boosting, random forest, and CNN baselines with the highest average F1 score among all compared methods. Crucially, the learned physics parameters converge to scientifically interpretable values -- Gutenberg-Richter b-value of 0.752 and Omori-Utsu parameters p=0.835, c=0.1948 days -- falling within established seismological ranges while enhancing rather than compromising predictive accuracy. The Poseidon dataset is publicly available at https://huggingface.co/datasets/BorisKriuk/Poseidon, providing pre-computed energy features, spatial grid indices, and standardized quality metrics to advance physics-informed seismic research.

In machine learning applications, it is well known that carefully designed learning rate (step size) schedules can significantly improve the convergence of commonly used first-order optimization algorithms. Therefore how to set step size adaptively becomes an important research question. A popular and effective method is the Polyak step size, which sets step size adaptively for gradient descent or stochastic gradient descent without the need to estimate the smoothness parameter of the objective function. However, there has not been a principled way to generalize the Polyak step size for algorithms with momentum accelerations. This paper presents a general framework to set the learning rate adaptively for first-order optimization methods with momentum, motivated by the derivation of Polyak step size. It is shown that the resulting methods are much less sensitive to the choice of momentum parameter and may avoid the oscillation of the heavy-ball method on ill-conditioned problems. These adaptive step sizes are further extended to the stochastic settings, which are attractive choices for stochastic gradient descent with momentum. Our methods are demonstrated to be more effective for stochastic gradient methods than prior adaptive step size algorithms in large-scale machine learning tasks.

We study the momentum-based minimization of a diffuse perimeter functional on Euclidean spaces and on graphs with applications to semi-supervised classification tasks in machine learning. While the gradient flow in the task at hand is a parabolic partial differential equation, the momentum-method corresponds to a damped hyperbolic PDE, leading to qualitatively and quantitatively different trajectories. Using a convex-concave splitting-based FISTA-type time discretization, we demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large. With large time steps, the PDE analysis offers only limited insight into the geometric behavior of solutions and typical hyperbolic phenomena like loss of regularity are not be observed in sample simulations.

For decades, classical point process models, such as the epidemic-type aftershock sequence (ETAS) model, have been widely used for forecasting the event times and locations of earthquakes. Recent advances have led to Neural Point Processes (NPPs), which promise greater flexibility and improvements over such classical models. However, the currently-used benchmark for NPPs does not represent an up-to-date challenge in the seismological community, since it contains data leakage and omits the largest earthquake sequence from the region. Additionally, initial earthquake forecasting benchmarks fail to compare NPPs with state-of-the-art forecasting models commonly used in seismology. To address these gaps, we introduce EarthquakeNPP: a benchmarking platform that curates and standardizes existing public resources: globally available earthquake catalogs, the ETAS model, and evaluation protocols from the seismology community. The datasets cover a range of small to large target regions within California, dating from 1971 to 2021, and include different methodologies for dataset generation. Benchmarking experiments, using both log-likelihood and generative evaluation metrics widely recognised in seismology, show that none of the five NPPs tested outperform ETAS. These findings suggest that current NPP implementations are not yet suitable for practical earthquake forecasting. Nonetheless, EarthquakeNPP provides a platform to foster future collaboration between the seismology and machine learning communities.

We introduce Gradient Descent with Adaptive Momentum Scaling (Grams), a novel optimization algorithm that decouples the direction and magnitude of parameter updates in deep learning. Unlike traditional optimizers that directly integrate momentum into updates, Grams separates the update direction, derived from current gradients, from momentum, which is used solely for adaptive magnitude scaling. This approach enables Grams to achieve improved loss descent compared to state-of-the-art cautious and momentum-based optimizers. We establish a global convergence guarantee for Grams and validate its effectiveness through extensive empirical evaluations. The results demonstrate Grams' superior performance, including faster convergence and better generalization, compared to widely-used optimizers such as Adam, Lion, and their cautious variants. Our results highlight Grams' potential as a transformative approach for efficient optimization in large-scale machine learning.

Is the standard weight decay in AdamW truly optimal? Although AdamW decouples weight decay from adaptive gradient scaling, a fundamental conflict remains: the Radial Tug-of-War. In deep learning, gradients tend to increase parameter norms to expand effective capacity while steering directions to learn features, whereas weight decay indiscriminately suppresses norm growth. This push--pull interaction induces radial oscillations, injecting noise into Adam's second-moment estimates and potentially degrading delicate tangential feature learning. We argue that magnitude and direction play distinct roles and should be decoupled in optimizer dynamics. We propose Orthogonal Dynamics Decoupling and instantiate it as AdamO: an SGD-style update handles the one-dimensional norm control, while Adam's adaptive preconditioning is confined to the tangential subspace. AdamO further incorporates curvature-adaptive radial step sizing and architecture-aware rules and projections for scale-invariant layers and low-dimensional parameters. Experiments on vision and language tasks show that AdamO improves generalization and stability over AdamW without introducing additional complex constraints.

Large pre-trained, zero-shot capable models have shown considerable success both for standard transfer and adaptation tasks, with particular robustness towards distribution shifts. In addition, subsequent fine-tuning can considerably improve performance on a selected downstream task. However, through naive fine-tuning, these zero-shot models lose their generalizability and robustness towards distribution shifts. This is a particular problem for tasks such as Continual Learning (CL), where continuous adaptation has to be performed as new task distributions are introduced sequentially. In this work, we showcase that where fine-tuning falls short to adapt such zero-shot capable models, simple momentum-based weight interpolation can provide consistent improvements for CL tasks in both memory-free and memory-based settings. In particular, we find improvements of over +4% on standard CL benchmarks, while reducing the error to the upper limit of jointly training on all tasks at once in parts by more than half, allowing the continual learner to inch closer to the joint training limits.

Sparse Mixture of Experts (SMoE) has become the key to unlocking unparalleled scalability in deep learning. SMoE has the potential to exponentially increase parameter count while maintaining the efficiency of the model by only activating a small subset of these parameters for a given sample. However, it has been observed that SMoE suffers from unstable training and has difficulty adapting to new distributions, leading to the model's lack of robustness to data contamination. To overcome these limitations, we first establish a connection between the dynamics of the expert representations in SMoEs and gradient descent on a multi-objective optimization problem. Leveraging our framework, we then integrate momentum into SMoE and propose a new family of SMoEs named MomentumSMoE. We theoretically prove and numerically demonstrate that MomentumSMoE is more stable and robust than SMoE. In particular, we verify the advantages of MomentumSMoE over SMoE on a variety of practical tasks including ImageNet-1K object recognition and WikiText-103 language modeling. We demonstrate the applicability of MomentumSMoE to many types of SMoE models, including those in the Sparse MoE model for vision (V-MoE) and the Generalist Language Model (GLaM). We also show that other advanced momentum-based optimization methods, such as Adam, can be easily incorporated into the MomentumSMoE framework for designing new SMoE models with even better performance, almost negligible additional computation cost, and simple implementations.

Open-ended text generation with autoregressive language models (LMs) is one of the core tasks in natural language processing. However, maximization-based decoding methods (e.g., greedy/beam search) often lead to the degeneration problem, i.e., the generated text is unnatural and contains undesirable repetitions. Existing solutions to this problem either introduce randomness prone to incoherence or require a look-ahead mechanism that demands extra computational overhead. In this study, we formulate open-ended text generation from a new perspective, i.e., we view it as an exploration process within a directed graph. Thereby, we understand the phenomenon of degeneration as circular loops within the directed graph. Based on our formulation, we propose a novel decoding method -- momentum decoding -- which encourages the LM to greedily explore new nodes outside the current graph. Meanwhile, it also allows the LM to return to the existing nodes with a momentum downgraded by a pre-defined resistance function. We extensively test our approach on three benchmarks from different domains through automatic and human evaluations. The results show that momentum decoding performs comparably with the current state of the art while enjoying notably improved inference speed and computation FLOPs. Furthermore, we conduct a detailed analysis to reveal the merits and inner workings of our approach. Our codes and other related resources are publicly available at https://github.com/gmftbyGMFTBY/MomentumDecoding.

Recent video diffusion models can synthesize visually compelling clips, yet often violate basic physical laws-objects float, accelerations drift, and collisions behave inconsistently-revealing a persistent gap between visual realism and physical realism. We propose NewtonRewards, the first physics-grounded post-training framework for video generation based on verifiable rewards. Instead of relying on human or VLM feedback, NewtonRewards extracts measurable proxies from generated videos using frozen utility models: optical flow serves as a proxy for velocity, while high-level appearance features serve as a proxy for mass. These proxies enable explicit enforcement of Newtonian structure through two complementary rewards: a Newtonian kinematic constraint enforcing constant-acceleration dynamics, and a mass conservation reward preventing trivial, degenerate solutions. We evaluate NewtonRewards on five Newtonian Motion Primitives (free fall, horizontal/parabolic throw, and ramp sliding down/up) using our newly constructed large-scale benchmark, NewtonBench-60K. Across all primitives in visual and physics metrics, NewtonRewards consistently improves physical plausibility, motion smoothness, and temporal coherence over prior post-training methods. It further maintains strong performance under out-of-distribution shifts in height, speed, and friction. Our results show that physics-grounded verifiable rewards offer a scalable path toward physics-aware video generation.

Hypernetworks, neural networks that predict the parameters of another neural network, are powerful models that have been successfully used in diverse applications from image generation to multi-task learning. Unfortunately, existing hypernetworks are often challenging to train. Training typically converges far more slowly than for non-hypernetwork models, and the rate of convergence can be very sensitive to hyperparameter choices. In this work, we identify a fundamental and previously unidentified problem that contributes to the challenge of training hypernetworks: a magnitude proportionality between the inputs and outputs of the hypernetwork. We demonstrate both analytically and empirically that this can lead to unstable optimization, thereby slowing down convergence, and sometimes even preventing any learning. We present a simple solution to this problem using a revised hypernetwork formulation that we call Magnitude Invariant Parametrizations (MIP). We demonstrate the proposed solution on several hypernetwork tasks, where it consistently stabilizes training and achieves faster convergence. Furthermore, we perform a comprehensive ablation study including choices of activation function, normalization strategies, input dimensionality, and hypernetwork architecture; and find that MIP improves training in all scenarios. We provide easy-to-use code that can turn existing networks into MIP-based hypernetworks.

Momentum is known to accelerate the convergence of gradient descent in strongly convex settings without stochastic gradient noise. In stochastic optimization, such as training neural networks, folklore suggests that momentum may help deep learning optimization by reducing the variance of the stochastic gradient update, but previous theoretical analyses do not find momentum to offer any provable acceleration. Theoretical results in this paper clarify the role of momentum in stochastic settings where the learning rate is small and gradient noise is the dominant source of instability, suggesting that SGD with and without momentum behave similarly in the short and long time horizons. Experiments show that momentum indeed has limited benefits for both optimization and generalization in practical training regimes where the optimal learning rate is not very large, including small- to medium-batch training from scratch on ImageNet and fine-tuning language models on downstream tasks.

Orthogonal gradient updates have emerged as a promising direction in optimization for machine learning. However, traditional approaches such as SVD/QR decomposition incur prohibitive computational costs of O(n^3) and underperform compared to well-tuned SGD with momentum, since momentum is applied only after strict orthogonalization. Recent advances, such as Muon, improve efficiency by applying momentum before orthogonalization and producing semi-orthogonal matrices via Newton-Schulz iterations, reducing complexity to O(n^2). Nevertheless, quadratic costs remain a bottleneck. In this work, we study the semi-orthogonal properties of momentum-based updates and develop a method to bound momentum updates under a spectral-norm trust region, preserving directional information without requiring explicit semi-orthogonalization. We propose AuON (Alternative Unit-norm momentum updates by Normalized nonlinear scaling), a linear-time optimizer that achieves strong performance without constructing semi-orthogonal matrices, while preserving structural alignment and reconditioning ill-posed updates. Our approach combines hyperbolic-cosine RMS scaling transformations with normalization, demonstrating both effectiveness and computational efficiency compared to Newton-Schulz methods. We further introduce a hybrid variant (Hybrid-AuON) that applies a single Newton-Schulz iteration. Experiments across vision and language benchmarks show that AuON and its hybrid variant achieve performance comparable to strong baselines such as AdamW and Muon. Code is available at: https://github.com/ryyzn9/AuON

Effective earthquake risk reduction relies on accurate site-specific evaluations. This requires models that can represent the influence of local site conditions on ground motion characteristics. In this context, data driven approaches that learn site controlled signatures from recorded ground motions offer a promising direction. We address strong ground motion generation from time-domain accelerometer records and introduce the TimesNet-Gen, a time-domain conditional generator. The approach uses a station specific latent bottleneck. We evaluate generation by comparing HVSR curves and fundamental site-frequency f_0 distributions between real and generated records per station, and summarize station specificity with a score based on the f_0 distribution confusion matrices. TimesNet-Gen achieves strong station-wise alignment and compares favorably with a spectrogram-based conditional VAE baseline for site-specific strong motion synthesis. Our codes are available via https://github.com/brsylmz23/TimesNet-Gen.

Magnitude pruning is a widely used strategy for reducing model size in pure supervised learning; however, it is less effective in the transfer learning regime that has become standard for state-of-the-art natural language processing applications. We propose the use of movement pruning, a simple, deterministic first-order weight pruning method that is more adaptive to pretrained model fine-tuning. We give mathematical foundations to the method and compare it to existing zeroth- and first-order pruning methods. Experiments show that when pruning large pretrained language models, movement pruning shows significant improvements in high-sparsity regimes. When combined with distillation, the approach achieves minimal accuracy loss with down to only 3% of the model parameters.

In this paper we introduce the concept of what we call "NUDAR" (NeUtrino Direction and Ranging), making the point that measurements of the observed energy and direction vectors can be employed to passively deduce the exact three-dimensional location and thermal power of geophysical and anthropogenic neutrino sources from even a single detector. We present the most precise background estimates to date, all handled in full three dimensions, as functions of depth and geographical location. For the present calculations, we consider a hypothetical 138 kiloton detector which can be transported to an ocean site and deployed to an operational depth. We present a Bayesian estimation framework to incorporate any a priori knowledge of the reactor that we are trying to detect, as well as the estimated uncertainty in the background and the oscillation parameters. Most importantly, we fully employ the knowledge of the reactor spectrum and the distance-dependent effects of neutrino oscillations on such spectra. The latter, in particular, makes possible determination of range from one location, given adequate signal statistics. Further, we explore the rich potential of improving detection with even modest improvements in individual neutrino direction determination. We conclude that a 300 MWth reactor can indeed be geolocated, and its operating power estimated with one or two detectors in the hundred kiloton class at ranges out to a few hundred kilometers. We note that such detectors would have natural and non-interfering utility for scientific studies of geo-neutrinos, neutrino oscillations, and astrophysical neutrinos. This motivates the development of cost effective methods of constructing and deploying such next generation detectors.

Understanding the physical world is essential for generalist AI agents. However, it remains unclear whether state-of-the-art vision perception models (e.g., large VLMs) can reason physical properties quantitatively. Existing evaluations are predominantly VQA-based and qualitative, offering limited insight into whether these models can infer the kinematic quantities of moving objects from video observations. To address this, we present QuantiPhy, the first benchmark designed to quantitatively measure a VLM's physical reasoning ability. Comprising more than 3.3K video-text instances with numerical ground truth, QuantiPhy evaluates a VLM's performance on estimating an object's size, velocity, and acceleration at a given timestamp, using one of these properties as an input prior. The benchmark standardizes prompts and scoring to assess numerical accuracy, enabling fair comparisons across models. Our experiments on state-of-the-art VLMs reveal a consistent gap between their qualitative plausibility and actual numerical correctness. We further provide an in-depth analysis of key factors like background noise, counterfactual priors, and strategic prompting and find that state-of-the-art VLMs lean heavily on pre-trained world knowledge rather than faithfully using the provided visual and textual inputs as references when reasoning kinematic properties quantitatively. QuantiPhy offers the first rigorous, scalable testbed to move VLMs beyond mere verbal plausibility toward a numerically grounded physical understanding.

Momentum based optimizers are central to a wide range of machine learning applications. These typically rely on an Exponential Moving Average (EMA) of gradients, which decays exponentially the present contribution of older gradients. This accounts for gradients being local linear approximations which lose their relevance as the iterate moves along the loss landscape. This work questions the use of a single EMA to accumulate past gradients and empirically demonstrates how this choice can be sub-optimal: a single EMA cannot simultaneously give a high weight to the immediate past, and a non-negligible weight to older gradients. Building on this observation, we propose AdEMAMix, a simple modification of the Adam optimizer with a mixture of two EMAs to better take advantage of past gradients. Our experiments on language modeling and image classification show -- quite surprisingly -- that gradients can stay relevant for tens of thousands of steps. They help to converge faster, and often to lower minima: e.g., a 1.3B parameter AdEMAMix LLM trained on 101B tokens performs comparably to an AdamW model trained on 197B tokens (+95%). Moreover, our method significantly slows-down model forgetting during training. Our work motivates further exploration of different types of functions to leverage past gradients, beyond EMAs.

As their size increases, Large Languages Models (LLMs) are natural candidates for network pruning methods: approaches that drop a subset of network weights while striving to preserve performance. Existing methods, however, require either retraining, which is rarely affordable for billion-scale LLMs, or solving a weight reconstruction problem reliant on second-order information, which may also be computationally expensive. In this paper, we introduce a novel, straightforward yet effective pruning method, termed Wanda (Pruning by Weights and activations), designed to induce sparsity in pretrained LLMs. Motivated by the recent observation of emergent large magnitude features in LLMs, our approach prunes weights with the smallest magnitudes multiplied by the corresponding input activations, on a per-output basis. Notably, Wanda requires no retraining or weight update, and the pruned LLM can be used as is. We conduct a thorough evaluation of our method Wanda on LLaMA and LLaMA-2 across various language benchmarks. Wanda significantly outperforms the established baseline of magnitude pruning and performs competitively against recent method involving intensive weight update. Code is available at https://github.com/locuslab/wanda.

Learning robotic manipulation policies directly in the real world can be expensive and time-consuming. While reinforcement learning (RL) policies trained in simulation present a scalable alternative, effective sim-to-real transfer remains challenging, particularly for tasks that require precise dynamics. To address this, we propose Phys2Real, a real-to-sim-to-real RL pipeline that combines vision-language model (VLM)-inferred physical parameter estimates with interactive adaptation through uncertainty-aware fusion. Our approach consists of three core components: (1) high-fidelity geometric reconstruction with 3D Gaussian splatting, (2) VLM-inferred prior distributions over physical parameters, and (3) online physical parameter estimation from interaction data. Phys2Real conditions policies on interpretable physical parameters, refining VLM predictions with online estimates via ensemble-based uncertainty quantification. On planar pushing tasks of a T-block with varying center of mass (CoM) and a hammer with an off-center mass distribution, Phys2Real achieves substantial improvements over a domain randomization baseline: 100% vs 79% success rate for the bottom-weighted T-block, 57% vs 23% in the challenging top-weighted T-block, and 15% faster average task completion for hammer pushing. Ablation studies indicate that the combination of VLM and interaction information is essential for success. Project website: https://phys2real.github.io/ .

We introduce novel variants of momentum by incorporating the variance of the stochastic loss function. The variance characterizes the confidence or uncertainty of the local features of the averaged loss surface across the i.i.d. subsets of the training data defined by the mini-batches. We show two applications of the gradient of the variance of the loss function. First, as a bias to the conventional momentum update to encourage conformity of the local features of the loss function (e.g. local minima) across mini-batches to improve generalization and the cumulative training progress made per epoch. Second, as an alternative direction for "exploration" in the parameter space, especially, for non-convex objectives, that exploits both the optimistic and pessimistic views of the loss function in the face of uncertainty. We also introduce a novel data-driven stochastic regularization technique through the parameter update rule that is model-agnostic and compatible with arbitrary architectures. We further establish connections to probability distributions over loss functions and the REINFORCE policy gradient update with baseline in RL. Finally, we incorporate the new variants of momentum proposed into Adam, and empirically show that our methods improve the rate of convergence of training based on our experiments on the MNIST and CIFAR-10 datasets.

Efficient stochastic optimization typically integrates an update direction that performs well in the deterministic regime with a mechanism adapting to stochastic perturbations. While Adam uses adaptive moment estimates to promote stability, Muon utilizes the weight layers' matrix structure via orthogonalized momentum, showing superior performance in large language model training. We propose a new optimizer and a diagonal extension, NAMO and NAMO-D, providing the first principled integration of orthogonalized momentum with norm-based Adam-type noise adaptation. NAMO scales orthogonalized momentum using a single adaptive stepsize, preserving orthogonality while improving upon Muon at negligible additional cost. NAMO-D instead right-multiplies orthogonalized momentum by a diagonal matrix with clamped entries. This design enables neuron-wise noise adaptation and aligns with the common near block-diagonal Hessian structure. Under standard assumptions, we establish optimal convergence rates for both algorithms in the deterministic setting and show that, in the stochastic setting, their convergence guarantees adapt to the noise level of stochastic gradients. Experiments on pretraining GPT-2 models demonstrate improved performance of both NAMO and NAMO-D compared to the AdamW and Muon baselines, with NAMO-D achieving further gains over NAMO via an additional clamping hyperparameter that balances the competing goals of maintaining a well-conditioned update direction and leveraging fine-grained noise adaptation.

In the training of large language models, momentum is widely used and often demonstrated to achieve significant acceleration. However, storing momentum typically presents memory challenges. In this paper, we propose AdaPM, an adaptive training strategy that leverages partial momentum to implement a memory-efficient optimizer. To this end, AdaPM utilizes a non-uniform momentum design: for most blocks, full momentum is not necessary to preserve the performance of the optimization. In the momentum design of AdaPM, to mitigate the bias and performance loss caused by partial momentum, we enhance the partial momentum by a bias correction technique. Empirically, we verify that our approach reduces memory by over 90% in momentum while maintaining both efficiency and performance for pretraining various language models ranging from 60M to 1.5B, as well as for supervised fine-tuning and RLHF. AdaPM can further reduce memory by up to 95% in optimizer states by combining the memory-efficient technique on the second-order statistic, saving over 30% GPU hours for pretraining GPT-2 1.5B.

We present Momentum Contrast (MoCo) for unsupervised visual representation learning. From a perspective on contrastive learning as dictionary look-up, we build a dynamic dictionary with a queue and a moving-averaged encoder. This enables building a large and consistent dictionary on-the-fly that facilitates contrastive unsupervised learning. MoCo provides competitive results under the common linear protocol on ImageNet classification. More importantly, the representations learned by MoCo transfer well to downstream tasks. MoCo can outperform its supervised pre-training counterpart in 7 detection/segmentation tasks on PASCAL VOC, COCO, and other datasets, sometimes surpassing it by large margins. This suggests that the gap between unsupervised and supervised representation learning has been largely closed in many vision tasks.

Gravitational waves from merging binary neutron stars carry characteristic information about their astrophysical properties, including masses and tidal deformabilities, that are needed to infer their radii. In this study, we use Bayesian inference to quantify the precision with which radius can inferred with upgrades in the current gravitational wave detectors and next-generation observatories such as the Einstein Telescope and Cosmic Explorer. We assign evidences for a set of plausible equations of state, which are then used as weights to obtain radius posteriors. We find that prior choices and the loudness of observed signals limit the precision and accuracy of inferred radii by current detectors. In contrast, next-generation observatories can resolve the radius precisely and accurately, across most of the mass range to within lesssim 5% for both soft and stiff equations of state. We also explore how the choice of the neutron star mass prior can influence the inferred masses and potentially affect radii measurements, finding that choosing an astrophysically motivated prior does not notably impact an individual neutron star's radius measurements.

Large Language Models (LLMs) achieve competitive performance across diverse natural language processing (NLP) tasks, yet pretraining is computationally demanding, making optimizer efficiency an important practical consideration. Muon accelerates LLM pretraining via orthogonal momentum updates that serve as a matrix analogue of the element-wise sign operator. Motivated by the recent perspective that Adam is a variance-adaptive sign update algorithm, we propose two variants of Muon, Muon-NSR and Muon-VS, which apply variance-adaptive normalization to momentum before orthogonalization. Muon-NSR applies noise-to-signal ratio (NSR) modulation, while Muon-VS performs variance-based scaling without introducing additional hyperparameters. Experiments on GPT-2 and LLaMA pretraining demonstrate that our proposed methods accelerate convergence and consistently achieve lower validation loss than both competitive, well-tuned AdamW and Muon baselines. For example, on the LLaMA-1.2B model, Muon-NSR and Muon-VS reduce the iterations required to reach the target validation loss by 1.36times relative to the well-tuned Muon following the recent benchmark.

Physics-informed Neural Networks (PINNs) have recently achieved remarkable progress in solving Partial Differential Equations (PDEs) in various fields by minimizing a weighted sum of PDE loss and boundary loss. However, there are several critical challenges in the training of PINNs, including the lack of theoretical frameworks and the imbalance between PDE loss and boundary loss. In this paper, we present an analysis of second-order non-homogeneous PDEs, which are classified into three categories and applicable to various common problems. We also characterize the connections between the training loss and actual error, guaranteeing convergence under mild conditions. The theoretical analysis inspires us to further propose MultiAdam, a scale-invariant optimizer that leverages gradient momentum to parameter-wisely balance the loss terms. Extensive experiment results on multiple problems from different physical domains demonstrate that our MultiAdam solver can improve the predictive accuracy by 1-2 orders of magnitude compared with strong baselines.

Predictive Coding (PC) is a theoretical framework in cognitive science suggesting that the human brain processes cognition through spatiotemporal prediction of the visual world. Existing studies have developed spatiotemporal prediction neural networks based on the PC theory, emulating its two core mechanisms: Correcting predictions from residuals and hierarchical learning. However, these models do not show the enhancement of prediction skills on real-world forecasting tasks and ignore the Precision Weighting mechanism of PC theory. The precision weighting mechanism posits that the brain allocates more attention to signals with lower precision, contributing to the cognitive ability of human brains. This work introduces the Cognitive Diffusion Probabilistic Models (CogDPM), which demonstrate the connection between diffusion probabilistic models and PC theory. CogDPM features a precision estimation method based on the hierarchical sampling capabilities of diffusion models and weight the guidance with precision weights estimated by the inherent property of diffusion models. We experimentally show that the precision weights effectively estimate the data predictability. We apply CogDPM to real-world prediction tasks using the United Kindom precipitation and ERA surface wind datasets. Our results demonstrate that CogDPM outperforms both existing domain-specific operational models and general deep prediction models by providing more proficient forecasting.

In deep learning, different kinds of deep networks typically need different optimizers, which have to be chosen after multiple trials, making the training process inefficient. To relieve this issue and consistently improve the model training speed across deep networks, we propose the ADAptive Nesterov momentum algorithm, Adan for short. Adan first reformulates the vanilla Nesterov acceleration to develop a new Nesterov momentum estimation (NME) method, which avoids the extra overhead of computing gradient at the extrapolation point. Then Adan adopts NME to estimate the gradient's first- and second-order moments in adaptive gradient algorithms for convergence acceleration. Besides, we prove that Adan finds an epsilon-approximate first-order stationary point within O(epsilon^{-3.5}) stochastic gradient complexity on the non-convex stochastic problems (e.g., deep learning problems), matching the best-known lower bound. Extensive experimental results show that Adan consistently surpasses the corresponding SoTA optimizers on vision, language, and RL tasks and sets new SoTAs for many popular networks and frameworks, e.g., ResNet, ConvNext, ViT, Swin, MAE, DETR, GPT-2, Transformer-XL, and BERT. More surprisingly, Adan can use half of the training cost (epochs) of SoTA optimizers to achieve higher or comparable performance on ViT, GPT-2, MAE, e.t.c., and also shows great tolerance to a large range of minibatch size, e.g., from 1k to 32k. Code is released at https://github.com/sail-sg/Adan, and has been used in multiple popular deep learning frameworks or projects.

We generalize gradient descent with momentum for optimization in differentiable games to have complex-valued momentum. We give theoretical motivation for our method by proving convergence on bilinear zero-sum games for simultaneous and alternating updates. Our method gives real-valued parameter updates, making it a drop-in replacement for standard optimizers. We empirically demonstrate that complex-valued momentum can improve convergence in realistic adversarial games - like generative adversarial networks - by showing we can find better solutions with an almost identical computational cost. We also show a practical generalization to a complex-valued Adam variant, which we use to train BigGAN to better inception scores on CIFAR-10.

We introduce two complementary techniques for efficient adaptive optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm adaptive step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) by reducing the second moment term's memory footprint from O(d) to O(d) through step-size sharing, where d is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian gradient noise, we prove a noise-adapted high-probability convergence guarantee showing improved dimensional dependence over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state's memory footprint by operating in a low-dimensional subspace while applying standard SGD in the orthogonal complement. We establish high-probability convergence rates under similar relaxed assumptions. Empirical evaluation on LLaMA models from 60M to 1B parameters demonstrates the effectiveness of our methods, where combining subset-norm with subspace-momentum achieves Adam's validation perplexity in approximately half the training tokens (6.8B vs 13.1B) while using only 20% of the Adam's optimizer-states memory footprint and requiring minimal additional hyperparameter tuning.

Heavy-ball momentum with decaying learning rates is widely used with SGD for optimizing deep learning models. In contrast to its empirical popularity, the understanding of its theoretical property is still quite limited, especially under the standard anisotropic gradient noise condition for quadratic regression problems. Although it is widely conjectured that heavy-ball momentum method can provide accelerated convergence and should work well in large batch settings, there is no rigorous theoretical analysis. In this paper, we fill this theoretical gap by establishing a non-asymptotic convergence bound for stochastic heavy-ball methods with step decay scheduler on quadratic objectives, under the anisotropic gradient noise condition. As a direct implication, we show that heavy-ball momentum can provide mathcal{O}(kappa) accelerated convergence of the bias term of SGD while still achieving near-optimal convergence rate with respect to the stochastic variance term. The combined effect implies an overall convergence rate within log factors from the statistical minimax rate. This means SGD with heavy-ball momentum is useful in the large-batch settings such as distributed machine learning or federated learning, where a smaller number of iterations can significantly reduce the number of communication rounds, leading to acceleration in practice.

Score-based diffusion modeling is a generative machine learning algorithm that can be used to sample from complex distributions. They achieve this by learning a score function, i.e., the gradient of the log-probability density of the data, and reversing a noising process using the same. Once trained, score-based diffusion models not only generate new samples but also enable zero-shot conditioning of the generated samples on observed data. This promises a novel paradigm for data and model fusion, wherein the implicitly learned distributions of pretrained score-based diffusion models can be updated given the availability of online data in a Bayesian formulation. In this article, we apply such a concept to the super-resolution of a high-dimensional dynamical system, given the real-time availability of low-resolution and experimentally observed sparse sensor measurements from multimodal data. Additional analysis on how score-based sampling can be used for uncertainty estimates is also provided. Our experiments are performed for a super-resolution task that generates the ERA5 atmospheric dataset given sparse observations from a coarse-grained representation of the same and/or from unstructured experimental observations of the IGRA radiosonde dataset. We demonstrate accurate recovery of the high dimensional state given multiple sources of low-fidelity measurements. We also discover that the generative model can balance the influence of multiple dataset modalities during spatiotemporal reconstructions.

Because large language models are expensive to pretrain on different datasets, using smaller-scale experiments to decide on data is crucial for reducing costs. Which benchmarks and methods of making decisions from observed performance at small scale most accurately predict the datasets that yield the best large models? To empower open exploration of this question, we release models, data, and evaluations in DataDecide -- the most extensive open suite of models over differences in data and scale. We conduct controlled pretraining experiments across 25 corpora with differing sources, deduplication, and filtering up to 100B tokens, model sizes up to 1B parameters, and 3 random seeds. We find that the ranking of models at a single, small size (e.g., 150M parameters) is a strong baseline for predicting best models at our larger target scale (1B) (~80% of com parisons correct). No scaling law methods among 8 baselines exceed the compute-decision frontier of single-scale predictions, but DataDecide can measure improvement in future scaling laws. We also identify that using continuous likelihood metrics as proxies in small experiments makes benchmarks including MMLU, ARC, HellaSwag, MBPP, and HumanEval >80% predictable at the target 1B scale with just 0.01% of the compute.

We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an epsilon-stationary solution of the bilevel problem after epsilon^{-7/2}, epsilon^{-5/2}, and epsilon^{-3/2} iterations (each iteration using O(1) samples) when stochastic noises are in both level objectives, only in the upper-level objective, and not present (deterministic settings), respectively. We further show that if we employ momentum-assisted gradient estimators, the iteration complexities can be improved to epsilon^{-5/2}, epsilon^{-4/2}, and epsilon^{-3/2}, respectively. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.

Score-based models can serve as expressive, data-driven priors for scientific inverse problems. In strong gravitational lensing, they enable posterior inference of a background galaxy from its distorted, multiply-imaged observation. Previous work, however, assumes that the lens mass distribution (and thus the forward operator) is known. We relax this assumption by jointly inferring the source and a parametric lens-mass profile, using a sampler based on GibbsDDRM but operating in continuous time. The resulting reconstructions yield residuals consistent with the observational noise, and the marginal posteriors of the lens parameters recover true values without systematic bias. To our knowledge, this is the first successful demonstration of joint source-and-lens inference with a score-based prior.

The loss functions of many learning problems contain multiple additive terms that can disagree and yield conflicting update directions. For Physics-Informed Neural Networks (PINNs), loss terms on initial/boundary conditions and physics equations are particularly interesting as they are well-established as highly difficult tasks. To improve learning the challenging multi-objective task posed by PINNs, we propose the ConFIG method, which provides conflict-free updates by ensuring a positive dot product between the final update and each loss-specific gradient. It also maintains consistent optimization rates for all loss terms and dynamically adjusts gradient magnitudes based on conflict levels. We additionally leverage momentum to accelerate optimizations by alternating the back-propagation of different loss terms. We provide a mathematical proof showing the convergence of the ConFIG method, and it is evaluated across a range of challenging PINN scenarios. ConFIG consistently shows superior performance and runtime compared to baseline methods. We also test the proposed method in a classic multi-task benchmark, where the ConFIG method likewise exhibits a highly promising performance. Source code is available at https://tum-pbs.github.io/ConFIG

Training a modern machine learning architecture on a new task requires extensive learning-rate tuning, which comes at a high computational cost. Here we develop new adaptive learning rates that can be used with any momentum method, and require less tuning to perform well. We first develop MoMo, a Momentum Model based adaptive learning rate for SGD-M (Stochastic gradient descent with momentum). MoMo uses momentum estimates of the batch losses and gradients sampled at each iteration to build a model of the loss function. Our model also makes use of any known lower bound of the loss function by using truncation, e.g. most losses are lower-bounded by zero. We then approximately minimize this model at each iteration to compute the next step. We show how MoMo can be used in combination with any momentum-based method, and showcase this by developing MoMo-Adam - which is Adam with our new model-based adaptive learning rate. Additionally, for losses with unknown lower bounds, we develop on-the-fly estimates of a lower bound, that are incorporated in our model. Through extensive numerical experiments, we demonstrate that MoMo and MoMo-Adam improve over SGD-M and Adam in terms of accuracy and robustness to hyperparameter tuning for training image classifiers on MNIST, CIFAR10, CIFAR100, Imagenet, recommender systems on the Criteo dataset, and a transformer model on the translation task IWSLT14.

We study the role of batch size in stochastic conditional gradient methods under a μ-Kurdyka-Łojasiewicz (μ-KL) condition. Focusing on momentum-based stochastic conditional gradient algorithms (e.g., Scion), we derive a new analysis that explicitly captures the interaction between stepsize, batch size, and stochastic noise. Our study reveals a regime-dependent behavior: increasing the batch size initially improves optimization accuracy but, beyond a critical threshold, the benefits saturate and can eventually degrade performance under a fixed token budget. Notably, the theory predicts the magnitude of the optimal stepsize and aligns well with empirical practices observed in large-scale training. Leveraging these insights, we derive principled guidelines for selecting the batch size and stepsize, and propose an adaptive strategy that increases batch size and sequence length during training while preserving convergence guarantees. Experiments on NanoGPT are consistent with the theoretical predictions and illustrate the emergence of the predicted scaling regimes. Overall, our results provide a theoretical framework for understanding batch size scaling in stochastic conditional gradient methods and offer guidance for designing efficient training schedules in large-scale optimization.

In medical healthcare, obtaining detailed annotations is challenging, highlighting the need for robust Vision-Language Models (VLMs). Pretrained VLMs enable fine-tuning on small datasets or zero-shot inference, achieving performance comparable to task-specific models. Contrastive learning (CL) is a key paradigm for training VLMs but inherently requires large batch sizes for effective learning, making it computationally demanding and often limited to well-resourced institutions. Moreover, with limited data in healthcare, it is important to prioritize knowledge extraction from both data and models during training to improve performance. Therefore, we focus on leveraging the momentum method combined with distillation to simultaneously address computational efficiency and knowledge exploitation. Our contributions can be summarized as follows: (1) leveraging momentum self-distillation to enhance multimodal learning, and (2) integrating momentum mechanisms with gradient accumulation to enlarge the effective batch size without increasing resource consumption. Our method attains competitive performance with state-of-the-art (SOTA) approaches in zero-shot classification, while providing a substantial boost in the few-shot adaption, achieving over 90% AUC-ROC and improving retrieval tasks by 2-3%. Importantly, our method achieves high training efficiency with a single GPU while maintaining reasonable training time. Our approach aims to advance efficient multimodal learning by reducing resource requirements while improving performance over SOTA methods. The implementation of our method is available at https://github.com/phphuc612/MSD .

As the number of seismic sensors grows, it is becoming increasingly difficult for analysts to pick seismic phases manually and comprehensively, yet such efforts are fundamental to earthquake monitoring. Despite years of improvements in automatic phase picking, it is difficult to match the performance of experienced analysts. A more subtle issue is that different seismic analysts may pick phases differently, which can introduce bias into earthquake locations. We present a deep-neural-network-based arrival-time picking method called "PhaseNet" that picks the arrival times of both P and S waves. Deep neural networks have recently made rapid progress in feature learning, and with sufficient training, have achieved super-human performance in many applications. PhaseNet uses three-component seismic waveforms as input and generates probability distributions of P arrivals, S arrivals, and noise as output. We engineer PhaseNet such that peaks in probability provide accurate arrival times for both P and S waves, and have the potential to increase the number of S-wave observations dramatically over what is currently available. This will enable both improved locations and improved shear wave velocity models. PhaseNet is trained on the prodigious available data set provided by analyst-labeled P and S arrival times from the Northern California Earthquake Data Center. The dataset we use contains more than seven million waveform samples extracted from over thirty years of earthquake recordings. We demonstrate that PhaseNet achieves much higher picking accuracy and recall rate than existing methods.

Lowering the memory requirement in full-parameter training on large models has become a hot research area. MeZO fine-tunes the large language models (LLMs) by just forward passes in a zeroth-order SGD optimizer (ZO-SGD), demonstrating excellent performance with the same GPU memory usage as inference. However, the simulated perturbation stochastic approximation for gradient estimate in MeZO leads to severe oscillations and incurs a substantial time overhead. Moreover, without momentum regularization, MeZO shows severe over-fitting problems. Lastly, the perturbation-irrelevant momentum on ZO-SGD does not improve the convergence rate. This study proposes ZO-AdaMU to resolve the above problems by adapting the simulated perturbation with momentum in its stochastic approximation. Unlike existing adaptive momentum methods, we relocate momentum on simulated perturbation in stochastic gradient approximation. Our convergence analysis and experiments prove this is a better way to improve convergence stability and rate in ZO-SGD. Extensive experiments demonstrate that ZO-AdaMU yields better generalization for LLMs fine-tuning across various NLP tasks than MeZO and its momentum variants.

Earthquake hazard analysis and design of spatially distributed infrastructure, such as power grids and energy pipeline networks, require scenario-specific ground-motion time histories with realistic frequency content and spatiotemporal coherence. However, producing the large ensembles needed for uncertainty quantification with physics-based simulations is computationally intensive and impractical for engineering workflows. To address this challenge, we introduce Ground-Motion Flow (GMFlow), a physics-inspired latent operator flow matching framework that generates realistic, large-scale regional ground-motion time-histories conditioned on physical parameters. Validated on simulated earthquake scenarios in the San Francisco Bay Area, GMFlow generates spatially coherent ground motion across more than 9 million grid points in seconds, achieving a 10,000-fold speedup over the simulation workflow, which opens a path toward rapid and uncertainty-aware hazard assessment for distributed infrastructure. More broadly, GMFlow advances mesh-agnostic functional generative modeling and could potentially be extended to the synthesis of large-scale spatiotemporal physical fields in diverse scientific domains.

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs). However, standard MLP-based PINNs often fail to converge when dealing with complex initial value problems, leading to solutions that violate causality and suffer from a spectral bias towards low-frequency components. To address these issues, we introduce NeuSA (Neuro-Spectral Architectures), a novel class of PINNs inspired by classical spectral methods, designed to solve linear and nonlinear PDEs with variable coefficients. NeuSA learns a projection of the underlying PDE onto a spectral basis, leading to a finite-dimensional representation of the dynamics which is then integrated with an adapted Neural ODE (NODE). This allows us to overcome spectral bias, by leveraging the high-frequency components enabled by the spectral representation; to enforce causality, by inheriting the causal structure of NODEs, and to start training near the target solution, by means of an initialization scheme based on classical methods. We validate NeuSA on canonical benchmarks for linear and nonlinear wave equations, demonstrating strong performance as compared to other architectures, with faster convergence, improved temporal consistency and superior predictive accuracy. Code and pretrained models are available in https://github.com/arthur-bizzi/neusa.

Implicitly defined, continuous, differentiable signal representations parameterized by neural networks have emerged as a powerful paradigm, offering many possible benefits over conventional representations. However, current network architectures for such implicit neural representations are incapable of modeling signals with fine detail, and fail to represent a signal's spatial and temporal derivatives, despite the fact that these are essential to many physical signals defined implicitly as the solution to partial differential equations. We propose to leverage periodic activation functions for implicit neural representations and demonstrate that these networks, dubbed sinusoidal representation networks or Sirens, are ideally suited for representing complex natural signals and their derivatives. We analyze Siren activation statistics to propose a principled initialization scheme and demonstrate the representation of images, wavefields, video, sound, and their derivatives. Further, we show how Sirens can be leveraged to solve challenging boundary value problems, such as particular Eikonal equations (yielding signed distance functions), the Poisson equation, and the Helmholtz and wave equations. Lastly, we combine Sirens with hypernetworks to learn priors over the space of Siren functions.

Video generators are increasingly evaluated as potential world models, which requires them to encode and understand physical laws. We investigate their representation of a fundamental law: gravity. Out-of-the-box video generators consistently generate objects falling at an effectively slower acceleration. However, these physical tests are often confounded by ambiguous metric scale. We first investigate if observed physical errors are artifacts of these ambiguities (e.g., incorrect frame rate assumptions). We find that even temporal rescaling cannot correct the high-variance gravity artifacts. To rigorously isolate the underlying physical representation from these confounds, we introduce a unit-free, two-object protocol that tests the timing ratio t_1^2/t_2^2 = h_1/h_2, a relationship independent of g, focal length, and scale. This relative test reveals violations of Galileo's equivalence principle. We then demonstrate that this physical gap can be partially mitigated with targeted specialization. A lightweight low-rank adaptor fine-tuned on only 100 single-ball clips raises g_{eff} from 1.81,m/s^2 to 6.43,m/s^2 (reaching 65% of terrestrial gravity). This specialist adaptor also generalizes zero-shot to two-ball drops and inclined planes, offering initial evidence that specific physical laws can be corrected with minimal data.

We develop tools for explicitly constructing categories enriched over generating data and that compose via ordinary scalar and matrix arithmetic arithmetic operations. We characterize meaningful size maps, weightings, and magnitude that reveal features analogous to outliers that these same notions have previously been shown to reveal in the context of metric spaces. Throughout, we provide examples of such "outlier detection" relevant to the analysis of computer programs, neural networks, cyber-physical systems, and networks of communications channels.

Reliable probability estimation is of crucial importance in many real-world applications where there is inherent (aleatoric) uncertainty. Probability-estimation models are trained on observed outcomes (e.g. whether it has rained or not, or whether a patient has died or not), because the ground-truth probabilities of the events of interest are typically unknown. The problem is therefore analogous to binary classification, with the difference that the objective is to estimate probabilities rather than predicting the specific outcome. This work investigates probability estimation from high-dimensional data using deep neural networks. There exist several methods to improve the probabilities generated by these models but they mostly focus on model (epistemic) uncertainty. For problems with inherent uncertainty, it is challenging to evaluate performance without access to ground-truth probabilities. To address this, we build a synthetic dataset to study and compare different computable metrics. We evaluate existing methods on the synthetic data as well as on three real-world probability estimation tasks, all of which involve inherent uncertainty: precipitation forecasting from radar images, predicting cancer patient survival from histopathology images, and predicting car crashes from dashcam videos. We also give a theoretical analysis of a model for high-dimensional probability estimation which reproduces several of the phenomena evinced in our experiments. Finally, we propose a new method for probability estimation using neural networks, which modifies the training process to promote output probabilities that are consistent with empirical probabilities computed from the data. The method outperforms existing approaches on most metrics on the simulated as well as real-world data.

The software package Volna-OP2 is a robust and efficient code capable of simulating the complete life cycle of a tsunami whilst harnessing the latest High Performance Computing (HPC) architectures. In this paper, a comprehensive error analysis and scalability study of the GPU version of the code is presented. A novel decomposition of the numerical errors into the dispersion and dissipation components is explored. Most tsunami codes exhibit amplitude smearing and/or phase lagging/leading, so the decomposition shown here is a new approach and novel tool for explaining these occurrences. It is the first time that the errors of a tsunami code have been assessed in this manner. To date, Volna-OP2 has been widely used by the tsunami modelling community. In particular its computational efficiency has allowed various sensitivity analyses and uncertainty quantification studies. Due to the number of simulations required, there is always a trade-off between accuracy and runtime when carrying out these statistical studies. The analysis presented in this paper will guide the user towards an acceptable level of accuracy within a given runtime.

We introduce MVN-Grad (Momentum on Variance-Normalized Gradients), an Adam-style optimizer that improves stability and performance by combining two complementary ideas: variance-based normalization and momentum applied after normalization. MVN-Grad scales each coordinate by an exponential moving average of gradient uncertainty and applies momentum to the resulting normalized gradients, eliminating the cross-time coupling between stale momentum and a stochastic normalizer present in standard Adam-type updates. We prove that this decoupling yields strictly smaller one-step conditional update variance than momentum-then-normalize variance methods under standard noise assumptions, and that MVN-Grad is robust to outliers: it has a uniformly bounded response to single gradient spikes. In low-variance regimes, we further show variance normalization avoids sign-type collapse associated with second-moment scaling and can yield accelerated convergence. Across CIFAR-100 image classification and GPT-style language modeling benchmarks, MVN-Grad matches or outperforms Adam, AdaBelief, and LaProp, delivering smoother training and improved generalization with no added overhead.

Extracting meaningful information from large seismic datasets often requires estimating the uncertainty associated with the results for quantitative analysis. This uncertainty arises from both the raw data and the manually labeled annotations. We introduce an uncertainty estimation framework designed to calculate the uncertainty from manually labeled data. This framework can efficiently output the true posterior from large datasets. We apply the framework to extract Rayleigh wave phase velocity dispersion and compute the posterior distribution of the dispersion results. We utilize 62,899 noise cross-correlation function (NCF) data from 438 stations located in Yunnan Province and manually label the Rayleigh phase velocity dispersion curves. Dispersion curve extraction presents two key challenges: (1) Researchers typically derive dispersion curves from spectrograms in the periodvelocity domain, limiting the ability to directly study the relationship between NCFs and dispersion curves; (2) Assessing uncertainty in manually labeled data remains difficult. To address these challenges, the framework takes the NCFs as input and directly output both the dispersion values and the posterior of the dispersion values when processing the NCF data. This approach allows us to construct a flexible deep neural network (DNN) architecture that balances accuracy and computational efficiency.

Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on the subspace dimension, indicating that diffusion models can circumvent the curse of data ambient dimensionality.

Training large neural network models requires extensive computational resources, often distributed across several nodes and accelerators. Recent findings suggest that it may be sufficient to only exchange the fast moving components of the gradients, while accumulating momentum locally (Decoupled Momentum, or DeMo). However, DeMo assumes that models fit on a single accelerator. We relax this assumption and introduce FlexDeMo, whereby nodes fully shard model parameters locally between different accelerators, while inter-node communication is reduced by synchronizing only fast-moving components instead of the full gradients -- resulting in a hybrid sharded data parallel training strategy. We further introduce a framework, denoted as DeToNATION, that generalizes DeMo, FlexDeMo, and other popular distributed training schemes such as DiLoCo -- introducing new variations of replication schemes and challenging choices made in DeMo. Our results across language and vision domains show that FlexDeMo attains similar validation loss as hybrid sharded data parallel training employing AdamW and full gradient synchronization, while being substantially faster. FlexDeMo is thus a promising distributed training scheme for the largest machine learning models.

Mutual information (MI) is a fundamental measure of statistical dependence between two variables, yet accurate estimation from finite data remains notoriously difficult. No estimator is universally reliable, and common approaches fail in the high-dimensional, undersampled regimes typical of modern experiments. Recent machine learning-based estimators show promise, but their accuracy depends sensitively on dataset size, structure, and hyperparameters, with no accepted tests to detect failures. We close these gaps through a systematic evaluation of classical and neural MI estimators across standard benchmarks and new synthetic datasets tailored to challenging high-dimensional, undersampled regimes. We contribute: (i) a practical protocol for reliable MI estimation with explicit checks for statistical consistency; (ii) confidence intervals (error bars around estimates) that existing neural MI estimator do not provide; and (iii) a new class of probabilistic critics designed for high-dimensional, high-information settings. We demonstrate the effectiveness of our protocol with computational experiments, showing that it consistently matches or surpasses existing methods while uniquely quantifying its own reliability. We show that reliable MI estimation is sometimes achievable even in severely undersampled, high-dimensional datasets, provided they admit accurate low-dimensional representations. This broadens the scope of applicability of neural MI estimators and clarifies when such estimators can be trusted.

Multimodal Large Language Models (MLLMs) have demonstrated remarkable capabilities in diverse reasoning tasks, yet their application to complex physics reasoning remains underexplored. Physics reasoning presents unique challenges, requiring grounding in physical conditions and the interpretation of multimodal information. Current physics benchmarks are limited, often focusing on text-only inputs or solely on problem-solving, thereby overlooking the critical intermediate steps of variable identification and process formulation. To address these limitations, we introduce PhysicsArena, the first multimodal physics reasoning benchmark designed to holistically evaluate MLLMs across three critical dimensions: variable identification, physical process formulation, and solution derivation. PhysicsArena aims to provide a comprehensive platform for assessing and advancing the multimodal physics reasoning abilities of MLLMs.

Adapting a pre-trained foundation model on downstream tasks should ensure robustness against distribution shifts without the need to retrain the whole model. Although existing weight interpolation methods are simple yet effective, we argue their static nature limits downstream performance while achieving efficiency. In this work, we propose DaWin, a training-free dynamic weight interpolation method that leverages the entropy of individual models over each unlabeled test sample to assess model expertise, and compute per-sample interpolation coefficients dynamically. Unlike previous works that typically rely on additional training to learn such coefficients, our approach requires no training. Then, we propose a mixture modeling approach that greatly reduces inference overhead raised by dynamic interpolation. We validate DaWin on the large-scale visual recognition benchmarks, spanning 14 tasks across robust fine-tuning -- ImageNet and derived five distribution shift benchmarks -- and multi-task learning with eight classification tasks. Results demonstrate that DaWin achieves significant performance gain in considered settings, with minimal computational overhead. We further discuss DaWin's analytic behavior to explain its empirical success.

We present Generative Anchored Fields (GAF), a generative model that learns independent endpoint predictors, J (noise) and K (data), from any point on a linear bridge. Unlike existing approaches that use a single trajectory or score predictor, GAF is trained to recover the bridge endpoints directly via coordinate learning. The velocity field v=K-J emerges from their time-conditioned disagreement. This factorization enables Transport Algebra: algebraic operations on multiple J/K heads for compositional control. With class-specific K_n heads, GAF defines directed transport maps between a shared base noise distribution and multiple data domains, allowing controllable interpolation, multi-class composition, and semantic editing. This is achieved either directly on the predicted data coordinates (K) using Iterative Endpoint Refinement (IER), a novel sampler that achieves high-quality generation in 5-8 steps, or on the emergent velocity field (v). We achieve strong sample quality (FID 7.51 on ImageNet 256times256 and 7.27 on CelebA-HQ 256times 256, without classifier-free guidance) while treating compositional generation as an architectural primitive. Code available at https://github.com/IDLabMedia/GAF.

Recent discoveries on neural network pruning reveal that, with a carefully chosen layerwise sparsity, a simple magnitude-based pruning achieves state-of-the-art tradeoff between sparsity and performance. However, without a clear consensus on "how to choose," the layerwise sparsities are mostly selected algorithm-by-algorithm, often resorting to handcrafted heuristics or an extensive hyperparameter search. To fill this gap, we propose a novel importance score for global pruning, coined layer-adaptive magnitude-based pruning (LAMP) score; the score is a rescaled version of weight magnitude that incorporates the model-level ell_2 distortion incurred by pruning, and does not require any hyperparameter tuning or heavy computation. Under various image classification setups, LAMP consistently outperforms popular existing schemes for layerwise sparsity selection. Furthermore, we observe that LAMP continues to outperform baselines even in weight-rewinding setups, while the connectivity-oriented layerwise sparsity (the strongest baseline overall) performs worse than a simple global magnitude-based pruning in this case. Code: https://github.com/jaeho-lee/layer-adaptive-sparsity

The ability of the deep learning model to recognize when a sample falls outside its learned distribution is critical for safe and reliable deployment. Recent state-of-the-art out-of-distribution (OOD) detection methods leverage activation shaping to improve the separation between in-distribution (ID) and OOD inputs. These approaches resort to sample-specific scaling but apply a static percentile threshold across all samples regardless of their nature, resulting in suboptimal ID-OOD separability. In this work, we propose AdaSCALE, an adaptive scaling procedure that dynamically adjusts the percentile threshold based on a sample's estimated OOD likelihood. This estimation leverages our key observation: OOD samples exhibit significantly more pronounced activation shifts at high-magnitude activations under minor perturbation compared to ID samples. AdaSCALE enables stronger scaling for likely ID samples and weaker scaling for likely OOD samples, yielding highly separable energy scores. Our approach achieves state-of-the-art OOD detection performance, outperforming the latest rival OptFS by 14.94 in near-OOD and 21.67 in far-OOD datasets in average FPR@95 metric on the ImageNet-1k benchmark across eight diverse architectures. The code is available at: https://github.com/sudarshanregmi/AdaSCALE/

Reconstructing subsurface ocean dynamics, such as vertical velocity fields, from incomplete surface observations poses a critical challenge in Earth science, a field long hampered by the lack of standardized, analysis-ready benchmarks. To systematically address this issue and catalyze research, we first build and release KD48, a high-resolution ocean dynamics benchmark derived from petascale simulations and curated with expert-driven denoising. Building on this benchmark, we introduce VISION, a novel reconstruction paradigm based on Dynamic Prompting designed to tackle the core problem of missing data in real-world observations. The essence of VISION lies in its ability to generate a visual prompt on-the-fly from any available subset of observations, which encodes both data availability and the ocean's physical state. More importantly, we design a State-conditioned Prompting module that efficiently injects this prompt into a universal backbone, endowed with geometry- and scale-aware operators, to guide its adaptive adjustment of computational strategies. This mechanism enables VISION to precisely handle the challenges posed by varying input combinations. Extensive experiments on the KD48 benchmark demonstrate that VISION not only substantially outperforms state-of-the-art models but also exhibits strong generalization under extreme data missing scenarios. By providing a high-quality benchmark and a robust model, our work establishes a solid infrastructure for ocean science research under data uncertainty. Our codes are available at: https://github.com/YuanGao-YG/VISION.

Modern foundational Multimodal Large Language Models (MLLMs) and video world models have advanced significantly in mathematical, common-sense, and visual reasoning, but their grasp of the underlying physics remains underexplored. Existing benchmarks attempting to measure this matter rely on synthetic, Visual Question Answer templates or focus on perceptual video quality that is tangential to measuring how well the video abides by physical laws. To address this fragmentation, we introduce PhysicsMind, a unified benchmark with both real and simulation environments that evaluates law-consistent reasoning and generation over three canonical principles: Center of Mass, Lever Equilibrium, and Newton's First Law. PhysicsMind comprises two main tasks: i) VQA tasks, testing whether models can reason and determine physical quantities and values from images or short videos, and ii) Video Generation(VG) tasks, evaluating if predicted motion trajectories obey the same center-of-mass, torque, and inertial constraints as the ground truth. A broad range of recent models and video generation models is evaluated on PhysicsMind and found to rely on appearance heuristics while often violating basic mechanics. These gaps indicate that current scaling and training are still insufficient for robust physical understanding, underscoring PhysicsMind as a focused testbed for physics-aware multimodal models. Our data will be released upon acceptance.

In mechanical structures like airplanes, cars and houses, noise is generated and transmitted through vibrations. To take measures to reduce this noise, vibrations need to be simulated with expensive numerical computations. Deep learning surrogate models present a promising alternative to classical numerical simulations as they can be evaluated magnitudes faster, while trading-off accuracy. To quantify such trade-offs systematically and foster the development of methods, we present a benchmark on the task of predicting the vibration of harmonically excited plates. The benchmark features a total of 12,000 plate geometries with varying forms of beadings, material, boundary conditions, load position and sizes with associated numerical solutions. To address the benchmark task, we propose a new network architecture, named Frequency-Query Operator, which predicts vibration patterns of plate geometries given a specific excitation frequency. Applying principles from operator learning and implicit models for shape encoding, our approach effectively addresses the prediction of highly variable frequency response functions occurring in dynamic systems. To quantify the prediction quality, we introduce a set of evaluation metrics and evaluate the method on our vibrating-plates benchmark. Our method outperforms DeepONets, Fourier Neural Operators and more traditional neural network architectures and can be used for design optimization. Code, dataset and visualizations: https://github.com/ecker-lab/Learning_Vibrating_Plates

The goal of predictive data attribution is to estimate how adding or removing a given set of training datapoints will affect model predictions. In convex settings, this goal is straightforward (i.e., via the infinitesimal jackknife). In large-scale (non-convex) settings, however, existing methods are far less successful -- current methods' estimates often only weakly correlate with ground truth. In this work, we present a new data attribution method (MAGIC) that combines classical methods and recent advances in metadifferentiation to (nearly) optimally estimate the effect of adding or removing training data on model predictions.

Video generation models have achieved remarkable progress in creating high-quality, photorealistic content. However, their ability to accurately simulate physical phenomena remains a critical and unresolved challenge. This paper presents PhyWorldBench, a comprehensive benchmark designed to evaluate video generation models based on their adherence to the laws of physics. The benchmark covers multiple levels of physical phenomena, ranging from fundamental principles like object motion and energy conservation to more complex scenarios involving rigid body interactions and human or animal motion. Additionally, we introduce a novel ""Anti-Physics"" category, where prompts intentionally violate real-world physics, enabling the assessment of whether models can follow such instructions while maintaining logical consistency. Besides large-scale human evaluation, we also design a simple yet effective method that could utilize current MLLM to evaluate the physics realism in a zero-shot fashion. We evaluate 12 state-of-the-art text-to-video generation models, including five open-source and five proprietary models, with a detailed comparison and analysis. we identify pivotal challenges models face in adhering to real-world physics. Through systematic testing of their outputs across 1,050 curated prompts-spanning fundamental, composite, and anti-physics scenarios-we identify pivotal challenges these models face in adhering to real-world physics. We then rigorously examine their performance on diverse physical phenomena with varying prompt types, deriving targeted recommendations for crafting prompts that enhance fidelity to physical principles.

Data attribution seeks to trace model outputs back to training data. With the recent development of diffusion models, data attribution has become a desired module to properly assign valuations for high-quality or copyrighted training samples, ensuring that data contributors are fairly compensated or credited. Several theoretically motivated methods have been proposed to implement data attribution, in an effort to improve the trade-off between computational scalability and effectiveness. In this work, we conduct extensive experiments and ablation studies on attributing diffusion models, specifically focusing on DDPMs trained on CIFAR-10 and CelebA, as well as a Stable Diffusion model LoRA-finetuned on ArtBench. Intriguingly, we report counter-intuitive observations that theoretically unjustified design choices for attribution empirically outperform previous baselines by a large margin, in terms of both linear datamodeling score and counterfactual evaluation. Our work presents a significantly more efficient approach for attributing diffusion models, while the unexpected findings suggest that at least in non-convex settings, constructions guided by theoretical assumptions may lead to inferior attribution performance. The code is available at https://github.com/sail-sg/D-TRAK.

We propose AdaMuon, a novel optimizer that combines element-wise adaptivity with orthogonal updates for large-scale neural network training. AdaMuon incorporates two tightly coupled mechanisms: (1) an element-wise second momentum estimator applied to orthogonalized update directions, and (2) a sign-stabilized orthogonal update, where the momentum is first sign-transformed before orthogonalization. These two components jointly enable variance-adaptive scaling while maintaining stable update geometry. In addition, AdaMuon employs an RMS-aligned rescaling strategy to match the root-mean-square update magnitude to Adam, allowing direct reuse of existing learning rate schedules without extra tuning. Experiments demonstrate that AdaMuon not only maintains stability but can surpass Adam by more than 40% training efficiency in large-scale scenarios.

Recent advances in video generation have shown remarkable potential for constructing world simulators. However, current models still struggle to produce physically consistent results, particularly when handling large-scale or complex dynamics. This limitation arises primarily because existing approaches respond isotropically to physical prompts and neglect the fine-grained alignment between generated content and localized physical cues. To address these challenges, we propose ProPhy, a Progressive Physical Alignment Framework that enables explicit physics-aware conditioning and anisotropic generation. ProPhy employs a two-stage Mixture-of-Physics-Experts (MoPE) mechanism for discriminative physical prior extraction, where Semantic Experts infer semantic-level physical principles from textual descriptions, and Refinement Experts capture token-level physical dynamics. This mechanism allows the model to learn fine-grained, physics-aware video representations that better reflect underlying physical laws. Furthermore, we introduce a physical alignment strategy that transfers the physical reasoning capabilities of vision-language models (VLMs) into the Refinement Experts, facilitating a more accurate representation of dynamic physical phenomena. Extensive experiments on physics-aware video generation benchmarks demonstrate that ProPhy produces more realistic, dynamic, and physically coherent results than existing state-of-the-art methods.

Large foundation models are typically trained on data from multiple domains, with the data mixture--the proportion of each domain used--playing a critical role in model performance. The standard approach to selecting this mixture relies on trial and error, which becomes impractical for large-scale pretraining. We propose a systematic method to determine the optimal data mixture for any target domain using scaling laws. Our approach accurately predicts the loss of a model of size N trained with D tokens and a specific domain weight vector h. We validate the universality of these scaling laws by demonstrating their predictive power in three distinct and large-scale settings: large language model (LLM), native multimodal model (NMM), and large vision models (LVM) pretraining. We further show that these scaling laws can extrapolate to new data mixtures and across scales: their parameters can be accurately estimated using a few small-scale training runs, and used to estimate the performance at larger scales and unseen domain weights. The scaling laws allow to derive the optimal domain weights for any target domain under a given training budget (N,D), providing a principled alternative to costly trial-and-error methods.

In this work, we show that the impact of model capacity varies across timesteps: it is crucial for the early and late stages but largely negligible during the intermediate stage. Accordingly, we propose FlowBlending, a stage-aware multi-model sampling strategy that employs a large model and a small model at capacity-sensitive stages and intermediate stages, respectively. We further introduce simple criteria to choose stage boundaries and provide a velocity-divergence analysis as an effective proxy for identifying capacity-sensitive regions. Across LTX-Video (2B/13B) and WAN 2.1 (1.3B/14B), FlowBlending achieves up to 1.65x faster inference with 57.35% fewer FLOPs, while maintaining the visual fidelity, temporal coherence, and semantic alignment of the large models. FlowBlending is also compatible with existing sampling-acceleration techniques, enabling up to 2x additional speedup. Project page is available at: https://jibin86.github.io/flowblending_project_page.

Rigorousness and clarity are both essential for interpretations of DNNs to engender human trust. Path methods are commonly employed to generate rigorous attributions that satisfy three axioms. However, the meaning of attributions remains ambiguous due to distinct path choices. To address the ambiguity, we introduce Concentration Principle, which centrally allocates high attributions to indispensable features, thereby endowing aesthetic and sparsity. We then present SAMP, a model-agnostic interpreter, which efficiently searches the near-optimal path from a pre-defined set of manipulation paths. Moreover, we propose the infinitesimal constraint (IC) and momentum strategy (MS) to improve the rigorousness and optimality. Visualizations show that SAMP can precisely reveal DNNs by pinpointing salient image pixels. We also perform quantitative experiments and observe that our method significantly outperforms the counterparts. Code: https://github.com/zbr17/SAMP.

Large-scale vision and language representation learning has shown promising improvements on various vision-language tasks. Most existing methods employ a transformer-based multimodal encoder to jointly model visual tokens (region-based image features) and word tokens. Because the visual tokens and word tokens are unaligned, it is challenging for the multimodal encoder to learn image-text interactions. In this paper, we introduce a contrastive loss to ALign the image and text representations BEfore Fusing (ALBEF) them through cross-modal attention, which enables more grounded vision and language representation learning. Unlike most existing methods, our method does not require bounding box annotations nor high-resolution images. In order to improve learning from noisy web data, we propose momentum distillation, a self-training method which learns from pseudo-targets produced by a momentum model. We provide a theoretical analysis of ALBEF from a mutual information maximization perspective, showing that different training tasks can be interpreted as different ways to generate views for an image-text pair. ALBEF achieves state-of-the-art performance on multiple downstream vision-language tasks. On image-text retrieval, ALBEF outperforms methods that are pre-trained on orders of magnitude larger datasets. On VQA and NLVR^2, ALBEF achieves absolute improvements of 2.37% and 3.84% compared to the state-of-the-art, while enjoying faster inference speed. Code and pre-trained models are available at https://github.com/salesforce/ALBEF/.

In subsurface imaging, learning the mapping from velocity maps to seismic waveforms (forward problem) and waveforms to velocity (inverse problem) is important for several applications. While traditional techniques for solving forward and inverse problems are computationally prohibitive, there is a growing interest in leveraging recent advances in deep learning to learn the mapping between velocity maps and seismic waveform images directly from data. Despite the variety of architectures explored in previous works, several open questions still remain unanswered such as the effect of latent space sizes, the importance of manifold learning, the complexity of translation models, and the value of jointly solving forward and inverse problems. We propose a unified framework to systematically characterize prior research in this area termed the Generalized Forward-Inverse (GFI) framework, building on the assumption of manifolds and latent space translations. We show that GFI encompasses previous works in deep learning for subsurface imaging, which can be viewed as specific instantiations of GFI. We also propose two new model architectures within the framework of GFI: Latent U-Net and Invertible X-Net, leveraging the power of U-Nets for domain translation and the ability of IU-Nets to simultaneously learn forward and inverse translations, respectively. We show that our proposed models achieve state-of-the-art (SOTA) performance for forward and inverse problems on a wide range of synthetic datasets, and also investigate their zero-shot effectiveness on two real-world-like datasets. Our code is available at https://github.com/KGML-lab/Generalized-Forward-Inverse-Framework-for-DL4SI

As an adaptive method, Shampoo employs a structured second-moment estimation, and its effectiveness has attracted growing attention. Prior work has primarily analyzed its estimation scheme through the Frobenius norm. Motivated by the natural connection between the second moment and a covariance matrix, we propose studying Shampoo's estimation as covariance estimation through the lens of Kullback-Leibler (KL) minimization. This alternative perspective reveals a previously hidden limitation, motivating improvements to Shampoo's design. Building on this insight, we develop a practical estimation scheme, termed KL-Shampoo, that eliminates Shampoo's reliance on Adam for stabilization, thereby removing the additional memory overhead introduced by Adam. Preliminary results show that KL-Shampoo improves Shampoo's performance, enabling it to stabilize without Adam and even outperform its Adam-stabilized variant, SOAP, in neural network pretraining.

In the era of mega-constellations, the need for accurate and publicly available information has become fundamental for satellite operators to guarantee the safety of spacecrafts and the Low Earth Orbit (LEO) space environment. This study critically evaluates the accuracy and reliability of publicly available ephemeris data for a LEO mega-constellation - Starlink. The goal of this work is twofold: (i) compare and analyze the quality of the data against high-precision numerical propagation. (ii) Leverage Physics-Informed Machine Learning to extract relevant satellite quantities, such as non-conservative forces, during the decay process. By analyzing two months of real orbital data for approximately 1500 Starlink satellites, we identify discrepancies between high precision numerical algorithms and the published ephemerides, recognizing the use of simplified dynamics at fixed thresholds, planned maneuvers, and limitations in uncertainty propagations. Furthermore, we compare data obtained from multiple sources to track and analyze deorbiting satellites over the same period. Empirically, we extract the acceleration profile of satellites during deorbiting and provide insights relating to the effects of non-conservative forces during reentry. For non-deorbiting satellites, the position Root Mean Square Error (RMSE) was approximately 300 m, while for deorbiting satellites it increased to about 600 m. Through this in-depth analysis, we highlight potential limitations in publicly available data for accurate and robust Space Situational Awareness (SSA), and importantly, we propose a data-driven model of satellite decay in mega-constellations.

Climbing grades are used to classify a climbing route based on its perceived difficulty, and have come to play a central role in the sport of rock climbing. Recently, the first statistically rigorous method for estimating climbing grades from whole-history ascent data was described, based on the dynamic Bradley-Terry model for games between players of time-varying ability. In this paper, we implement inference under the whole-history rating model using Markov chain Monte Carlo and apply the method to a curated data set made up of climbers who climb regularly. We use these data to get an estimate of the model's fundamental scale parameter m, which defines the proportional increase in difficulty associated with an increment of grade. We show that the data conform to assumptions that the climbing grade scale is a logarithmic scale of difficulty, like decibels or stellar magnitude. We estimate that an increment in Ewbank, French and UIAA climbing grade systems corresponds to 2.1, 2.09 and 2.13 times increase in difficulty respectively, assuming a logistic model of probability of success as a function of grade. Whereas we find that the Vermin scale for bouldering (V-grade scale) corresponds to a 3.17 increase in difficulty per grade increment. In addition, we highlight potential connections between the logarithmic properties of climbing grade scales and the psychophysical laws of Weber and Fechner.

Earth system models suffer from various structural and parametric errors in their representation of nonlinear, multi-scale processes, leading to uncertainties in their long-term projections. The effects of many of these errors (particularly those due to fast physics) can be quantified in short-term simulations, e.g., as differences between the predicted and observed states (analysis increments). With the increase in the availability of high-quality observations and simulations, learning nudging from these increments to correct model errors has become an active research area. However, most studies focus on using neural networks, which while powerful, are hard to interpret, are data-hungry, and poorly generalize out-of-distribution. Here, we show the capabilities of Model Error Discovery with Interpretability and Data Assimilation (MEDIDA), a general, data-efficient framework that uses sparsity-promoting equation-discovery techniques to learn model errors from analysis increments. Using two-layer quasi-geostrophic turbulence as the test case, MEDIDA is shown to successfully discover various linear and nonlinear structural/parametric errors when full observations are available. Discovery from spatially sparse observations is found to require highly accurate interpolation schemes. While NNs have shown success as interpolators in recent studies, here, they are found inadequate due to their inability to accurately represent small scales, a phenomenon known as spectral bias. We show that a general remedy, adding a random Fourier feature layer to the NN, resolves this issue enabling MEDIDA to successfully discover model errors from sparse observations. These promising results suggest that with further development, MEDIDA could be scaled up to models of the Earth system and real observations.

The solar wind speed at Earth is one of the most important parameters regarding the effects of space weather on society. Thus far, most approaches for predicting the solar wind speed produce a single-value time series without uncertainty, or utilize ensemble methods which require custom calibration development. In this study, a method is developed that produces calibrated probabilistic forecasts of the solar wind speed using skew normal distributions and a novel extension of analog ensembles. In our extension, the single-value predictions from a baseline model of the next Δt days are used along with Δwindow hours of recent observations and single-value predictions to create a forecasting scenario vector that is compared against a historical database for outcomes. The baseline model used is the combined Air Force Data Assimilative Photospheric Flux Transport-Wang Sheeley Arge (ADAPT-WSA) model and the WSA point parcel simulation, but the method is directly applicable to other deterministic models including components such as Enlil or the Heliospheric Upwind Extrapolation with time dependence model (HUXt). The approach works notably well on the benchmark of whether observations fall within the p^{th} percentile p% of the time (for p between 0 and 100). Falling back on the mean or median of the predicted distribution as a non-probabilistic prediction yields a direct improvement in root-mean-square error (RMSE) over the original WSA point parcel simulation, and is shown to beat approx 1 solar rotation recurrence for 1-5 day ahead forecasts.

Seismic processing transforms raw data into subsurface images essential for geophysical applications. Traditional methods face challenges, such as noisy data, and manual parameter tuning, among others. Recently deep learning approaches have proposed alternative solutions to some of these problems. However, important challenges of existing deep learning approaches are spatially inconsistent results across neighboring seismic gathers and lack of user-control. We address these limitations by introducing ContextSeisNet, an in-context learning model, to seismic demultiple processing. Our approach conditions predictions on a support set of spatially related example pairs: neighboring common-depth point gathers from the same seismic line and their corresponding labels. This allows the model to learn task-specific processing behavior at inference time by observing how similar gathers should be processed, without any retraining. This method provides both flexibility through user-defined examples and improved lateral consistency across seismic lines. On synthetic data, ContextSeisNet outperforms a U-Net baseline quantitatively and demonstrates enhanced spatial coherence between neighboring gathers. On field data, our model achieves superior lateral consistency compared to both traditional Radon demultiple and the U-Net baseline. Relative to the U-Net, ContextSeisNet also delivers improved near-offset performance and more complete multiple removal. Notably, ContextSeisNet achieves comparable field data performance despite being trained on 90% less data, demonstrating substantial data efficiency. These results establish ContextSeisNet as a practical approach for spatially consistent seismic demultiple with potential applicability to other seismic processing tasks.

Standard methods for detecting discontinuities in conditional means are not applicable to outcomes that are complex, non-Euclidean objects like distributions, networks, or covariance matrices. This article develops a nonparametric test for jumps in conditional means when outcomes lie in a non-Euclidean metric space. Using local Fr\'echet regressionx2014which generalizes standard regression to metric-space valued datax2014the method estimates a mean path on either side of a candidate cutoff, extending existing k-sample tests to a flexible regression setting. Key theoretical contributions include a central limit theorem for the local estimator of the conditional Fr\'echet variance and the asymptotic validity and consistency of the proposed test. Simulations confirm nominal size control and robust power in finite samples. Two applications demonstrate the method's value by revealing effects invisible to scalar-based tests. First, I detect a sharp change in work-from-home compositions at Washington State's income threshold for non-compete enforceability during COVID-19, highlighting remote work's role as a bargaining margin. Second, I find that countries restructure their input-output networks after losing preferential US trade access. These findings underscore that analyzing regression functions within their native metric spaces can reveal structural discontinuities that scalar summaries would miss.

This paper presents MetricGrids, a novel grid-based neural representation that combines elementary metric grids in various metric spaces to approximate complex nonlinear signals. While grid-based representations are widely adopted for their efficiency and scalability, the existing feature grids with linear indexing for continuous-space points can only provide degenerate linear latent space representations, and such representations cannot be adequately compensated to represent complex nonlinear signals by the following compact decoder. To address this problem while keeping the simplicity of a regular grid structure, our approach builds upon the standard grid-based paradigm by constructing multiple elementary metric grids as high-order terms to approximate complex nonlinearities, following the Taylor expansion principle. Furthermore, we enhance model compactness with hash encoding based on different sparsities of the grids to prevent detrimental hash collisions, and a high-order extrapolation decoder to reduce explicit grid storage requirements. experimental results on both 2D and 3D reconstructions demonstrate the superior fitting and rendering accuracy of the proposed method across diverse signal types, validating its robustness and generalizability. Code is available at https://github.com/wangshu31/MetricGrids}{https://github.com/wangshu31/MetricGrids.

Crowdsourcing annotations has created a paradigm shift in the availability of labeled data for machine learning. Availability of large datasets has accelerated progress in common knowledge applications involving visual and language data. However, specialized applications that require expert labels lag in data availability. One such application is fault segmentation in subsurface imaging. Detecting, tracking, and analyzing faults has broad societal implications in predicting fluid flows, earthquakes, and storing excess atmospheric CO_2. However, delineating faults with current practices is a labor-intensive activity that requires precise analysis of subsurface imaging data by geophysicists. In this paper, we propose the CRACKS dataset to detect and segment faults in subsurface images by utilizing crowdsourced resources. We leverage Amazon Mechanical Turk to obtain fault delineations from sections of the Netherlands North Sea subsurface images from (i) 26 novices who have no exposure to subsurface data and were shown a video describing and labeling faults, (ii) 8 practitioners who have previously interacted and worked on subsurface data, (iii) one geophysicist to label 7636 faults in the region. Note that all novices, practitioners, and the expert segment faults on the same subsurface volume with disagreements between and among the novices and practitioners. Additionally, each fault annotation is equipped with the confidence level of the annotator. The paper provides benchmarks on detecting and segmenting the expert labels, given the novice and practitioner labels. Additional details along with the dataset links and codes are available at https://alregib.ece.gatech.edu/cracks-crowdsourcing-resources-for-analysis-and-categorization-of-key-subsurface-faults/{link}.

Predicting the evolution of complex physical systems remains a central problem in science and engineering. Despite rapid progress in scientific Machine Learning (ML) models, a critical bottleneck is the lack of expensive real-world data, resulting in most current models being trained and validated on simulated data. Beyond limiting the development and evaluation of scientific ML, this gap also hinders research into essential tasks such as sim-to-real transfer. We introduce RealPDEBench, the first benchmark for scientific ML that integrates real-world measurements with paired numerical simulations. RealPDEBench consists of five datasets, three tasks, eight metrics, and ten baselines. We first present five real-world measured datasets with paired simulated datasets across different complex physical systems. We further define three tasks, which allow comparisons between real-world and simulated data, and facilitate the development of methods to bridge the two. Moreover, we design eight evaluation metrics, spanning data-oriented and physics-oriented metrics, and finally benchmark ten representative baselines, including state-of-the-art models, pretrained PDE foundation models, and a traditional method. Experiments reveal significant discrepancies between simulated and real-world data, while showing that pretraining with simulated data consistently improves both accuracy and convergence. In this work, we hope to provide insights from real-world data, advancing scientific ML toward bridging the sim-to-real gap and real-world deployment. Our benchmark, datasets, and instructions are available at https://realpdebench.github.io/.

Stochastic Gradient Descent (SGD) and its momentum variants form the backbone of deep learning optimization, yet the underlying dynamics of their gradient behavior remain insufficiently understood. In this work, we reinterpret gradient updates through the lens of signal processing and reveal that fixed momentum coefficients inherently distort the balance between bias and variance, leading to skewed or suboptimal parameter updates. To address this, we propose SGDF (SGD with Filter), an optimizer inspired by the principles of Optimal Linear Filtering. SGDF computes an online, time-varying gain to dynamically refine gradient estimation by minimizing the mean-squared error, thereby achieving an optimal trade-off between noise suppression and signal preservation. Furthermore, our approach could extend to other optimizers, showcasing its broad applicability to optimization frameworks. Extensive experiments across diverse architectures and benchmarks demonstrate SGDF surpasses conventional momentum methods and achieves performance on par with or surpassing state-of-the-art optimizers.

While text-to-video (T2V) generation has achieved remarkable progress in photorealism, generating intent-aligned videos that faithfully obey physics principles remains a core challenge. In this work, we systematically study Newtonian motion-controlled text-to-video generation and evaluation, emphasizing physical precision and motion coherence. We introduce MoReGen, a motion-aware, physics-grounded T2V framework that integrates multi-agent LLMs, physics simulators, and renderers to generate reproducible, physically accurate videos from text prompts in the code domain. To quantitatively assess physical validity, we propose object-trajectory correspondence as a direct evaluation metric and present MoReSet, a benchmark of 1,275 human-annotated videos spanning nine classes of Newtonian phenomena with scene descriptions, spatiotemporal relations, and ground-truth trajectories. Using MoReSet, we conduct experiments on existing T2V models, evaluating their physical validity through both our MoRe metrics and existing physics-based evaluators. Our results reveal that state-of-the-art models struggle to maintain physical validity, while MoReGen establishes a principled direction toward physically coherent video synthesis.

Simulation-based inference with conditional neural density estimators is a powerful approach to solving inverse problems in science. However, these methods typically treat the underlying forward model as a black box, with no way to exploit geometric properties such as equivariances. Equivariances are common in scientific models, however integrating them directly into expressive inference networks (such as normalizing flows) is not straightforward. We here describe an alternative method to incorporate equivariances under joint transformations of parameters and data. Our method -- called group equivariant neural posterior estimation (GNPE) -- is based on self-consistently standardizing the "pose" of the data while estimating the posterior over parameters. It is architecture-independent, and applies both to exact and approximate equivariances. As a real-world application, we use GNPE for amortized inference of astrophysical binary black hole systems from gravitational-wave observations. We show that GNPE achieves state-of-the-art accuracy while reducing inference times by three orders of magnitude.

We propose a principled and effective framework for one-step generative modeling. We introduce the notion of average velocity to characterize flow fields, in contrast to instantaneous velocity modeled by Flow Matching methods. A well-defined identity between average and instantaneous velocities is derived and used to guide neural network training. Our method, termed the MeanFlow model, is self-contained and requires no pre-training, distillation, or curriculum learning. MeanFlow demonstrates strong empirical performance: it achieves an FID of 3.43 with a single function evaluation (1-NFE) on ImageNet 256x256 trained from scratch, significantly outperforming previous state-of-the-art one-step diffusion/flow models. Our study substantially narrows the gap between one-step diffusion/flow models and their multi-step predecessors, and we hope it will motivate future research to revisit the foundations of these powerful models.

In several recently proposed stochastic optimization methods (e.g. RMSProp, Adam, Adadelta), parameter updates are scaled by the inverse square roots of exponential moving averages of squared past gradients. Maintaining these per-parameter second-moment estimators requires memory equal to the number of parameters. For the case of neural network weight matrices, we propose maintaining only the per-row and per-column sums of these moving averages, and estimating the per-parameter second moments based on these sums. We demonstrate empirically that this method produces similar results to the baseline. Secondly, we show that adaptive methods can produce larger-than-desired updates when the decay rate of the second moment accumulator is too slow. We propose update clipping and a gradually increasing decay rate scheme as remedies. Combining these methods and dropping momentum, we achieve comparable results to the published Adam regime in training the Transformer model on the WMT 2014 English-German machine translation task, while using very little auxiliary storage in the optimizer. Finally, we propose scaling the parameter updates based on the scale of the parameters themselves.

Machine learning-based modeling of physical systems has experienced increased interest in recent years. Despite some impressive progress, there is still a lack of benchmarks for Scientific ML that are easy to use but still challenging and representative of a wide range of problems. We introduce PDEBench, a benchmark suite of time-dependent simulation tasks based on Partial Differential Equations (PDEs). PDEBench comprises both code and data to benchmark the performance of novel machine learning models against both classical numerical simulations and machine learning baselines. Our proposed set of benchmark problems contribute the following unique features: (1) A much wider range of PDEs compared to existing benchmarks, ranging from relatively common examples to more realistic and difficult problems; (2) much larger ready-to-use datasets compared to prior work, comprising multiple simulation runs across a larger number of initial and boundary conditions and PDE parameters; (3) more extensible source codes with user-friendly APIs for data generation and baseline results with popular machine learning models (FNO, U-Net, PINN, Gradient-Based Inverse Method). PDEBench allows researchers to extend the benchmark freely for their own purposes using a standardized API and to compare the performance of new models to existing baseline methods. We also propose new evaluation metrics with the aim to provide a more holistic understanding of learning methods in the context of Scientific ML. With those metrics we identify tasks which are challenging for recent ML methods and propose these tasks as future challenges for the community. The code is available at https://github.com/pdebench/PDEBench.

We propose a new approach for propagating stable probability distributions through neural networks. Our method is based on local linearization, which we show to be an optimal approximation in terms of total variation distance for the ReLU non-linearity. This allows propagating Gaussian and Cauchy input uncertainties through neural networks to quantify their output uncertainties. To demonstrate the utility of propagating distributions, we apply the proposed method to predicting calibrated confidence intervals and selective prediction on out-of-distribution data. The results demonstrate a broad applicability of propagating distributions and show the advantages of our method over other approaches such as moment matching.

Modeling stochastic dynamics from discrete observations is a key interdisciplinary challenge. Existing methods often fail to estimate the continuous evolution of probability densities from trajectories or face the curse of dimensionality. To address these limitations, we presents a novel paradigm: modeling dynamics directly in the weight space of a neural network by projecting the evolving probability distribution. We first theoretically establish the connection between dynamic optimal transport in measure space and an equivalent energy functional in weight space. Subsequently, we design WeightFlow, which constructs the neural network weights into a graph and learns its evolution via a graph controlled differential equation. Experiments on interdisciplinary datasets demonstrate that WeightFlow improves performance by an average of 43.02\% over state-of-the-art methods, providing an effective and scalable solution for modeling high-dimensional stochastic dynamics.

We consider the problem of forecasting motion from a single image, i.e., predicting how objects in the world are likely to move, without the ability to observe other parameters such as the object velocities or the forces applied to them. We formulate this task as conditional generation of dense trajectory grids with a model that closely follows the architecture of modern video generators but outputs motion trajectories instead of pixels. This approach captures scene-wide dynamics and uncertainty, yielding more accurate and diverse predictions than prior regressors and generators. We extensively evaluate our method on simulated data, demonstrate its effectiveness on downstream applications such as robotics, and show promising accuracy on real-world intuitive physics datasets. Although recent state-of-the-art video generators are often regarded as world models, we show that they struggle with forecasting motion from a single image, even in simple physical scenarios such as falling blocks or mechanical object interactions, despite fine-tuning on such data. We show that this limitation arises from the overhead of generating pixels rather than directly modeling motion.

3D modeling is moving from virtual to physical. Existing 3D generation primarily emphasizes geometries and textures while neglecting physical-grounded modeling. Consequently, despite the rapid development of 3D generative models, the synthesized 3D assets often overlook rich and important physical properties, hampering their real-world application in physical domains like simulation and embodied AI. As an initial attempt to address this challenge, we propose PhysX, an end-to-end paradigm for physical-grounded 3D asset generation. 1) To bridge the critical gap in physics-annotated 3D datasets, we present PhysXNet - the first physics-grounded 3D dataset systematically annotated across five foundational dimensions: absolute scale, material, affordance, kinematics, and function description. In particular, we devise a scalable human-in-the-loop annotation pipeline based on vision-language models, which enables efficient creation of physics-first assets from raw 3D assets.2) Furthermore, we propose PhysXGen, a feed-forward framework for physics-grounded image-to-3D asset generation, injecting physical knowledge into the pre-trained 3D structural space. Specifically, PhysXGen employs a dual-branch architecture to explicitly model the latent correlations between 3D structures and physical properties, thereby producing 3D assets with plausible physical predictions while preserving the native geometry quality. Extensive experiments validate the superior performance and promising generalization capability of our framework. All the code, data, and models will be released to facilitate future research in generative physical AI.

Humans possess an exceptional ability to imagine 4D scenes, encompassing both motion and 3D geometry, from a single still image. This ability is rooted in our accumulated observations of similar scenes and an intuitive understanding of physics. In this paper, we aim to replicate this capacity in neural networks, specifically focusing on natural fluid imagery. Existing methods for this task typically employ simplistic 2D motion estimators to animate the image, leading to motion predictions that often defy physical principles, resulting in unrealistic animations. Our approach introduces a novel method for generating 4D scenes with physics-consistent animation from a single image. We propose the use of a physics-informed neural network that predicts motion for each surface point, guided by a loss term derived from fundamental physical principles, including the Navier-Stokes equations. To capture appearance, we predict feature-based 3D Gaussians from the input image and its estimated depth, which are then animated using the predicted motions and rendered from any desired camera perspective. Experimental results highlight the effectiveness of our method in producing physically plausible animations, showcasing significant performance improvements over existing methods. Our project page is https://physfluid.github.io/ .

Existing benchmarks for Earth science multimodal learning exhibit critical limitations in systematic coverage of geosystem components and cross-sphere interactions, often constrained to isolated subsystems (only in Human-activities sphere or atmosphere) with limited evaluation dimensions (less than 16 tasks). To address these gaps, we introduce OmniEarth-Bench, the first comprehensive multimodal benchmark spanning all six Earth science spheres (atmosphere, lithosphere, Oceansphere, cryosphere, biosphere and Human-activities sphere) and cross-spheres with one hundred expert-curated evaluation dimensions. Leveraging observational data from satellite sensors and in-situ measurements, OmniEarth-Bench integrates 29,779 annotations across four tiers: perception, general reasoning, scientific knowledge reasoning and chain-of-thought (CoT) reasoning. This involves the efforts of 2-5 experts per sphere to establish authoritative evaluation dimensions and curate relevant observational datasets, 40 crowd-sourcing annotators to assist experts for annotations, and finally, OmniEarth-Bench is validated via hybrid expert-crowd workflows to reduce label ambiguity. Experiments on 9 state-of-the-art MLLMs reveal that even the most advanced models struggle with our benchmarks, where none of them reach 35\% accuracy. Especially, in some cross-spheres tasks, the performance of leading models like GPT-4o drops to 0.0\%. OmniEarth-Bench sets a new standard for geosystem-aware AI, advancing both scientific discovery and practical applications in environmental monitoring and disaster prediction. The dataset, source code, and trained models were released.

Videos often capture objects, their visible properties, their motion, and the interactions between different objects. Objects also have physical properties such as mass, which the imaging pipeline is unable to directly capture. However, these properties can be estimated by utilizing cues from relative object motion and the dynamics introduced by collisions. In this paper, we introduce CRIPP-VQA, a new video question answering dataset for reasoning about the implicit physical properties of objects in a scene. CRIPP-VQA contains videos of objects in motion, annotated with questions that involve counterfactual reasoning about the effect of actions, questions about planning in order to reach a goal, and descriptive questions about visible properties of objects. The CRIPP-VQA test set enables evaluation under several out-of-distribution settings -- videos with objects with masses, coefficients of friction, and initial velocities that are not observed in the training distribution. Our experiments reveal a surprising and significant performance gap in terms of answering questions about implicit properties (the focus of this paper) and explicit properties of objects (the focus of prior work).

Recent diffusion and flow matching models have demonstrated strong capabilities in image generation and editing by progressively removing noise through iterative sampling. While this enables flexible inversion for semantic-preserving edits, few-step sampling regimes suffer from poor forward process approximation, leading to degraded editing quality. Existing few-step inversion methods often rely on pretrained generators and auxiliary modules, limiting scalability and generalization across different architectures. To address these limitations, we propose BiFM (Bidirectional Flow Matching), a unified framework that jointly learns generation and inversion within a single model. BiFM directly estimates average velocity fields in both ``image to noise" and ``noise to image" directions, constrained by a shared instantaneous velocity field derived from either predefined schedules or pretrained multi-step diffusion models. Additionally, BiFM introduces a novel training strategy using continuous time-interval supervision, stabilized by a bidirectional consistency objective and a lightweight time-interval embedding. This bidirectional formulation also enables one-step inversion and can integrate seamlessly into popular diffusion and flow matching backbones. Across diverse image editing and generation tasks, BiFM consistently outperforms existing few-step approaches, achieving superior performance and editability.

Few images on the Web receive alt-text descriptions that would make them accessible to blind and low vision (BLV) users. Image-based NLG systems have progressed to the point where they can begin to address this persistent societal problem, but these systems will not be fully successful unless we evaluate them on metrics that guide their development correctly. Here, we argue against current referenceless metrics -- those that don't rely on human-generated ground-truth descriptions -- on the grounds that they do not align with the needs of BLV users. The fundamental shortcoming of these metrics is that they do not take context into account, whereas contextual information is highly valued by BLV users. To substantiate these claims, we present a study with BLV participants who rated descriptions along a variety of dimensions. An in-depth analysis reveals that the lack of context-awareness makes current referenceless metrics inadequate for advancing image accessibility. As a proof-of-concept, we provide a contextual version of the referenceless metric CLIPScore which begins to address the disconnect to the BLV data. An accessible HTML version of this paper is available at https://elisakreiss.github.io/contextual-description-evaluation/paper/reflessmetrics.html

Deep neural networks conventionally employ end-to-end backpropagation for their training process, which lacks biological credibility and triggers a locking dilemma during network parameter updates, leading to significant GPU memory use. Supervised local learning, which segments the network into multiple local blocks updated by independent auxiliary networks. However, these methods cannot replace end-to-end training due to lower accuracy, as gradients only propagate within their local block, creating a lack of information exchange between blocks. To address this issue and establish information transfer across blocks, we propose a Momentum Auxiliary Network (MAN) that establishes a dynamic interaction mechanism. The MAN leverages an exponential moving average (EMA) of the parameters from adjacent local blocks to enhance information flow. This auxiliary network, updated through EMA, helps bridge the informational gap between blocks. Nevertheless, we observe that directly applying EMA parameters has certain limitations due to feature discrepancies among local blocks. To overcome this, we introduce learnable biases, further boosting performance. We have validated our method on four image classification datasets (CIFAR-10, STL-10, SVHN, ImageNet), attaining superior performance and substantial memory savings. Notably, our method can reduce GPU memory usage by more than 45\% on the ImageNet dataset compared to end-to-end training, while achieving higher performance. The Momentum Auxiliary Network thus offers a new perspective for supervised local learning. Our code is available at: https://github.com/JunhaoSu0/MAN.

We present ACAR (Adaptive Complexity and Attribution Routing), a measurement framework for studying multi-model orchestration under auditable conditions. ACAR uses self-consistency variance (sigma) computed from N=3 probe samples to route tasks across single-model, two-model, and three-model execution modes. The system is implemented on top of TEAMLLM, a deterministic execution substrate with immutable artifacts and complete decision traces. We evaluate ACAR on 1,510 tasks spanning four benchmarks: MathArena, Reasoning Gym, LiveCodeBench, and SuperGPQA, using Claude Sonnet 4, GPT-4o, and Gemini 2.0 Flash, producing more than 7,550 auditable runs. Results show that sigma-based routing achieves 55.6 percent accuracy, exceeding the two-model baseline of 54.4 percent while avoiding full ensembling on 54.2 percent of tasks. The routing mechanism is model-agnostic and requires no learned components. We also document negative results. First, retrieval augmentation reduced accuracy by 3.4 percentage points, as median retrieval similarity was only 0.167, demonstrating that experience injection without semantic alignment introduces noise rather than grounding. Second, when models agree on incorrect answers (sigma equals zero), no downstream ensemble can recover; this agreement-but-wrong failure mode is intrinsic to self-consistency and bounds achievable accuracy at approximately eight percentage points below full ensembling. Third, attribution estimates based on proxy signals such as response similarity and entropy showed weak correlation with ground-truth leave-one-out values, indicating that practical attribution requires explicit counterfactual computation. This work documents which assumptions fail in practice and provides falsifiable baselines for future research on routing, retrieval, and multi-model attribution.

Recent advances in image and video generation raise hopes that these models possess world modeling capabilities, the ability to generate realistic, physically plausible videos. This could revolutionize applications in robotics, autonomous driving, and scientific simulation. However, before treating these models as world models, we must ask: Do they adhere to physical conservation laws? To answer this, we introduce Morpheus, a benchmark for evaluating video generation models on physical reasoning. It features 80 real-world videos capturing physical phenomena, guided by conservation laws. Since artificial generations lack ground truth, we assess physical plausibility using physics-informed metrics evaluated with respect to infallible conservation laws known per physical setting, leveraging advances in physics-informed neural networks and vision-language foundation models. Our findings reveal that even with advanced prompting and video conditioning, current models struggle to encode physical principles despite generating aesthetically pleasing videos. All data, leaderboard, and code are open-sourced at our project page.

A primary bottleneck in large-scale text-to-video generation today is physical consistency and controllability. Despite recent advances, state-of-the-art models often produce unrealistic motions, such as objects falling upward, or abrupt changes in velocity and direction. Moreover, these models lack precise parameter control, struggling to generate physically consistent dynamics under different initial conditions. We argue that this fundamental limitation stems from current models learning motion distributions solely from appearance, while lacking an understanding of the underlying dynamics. In this work, we propose NewtonGen, a framework that integrates data-driven synthesis with learnable physical principles. At its core lies trainable Neural Newtonian Dynamics (NND), which can model and predict a variety of Newtonian motions, thereby injecting latent dynamical constraints into the video generation process. By jointly leveraging data priors and dynamical guidance, NewtonGen enables physically consistent video synthesis with precise parameter control.

For deploying foundation models, practitioners increasingly need prescriptive scaling laws: given a pre training compute budget, what downstream accuracy is attainable with contemporary post training practice, and how stable is that mapping as the field evolves? Using large scale observational evaluations with 5k observational and 2k newly sampled data on model performance, we estimate capability boundaries, high conditional quantiles of benchmark scores as a function of log pre training FLOPs, via smoothed quantile regression with a monotone, saturating sigmoid parameterization. We validate the temporal reliability by fitting on earlier model generations and evaluating on later releases. Across various tasks, the estimated boundaries are mostly stable, with the exception of math reasoning that exhibits a consistently advancing boundary over time. We then extend our approach to analyze task dependent saturation and to probe contamination related shifts on math reasoning tasks. Finally, we introduce an efficient algorithm that recovers near full data frontiers using roughly 20% of evaluation budget. Together, our work releases the Proteus 2k, the latest model performance evaluation dataset, and introduces a practical methodology for translating compute budgets into reliable performance expectations and for monitoring when capability boundaries shift across time.

Foundation models have transformed machine learning for language and vision, but achieving comparable impact in physical simulation remains a challenge. Data heterogeneity and unstable long-term dynamics inhibit learning from sufficiently diverse dynamics, while varying resolutions and dimensionalities challenge efficient training on modern hardware. Through empirical and theoretical analysis, we incorporate new approaches to mitigate these obstacles, including a harmonic-analysis-based stabilization method, load-balanced distributed 2D and 3D training strategies, and compute-adaptive tokenization. Using these tools, we develop Walrus, a transformer-based foundation model developed primarily for fluid-like continuum dynamics. Walrus is pretrained on nineteen diverse scenarios spanning astrophysics, geoscience, rheology, plasma physics, acoustics, and classical fluids. Experiments show that Walrus outperforms prior foundation models on both short and long term prediction horizons on downstream tasks and across the breadth of pretraining data, while ablation studies confirm the value of our contributions to forecast stability, training throughput, and transfer performance over conventional approaches. Code and weights are released for community use.

In the face of escalating climate changes, typhoon intensities and their ensuing damage have surged. Accurate trajectory prediction is crucial for effective damage control. Traditional physics-based models, while comprehensive, are computationally intensive and rely heavily on the expertise of forecasters. Contemporary data-driven methods often rely on reanalysis data, which can be considered to be the closest to the true representation of weather conditions. However, reanalysis data is not produced in real-time and requires time for adjustment because prediction models are calibrated with observational data. This reanalysis data, such as ERA5, falls short in challenging real-world situations. Optimal preparedness necessitates predictions at least 72 hours in advance, beyond the capabilities of standard physics models. In response to these constraints, we present an approach that harnesses real-time Unified Model (UM) data, sidestepping the limitations of reanalysis data. Our model provides predictions at 6-hour intervals for up to 72 hours in advance and outperforms both state-of-the-art data-driven methods and numerical weather prediction models. In line with our efforts to mitigate adversities inflicted by typhoons, we release our preprocessed PHYSICS TRACK dataset, which includes ERA5 reanalysis data, typhoon best-track, and UM forecast data.

Neural operators have proven to be a promising approach for modeling spatiotemporal systems in the physical sciences. However, training these models for large systems can be quite challenging as they incur significant computational and memory expense -- these systems are often forced to rely on autoregressive time-stepping of the neural network to predict future temporal states. While this is effective in managing costs, it can lead to uncontrolled error growth over time and eventual instability. We analyze the sources of this autoregressive error growth using prototypical neural operator models for physical systems and explore ways to mitigate it. We introduce architectural and application-specific improvements that allow for careful control of instability-inducing operations within these models without inflating the compute/memory expense. We present results on several scientific systems that include Navier-Stokes fluid flow, rotating shallow water, and a high-resolution global weather forecasting system. We demonstrate that applying our design principles to neural operators leads to significantly lower errors for long-term forecasts as well as longer time horizons without qualitative signs of divergence compared to the original models for these systems. We open-source our https://github.com/mikemccabe210/stabilizing_neural_operators{code} for reproducibility.

Vision Language Models (VLMs) perform well on standard video tasks but struggle with physics-driven reasoning involving motion dynamics and spatial interactions. This limitation reduces their ability to interpret real or AI-generated content (AIGC) videos and to generate physically consistent content. We present an approach that addresses this gap by translating physical-world context cues into interpretable representations aligned with VLMs' perception, comprehension, and reasoning. We introduce MASS-Bench, a comprehensive benchmark consisting of 4,350 real-world and AIGC videos and 8,361 free-form video question-answering pairs focused on physics-related comprehension tasks, with detailed annotations including visual detections, sub-segment grounding, and full-sequence 3D motion tracking of entities. We further present MASS, a model-agnostic method that injects spatial-temporal signals into the VLM language space via depth-based 3D encoding and visual grounding, coupled with a motion tracker for object dynamics. To strengthen cross-modal alignment and reasoning, we apply reinforcement fine-tuning. Experiments and ablations show that our refined VLMs outperform comparable and larger baselines, as well as prior state-of-the-art models, by 8.7% and 6.0%, achieving performance comparable to close-source SoTA VLMs such as Gemini-2.5-Flash on physics reasoning and comprehension. These results validate the effectiveness of our approach.

Despite the success of physics-informed neural networks (PINNs) in approximating partial differential equations (PDEs), PINNs can sometimes fail to converge to the correct solution in problems involving complicated PDEs. This is reflected in several recent studies on characterizing the "failure modes" of PINNs, although a thorough understanding of the connection between PINN failure modes and sampling strategies is missing. In this paper, we provide a novel perspective of failure modes of PINNs by hypothesizing that training PINNs relies on successful "propagation" of solution from initial and/or boundary condition points to interior points. We show that PINNs with poor sampling strategies can get stuck at trivial solutions if there are propagation failures, characterized by highly imbalanced PDE residual fields. To mitigate propagation failures, we propose a novel Retain-Resample-Release sampling (R3) algorithm that can incrementally accumulate collocation points in regions of high PDE residuals with little to no computational overhead. We provide an extension of R3 sampling to respect the principle of causality while solving time-dependent PDEs. We theoretically analyze the behavior of R3 sampling and empirically demonstrate its efficacy and efficiency in comparison with baselines on a variety of PDE problems.