Daily Papers

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

In the Lagrange-Newton method, where Newton's method is applied to a Lagrangian function that includes equality constraints, all stationary points are saddle points. It is therefore not possible to use a line-search method based on the value of the objective function; instead, the line search can operate on merit functions. In this report, we explore an alternative line-search method which is applicable to this case; it particulary addresses the damping of the step length in tight valleys. We propose a line-search criterion based on the divergence of the field of Newton step vectors. The visualization of the criterion for two-dimensional test functions reveals a network of ravines with flat bottom at the zero points of the criterion. The ravines are typically connected to stationary points. To traverse this ravine network in order to approach a stationary point, a zigzag strategy is devised. Numerical experiments demonstrate that the novel line-search strategy succeeds from most starting points in all test functions, but only exhibits the desired damping of the step length in some situations. At the present stage it is therefore difficult to appraise the utility of this contribution.

In physics, Lagrangians provide a systematic way to describe laws governing physical systems. In the context of particle physics, they encode the interactions and behavior of the fundamental building blocks of our universe. By treating Lagrangians as complex, rule-based constructs similar to linguistic expressions, we trained a transformer model -- proven to be effective in natural language tasks -- to predict the Lagrangian corresponding to a given list of particles. We report on the transformer's performance in constructing Lagrangians respecting the Standard Model SU(3)times SU(2)times U(1) gauge symmetries. The resulting model is shown to achieve high accuracies (over 90\%) with Lagrangians up to six matter fields, with the capacity to generalize beyond the training distribution, albeit within architectural constraints. We show through an analysis of input embeddings that the model has internalized concepts such as group representations and conjugation operations as it learned to generate Lagrangians. We make the model and training datasets available to the community. An interactive demonstration can be found at: https://huggingface.co/spaces/JoseEliel/generate-lagrangians.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle to efficiently capture regularities in the world, but the role of symmetry breaking is not well understood. Here, we develop a theoretical framework to study the "geometry of learning dynamics" in neural networks, and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. To build this understanding, we model the discrete learning dynamics of gradient descent using a continuous-time Lagrangian formulation, in which the learning rule corresponds to the kinetic energy and the loss function corresponds to the potential energy. Then, we identify "kinetic symmetry breaking" (KSB), the condition when the kinetic energy explicitly breaks the symmetry of the potential function. We generalize Noether's theorem known in physics to take into account KSB and derive the resulting motion of the Noether charge: "Noether's Learning Dynamics" (NLD). Finally, we apply NLD to neural networks with normalization layers and reveal how KSB introduces a mechanism of "implicit adaptive optimization", establishing an analogy between learning dynamics induced by normalization layers and RMSProp. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.

Accurately predicting the future fluid is important to extensive areas, such as meteorology, oceanology and aerodynamics. However, since the fluid is usually observed from an Eulerian perspective, its active and intricate dynamics are seriously obscured and confounded in static grids, bringing horny challenges to the prediction. This paper introduces a new Lagrangian-guided paradigm to tackle the tanglesome fluid dynamics. Instead of solely predicting the future based on Eulerian observations, we propose the Eulerian-Lagrangian Dual Recurrent Network (EuLagNet), which captures multiscale fluid dynamics by tracking movements of adaptively sampled key particles on multiple scales and integrating dynamics information over time. Concretely, a EuLag Block is presented to communicate the learned Eulerian and Lagrangian features at each moment and scale, where the motion of tracked particles is inferred from Eulerian observations and their accumulated dynamics information is incorporated into Eulerian fields to guide future prediction. Tracking key particles not only provides a clear and interpretable clue for fluid dynamics but also makes our model free from modeling complex correlations among massive grids for better efficiency. Experimentally, EuLagNet excels in three challenging fluid prediction tasks, covering both 2D and 3D, simulated and real-world fluids.

Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

Simulating physical systems governed by Lagrangian dynamics often entails solving partial differential equations (PDEs) over high-resolution spatial domains, leading to significant computational expense. Reduced-order modeling (ROM) mitigates this cost by evolving low-dimensional latent representations of the underlying system. While neural ROMs enable querying solutions from latent states at arbitrary spatial points, their latent states typically represent the global domain and struggle to capture localized, highly dynamic behaviors such as fluids. We propose a sampling-based reduction framework that evolves Lagrangian systems directly in physical space over the particles themselves, reducing the number of active degrees of freedom via data-driven neural PDE operators. To enable querying at arbitrary spatial locations, we introduce a learnable kernel parameterization that uses local spatial information from time-evolved sample particles to infer the underlying solution manifold. Empirically, our approach achieves a 6.6x to 32x reduction in input dimensionality while maintaining high-fidelity evaluations across diverse Lagrangian regimes, including fluid flows, granular media, and elastoplastic dynamics. We refer to this framework as GIOROM (Geometry-Informed Reduced-Order Modeling). All code and data are available at: https://github.com/HrishikeshVish/GIOROM

Physical simulators have been widely used in robot planning and control. Among them, differentiable simulators are particularly favored, as they can be incorporated into gradient-based optimization algorithms that are efficient in solving inverse problems such as optimal control and motion planning. Simulating deformable objects is, however, more challenging compared to rigid body dynamics. The underlying physical laws of deformable objects are more complex, and the resulting systems have orders of magnitude more degrees of freedom and therefore they are significantly more computationally expensive to simulate. Computing gradients with respect to physical design or controller parameters is typically even more computationally challenging. In this paper, we propose a real-time, differentiable hybrid Lagrangian-Eulerian physical simulator for deformable objects, ChainQueen, based on the Moving Least Squares Material Point Method (MLS-MPM). MLS-MPM can simulate deformable objects including contact and can be seamlessly incorporated into inference, control and co-design systems. We demonstrate that our simulator achieves high precision in both forward simulation and backward gradient computation. We have successfully employed it in a diverse set of control tasks for soft robots, including problems with nearly 3,000 decision variables.

Initial value problems -- a system of ordinary differential equations and corresponding initial conditions -- can be used to describe many physical phenomena including those arise in classical mechanics. We have developed a novel approach to solve physics-based initial value problems using unsupervised machine learning. We propose a deep learning framework that models the dynamics of a variety of mechanical systems through neural networks. Our framework is flexible, allowing us to solve non-linear, coupled, and chaotic dynamical systems. We demonstrate the effectiveness of our approach on systems including a free particle, a particle in a gravitational field, a classical pendulum, and the H\'enon--Heiles system (a pair of coupled harmonic oscillators with a non-linear perturbation, used in celestial mechanics). Our results show that deep neural networks can successfully approximate solutions to these problems, producing trajectories which conserve physical properties such as energy and those with stationary action. We note that probabilistic activation functions, as defined in this paper, are required to learn any solutions of initial value problems in their strictest sense, and we introduce coupled neural networks to learn solutions of coupled systems.

We present MovingParts, a NeRF-based method for dynamic scene reconstruction and part discovery. We consider motion as an important cue for identifying parts, that all particles on the same part share the common motion pattern. From the perspective of fluid simulation, existing deformation-based methods for dynamic NeRF can be seen as parameterizing the scene motion under the Eulerian view, i.e., focusing on specific locations in space through which the fluid flows as time passes. However, it is intractable to extract the motion of constituting objects or parts using the Eulerian view representation. In this work, we introduce the dual Lagrangian view and enforce representations under the Eulerian/Lagrangian views to be cycle-consistent. Under the Lagrangian view, we parameterize the scene motion by tracking the trajectory of particles on objects. The Lagrangian view makes it convenient to discover parts by factorizing the scene motion as a composition of part-level rigid motions. Experimentally, our method can achieve fast and high-quality dynamic scene reconstruction from even a single moving camera, and the induced part-based representation allows direct applications of part tracking, animation, 3D scene editing, etc.

Grey-box methods for system identification combine deep learning with physics-informed constraints, capturing complex dependencies while improving out-of-distribution generalization. Yet, despite the growing importance of floating-base systems such as humanoids and quadrupeds, current grey-box models ignore their specific physical constraints. For instance, the inertia matrix is not only positive definite but also exhibits branch-induced sparsity and input independence. Moreover, the 6x6 composite spatial inertia of the floating base inherits properties of single-rigid-body inertia matrices. As we show, this includes the triangle inequality on the eigenvalues of the composite rotational inertia. To address the lack of physical consistency in deep learning models of floating-base systems, we introduce a parameterization of inertia matrices that satisfies all these constraints. Inspired by Deep Lagrangian Networks (DeLaN), we train neural networks to predict physically plausible inertia matrices that minimize inverse dynamics error under Lagrangian mechanics. For evaluation, we collected and released a dataset on multiple quadrupeds and humanoids. In these experiments, our Floating-Base Deep Lagrangian Networks (FeLaN) achieve highly competitive performance on both simulated and real robots, while providing greater physical interpretability.

Foundations of a new projection-based model reduction approach for convection dominated nonlinear fluid flows are summarized. In this method the evolution of the flow is approximated in the Lagrangian frame of reference. Global basis functions are used to approximate both the state and the position of the Lagrangian computational domain. It is demonstrated that in this framework, certain wave-like solutions exhibit low-rank structure and thus, can be efficiently compressed using relatively few global basis. The proposed approach is successfully demonstrated for the reduction of several simple but representative problems.

Flow-based generative models achieve state-of-the-art sample quality, but require the expensive solution of a differential equation at inference time. Flow map models, commonly known as consistency models, encompass many recent efforts to improve inference-time efficiency by learning the solution operator of this differential equation. Yet despite their promise, these models lack a unified description that clearly explains how to learn them efficiently in practice. Here, building on the methodology proposed in Boffi et. al. (2024), we present a systematic algorithmic framework for directly learning the flow map associated with a flow or diffusion model. By exploiting a relationship between the velocity field underlying a continuous-time flow and the instantaneous rate of change of the flow map, we show how to convert any distillation scheme into a direct training algorithm via self-distillation, eliminating the need for pre-trained teachers. We introduce three algorithmic families based on different mathematical characterizations of the flow map: Eulerian, Lagrangian, and Progressive methods, which we show encompass and extend all known distillation and direct training schemes for consistency models. We find that the novel class of Lagrangian methods, which avoid both spatial derivatives and bootstrapping from small steps by design, achieve significantly more stable training and higher performance than more standard Eulerian and Progressive schemes. Our methodology unifies existing training schemes under a single common framework and reveals new design principles for accelerated generative modeling. Associated code is available at https://github.com/nmboffi/flow-maps.

Humans manipulate various kinds of fluids in their everyday life: creating latte art, scooping floating objects from water, rolling an ice cream cone, etc. Using robots to augment or replace human labors in these daily settings remain as a challenging task due to the multifaceted complexities of fluids. Previous research in robotic fluid manipulation mostly consider fluids governed by an ideal, Newtonian model in simple task settings (e.g., pouring). However, the vast majority of real-world fluid systems manifest their complexities in terms of the fluid's complex material behaviors and multi-component interactions, both of which were well beyond the scope of the current literature. To evaluate robot learning algorithms on understanding and interacting with such complex fluid systems, a comprehensive virtual platform with versatile simulation capabilities and well-established tasks is needed. In this work, we introduce FluidLab, a simulation environment with a diverse set of manipulation tasks involving complex fluid dynamics. These tasks address interactions between solid and fluid as well as among multiple fluids. At the heart of our platform is a fully differentiable physics simulator, FluidEngine, providing GPU-accelerated simulations and gradient calculations for various material types and their couplings. We identify several challenges for fluid manipulation learning by evaluating a set of reinforcement learning and trajectory optimization methods on our platform. To address these challenges, we propose several domain-specific optimization schemes coupled with differentiable physics, which are empirically shown to be effective in tackling optimization problems featured by fluid system's non-convex and non-smooth properties. Furthermore, we demonstrate reasonable sim-to-real transfer by deploying optimized trajectories in real-world settings.

This paper aims to extend the Structured Knowledge Accumulation (SKA) framework recently proposed by mahi2025ska. We introduce two core concepts: the Tensor Net function and the characteristic time property of neural learning. First, we reinterpret the learning rate as a time step in a continuous system. This transforms neural learning from discrete optimization into continuous-time evolution. We show that learning dynamics remain consistent when the product of learning rate and iteration steps stays constant. This reveals a time-invariant behavior and identifies an intrinsic timescale of the network. Second, we define the Tensor Net function as a measure that captures the relationship between decision probabilities, entropy gradients, and knowledge change. Additionally, we define its zero-crossing as the equilibrium state between decision probabilities and entropy gradients. We show that the convergence of entropy and knowledge flow provides a natural stopping condition, replacing arbitrary thresholds with an information-theoretic criterion. We also establish that SKA dynamics satisfy a variational principle based on the Euler-Lagrange equation. These findings extend SKA into a continuous and self-organizing learning model. The framework links computational learning with physical systems that evolve by natural laws. By understanding learning as a time-based process, we open new directions for building efficient, robust, and biologically-inspired AI systems.

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of R^n, under compactly supported variations. The critical point solves a fourth order nonlinear equation in double divergence form. We show that for smooth convex functionals, a W^{2,infty} critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most n-p_0, for some p_0 in (2,3). We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.

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.

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.

Advances in time-resolved Particle Tracking Velocimetry (4D-PTV) techniques have been consistently revealed more accurate Lagrangian particle motions. A novel track initialisation technique as a complementary part of 4D-PTV, based on local temporal and spatial coherency of neighbour trajectories, is proposed. The proposed Lagrangian Coherent Track Initialisation (LCTI) applies physics-based Finite Time Lyapunov Exponent (FTLE) to build four frame coherent tracks. We locally determine the boundaries (i.e., ridges) of Lagrangian Coherent Structures (LCS) among neighbour trajectories by using FTLE to distinguish clusters of coherent motions. To evaluate the proposed technique, we created an open-access synthetic Lagrangian and Eulerian dataset of the wake downstream of a smooth cylinder at a Reynolds number equal to 3900 obtained from 3D Direct Numerical Simulation (DNS). The dataset is available to the public. Performance of the proposed method based on three characteristic parameters, temporal scale, particle concentration (i.e., density), and noise ratio, showed robust behaviour in finding true tracks compared to the recent initialisation algorithms. Sensitivity of LCTI to the number of untracked and wrong tracks are also discussed. We address the capability of using the proposed method as a function of a 4D-PTV scheme in the Lagrangian Particle Tracking challenge for a flow with high particle densities. Finally, the LCTI behaviour was assessed in a real jet impingement experiment. LCTI was found to be a reliable tracking tool in complex flow motions, with a strength revealed for flows with high particle concentrations.

The precision, stability, and performance of lightweight high-strength steel structures in heavy machinery is affected by their highly nonlinear dynamics. This, in turn, makes control more difficult, simulation more computationally intensive, and achieving real-time autonomy, using standard approaches, impossible. Machine learning through data-driven, physics-informed and physics-inspired networks, however, promises more computationally efficient and accurate solutions to nonlinear dynamic problems. This study proposes a novel framework that has been developed to estimate real-time structural deflection in hydraulically actuated three-dimensional systems. It is based on SLIDE, a machine-learning-based method to estimate dynamic responses of mechanical systems subjected to forced excitations.~Further, an algorithm is introduced for the data acquisition from a hydraulically actuated system using randomized initial configurations and hydraulic pressures.~The new framework was tested on a hydraulically actuated flexible boom with various sensor combinations and lifting various payloads. The neural network was successfully trained in less time using standard parameters from PyTorch, ADAM optimizer, the various sensor inputs, and minimal output data. The SLIDE-trained neural network accelerated deflection estimation solutions by a factor of 10^7 in reference to flexible multibody simulation batches and provided reasonable accuracy. These results support the studies goal of providing robust, real-time solutions for control, robotic manipulators, structural health monitoring, and automation problems.

This paper introduces a novel Hamiltonian-inspired neural network approach to credit scoring, designed to address the challenges of class imbalance and out-of-time (OOT) prediction in financial risk management. Drawing from concepts in Hamiltonian mechanics, we develop a symplectic optimizer and a new loss function to capture the complex dynamics of credit risk evolution. Using the Freddie Mac Single-Family Loan-Level Dataset, we evaluate our model's performance against other machine learning approaches. Our method shows superior discriminative power in OOT scenarios, as measured by the Area Under the Curve (AUC), indicating better ranking ability and robustness to class imbalance. The Hamiltonian-inspired approach shows particular strength in maintaining consistent performance between in-sample and OOT test sets, suggesting improved generalization to future, unseen data. These findings suggest that physics-inspired techniques offer a promising direction for developing more robust and reliable credit scoring models, particularly in uncertain economic situations.

Machine learning frameworks for physical problems must capture and enforce physical constraints that preserve the structure of dynamical systems. Many existing approaches achieve this by integrating physical operators into neural networks. While these methods offer theoretical guarantees, they face two key limitations: (i) they primarily model local relations between adjacent time steps, overlooking longer-range or higher-level physical interactions, and (ii) they focus on forward simulation while neglecting broader physical reasoning tasks. We propose the Denoising Hamiltonian Network (DHN), a novel framework that generalizes Hamiltonian mechanics operators into more flexible neural operators. DHN captures non-local temporal relationships and mitigates numerical integration errors through a denoising mechanism. DHN also supports multi-system modeling with a global conditioning mechanism. We demonstrate its effectiveness and flexibility across three diverse physical reasoning tasks with distinct inputs and outputs.

Foundation models are premised on the idea that sequence prediction can uncover deeper domain understanding, much like how Kepler's predictions of planetary motion later led to the discovery of Newtonian mechanics. However, evaluating whether these models truly capture deeper structure remains a challenge. We develop a technique for evaluating foundation models that examines how they adapt to synthetic datasets generated from some postulated world model. Our technique measures whether the foundation model's inductive bias aligns with the world model, and so we refer to it as an inductive bias probe. Across multiple domains, we find that foundation models can excel at their training tasks yet fail to develop inductive biases towards the underlying world model when adapted to new tasks. We particularly find that foundation models trained on orbital trajectories consistently fail to apply Newtonian mechanics when adapted to new physics tasks. Further analysis reveals that these models behave as if they develop task-specific heuristics that fail to generalize.

We introduce Riemannian Lyapunov Optimizers (RLOs), a family of optimization algorithms that unifies classic optimizers within one geometric framework. Unlike heuristic improvements to existing optimizers, RLOs are systematically derived from a novel control-theoretic framework that reinterprets optimization as an extended state discrete-time controlled dynamical system on a Riemannian parameter manifold. Central to this framework is the identification of a Normally Attracting Invariant Manifold (NAIM), which organizes training dynamics into two distinct stages: rapid alignment of the speed state to a target graph, followed by controlled evolution within it. We formalize this by constructing a strict Lyapunov function that certifies convergence to a target manifold. This perspective yields a constructive ``optimizer generator" that not only recovers classic algorithms but enables the principled design of RLOs. We validate our theory via geometric diagnostics and demonstrate that grounding optimizer design in control theory yields state-of-the-art performance in large-scale benchmarks. Overall, RLOs bridge control theory and modern machine learning optimization, providing a unified language and a systematic toolkit for designing stable, effective optimizers.

We revisit the geometric theory of defects. In the differential-geometric models of defects that have been adopted since the 1950s, dislocations have been associated with torsion, disclinations with the full curvature, and point defects with the first kind trace of non-metricity. The mainstream formulation exhibits several conceptual and technical shortcomings, most notably a hierarchy inconsistency, the non-exictence of a genuine metric formulation, and the potential emergence of Ostrogradsky-type instabilities. These issues have motivated us to develop a new framework, namely a generalized teleparallel geometric theory of defects. In our model, dislocations are identified with the trace of torsion, disclinations with the second kind trace of the non-metricity, and point defects with the first kind trace of the non-metricity. In addition, we retain the scalar part torsion as a free parameter for describing some possible unknown degrees of freedom in the theory of defects. The proposed geometric theory of defects is free from all of the aforementioned drawbacks and is therefore worthy of further investigation. To ensure the coherence and completeness of the discussion, we begin our analysis with elastic deformations, then summarize the existing metric-affine geometric theory of defects, and finally proceed to our original contribution, namely the new theory introduced here. We formulate the entire theory in Eulerian coordinates. Naturally, all results can be reformulated in Lagrangian coordinates as well. All analyses and formulae are expressed in the language of exterior algebra and are carried out in coordinate-independent orthonormal frames.

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

Minimax problems are notoriously challenging to optimize. However, we demonstrate that the two-timescale extragradient can be a viable solution. By utilizing dynamical systems theory, we show that it converges to points that satisfy the second-order necessary condition of local minimax points, under a mild condition. This work surpasses all previous results as we eliminate a crucial assumption that the Hessian, with respect to the maximization variable, is nondegenerate.

This paper presents Dual Lagrangian Learning (DLL), a principled learning methodology for dual conic optimization proxies. DLL leverages conic duality and the representation power of ML models to provide high-duality, dual-feasible solutions, and therefore valid Lagrangian dual bounds, for linear and nonlinear conic optimization problems. The paper introduces a systematic dual completion procedure, differentiable conic projection layers, and a self-supervised learning framework based on Lagrangian duality. It also provides closed-form dual completion formulae for broad classes of conic problems, which eliminate the need for costly implicit layers. The effectiveness of DLL is demonstrated on linear and nonlinear conic optimization problems. The proposed methodology significantly outperforms a state-of-the-art learning-based method, and achieves 1000x speedups over commercial interior-point solvers with optimality gaps under 0.5\% on average.

The quantitative prediction of the intensity of rainfall events (light or heavy) has remained a challenge in Numerical Weather Prediction (NWP) models. For the first time the mean coefficient of diffusional growth rates are calculated using an Eulerian-Lagrangian particle-based small-scale model on in situ airborne measurement data of Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX) during monsoon over Indian sub-continent. The results show that diffusional growth rates varies in the range of 0.00025 - 0.0015(cm/s). The generic problem of the overestimation of light rain in NWP models might be related with the choice of cm in the model. It is also shown from DNS experiment using Eulerian-Lagrangian particle-based small-scale model that the relative dispersion is constrained with average values in the range of ~ 0.2 - 0.37 (~ 0.1- 0.26) in less humid (more humid) conditions. This is in agreement with in situ airborne observation (dispersion ~ 0.36) and previous study over Indian sub-continent. The linear relationship between relative dispersion and cloud droplet number concentration (NC) is obtained from this study using CAIPEEX observation over Indian subcontinent. The dispersion based autoconversion-scheme for Indian region must be useful for the Indian summer monsoon precipitation calculation in the general circulation model. The present study also provide valuable guidance for the parameterization of effective radius, important for radiation scheme.

Recent advances in deep learning for physics have focused on discovering shared representations of target systems by incorporating physics priors or inductive biases into neural networks. While effective, these methods are limited to the system domain, where the type of system remains consistent and thus cannot ensure the adaptation to new, or unseen physical systems governed by different laws. For instance, a neural network trained on a mass-spring system cannot guarantee accurate predictions for the behavior of a two-body system or any other system with different physical laws. In this work, we take a significant leap forward by targeting cross domain generalization within the field of Hamiltonian dynamics. We model our system with a graph neural network and employ a meta learning algorithm to enable the model to gain experience over a distribution of tasks and make it adapt to new physics. Our approach aims to learn a unified Hamiltonian representation that is generalizable across multiple system domains, thereby overcoming the limitations of system-specific models. Our results demonstrate that the meta-trained model not only adapts effectively to new systems but also captures a generalized Hamiltonian representation that is consistent across different physical domains. Overall, through the use of meta learning, we offer a framework that achieves cross domain generalization, providing a step towards a unified model for understanding a wide array of dynamical systems via deep learning.

Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi-analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid-based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation -- the Legendre equation, the hypergeometric equation, the solution for black hole space-times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.

AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN). We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP in terms of accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

Imposing known physical constraints, such as conservation laws, during neural network training introduces an inductive bias that can improve accuracy, reliability, convergence, and data efficiency for modeling physical dynamics. While such constraints can be softly imposed via loss function penalties, recent advancements in differentiable physics and optimization improve performance by incorporating PDE-constrained optimization as individual layers in neural networks. This enables a stricter adherence to physical constraints. However, imposing hard constraints significantly increases computational and memory costs, especially for complex dynamical systems. This is because it requires solving an optimization problem over a large number of points in a mesh, representing spatial and temporal discretizations, which greatly increases the complexity of the constraint. To address this challenge, we develop a scalable approach to enforce hard physical constraints using Mixture-of-Experts (MoE), which can be used with any neural network architecture. Our approach imposes the constraint over smaller decomposed domains, each of which is solved by an "expert" through differentiable optimization. During training, each expert independently performs a localized backpropagation step by leveraging the implicit function theorem; the independence of each expert allows for parallelization across multiple GPUs. Compared to standard differentiable optimization, our scalable approach achieves greater accuracy in the neural PDE solver setting for predicting the dynamics of challenging non-linear systems. We also improve training stability and require significantly less computation time during both training and inference stages.

Differentiable physics enables efficient gradient-based optimizations of neural network (NN) controllers. However, existing work typically only delivers NN controllers with limited capability and generalizability. We present a practical learning framework that outputs unified NN controllers capable of tasks with significantly improved complexity and diversity. To systematically improve training robustness and efficiency, we investigated a suite of improvements over the baseline approach, including periodic activation functions, and tailored loss functions. In addition, we find our adoption of batching and an Adam optimizer effective in training complex locomotion tasks. We evaluate our framework on differentiable mass-spring and material point method (MPM) simulations, with challenging locomotion tasks and multiple robot designs. Experiments show that our learning framework, based on differentiable physics, delivers better results than reinforcement learning and converges much faster. We demonstrate that users can interactively control soft robot locomotion and switch among multiple goals with specified velocity, height, and direction instructions using a unified NN controller trained in our system. Code is available at https://github.com/erizmr/Complex-locomotion-skill-learning-via-differentiable-physics.

We present formulation and open-source tools to achieve in-material model predictive control of sensor/actuator systems using learned forward kinematics and on-device computation. Microcontroller units (MCUs) that compute the prediction and control task while colocated with the sensors and actuators enable in-material untethered behaviors. In this approach, small parameter size neural network models learn forward kinematics offline. Our open-source compiler, nn4mc, generates code to offload these predictions onto MCUs. A Newton-Raphson solver then computes the control input in real time. We first benchmark this nonlinear control approach against a PID controller on a mass-spring-damper simulation. We then study experimental results on two experimental rigs with different sensing, actuation and computational hardware: a tendon-based platform with embedded LightLace sensors and a HASEL-based platform with magnetic sensors. Experimental results indicate effective high-bandwidth tracking of reference paths (greater than or equal to 120 Hz) with a small memory footprint (less than or equal to 6.4% of flash memory). The measured path following error does not exceed 2mm in the tendon-based platform. The simulated path following error does not exceed 1mm in the HASEL-based platform. The mean power consumption of this approach in an ARM Cortex-M4f device is 45.4 mW. This control approach is also compatible with Tensorflow Lite models and equivalent on-device code. In-material intelligence enables a new class of composites that infuse autonomy into structures and systems with refined artificial proprioception.

Ever since the original algorithm by Nesterov (1983), the true nature of the acceleration phenomenon has remained elusive, with various interpretations of why the method is actually faster. The diagnosis of the algorithm through the lens of Ordinary Differential Equations (ODEs) and the corresponding dynamical system formulation to explain the underlying dynamics has a rich history. In the literature, the ODEs that explain algorithms are typically derived by considering the limiting case of the algorithm maps themselves, that is, an ODE formulation follows the development of an algorithm. This obfuscates the underlying higher order principles and thus provides little evidence of the working of the algorithm. Such has been the case with Nesterov algorithm and the various analogies used to describe the acceleration phenomena, viz, momentum associated with the rolling of a Heavy-Ball down a slope, Hessian damping etc. The main focus of our work is to ideate the genesis of the Nesterov algorithm from the viewpoint of dynamical systems leading to demystifying the mathematical rigour behind the algorithm. Instead of reverse engineering ODEs from discrete algorithms, this work explores tools from the recently developed control paradigm titled Passivity and Immersion approach and the Geometric Singular Perturbation theory which are applied to arrive at the formulation of a dynamical system that explains and models the acceleration phenomena. This perspective helps to gain insights into the various terms present and the sequence of steps used in Nesterovs accelerated algorithm for the smooth strongly convex and the convex case. The framework can also be extended to derive the acceleration achieved using the triple momentum method and provides justifications for the non-convergence to the optimal solution in the Heavy-Ball method.

Controlling a robot based on physics-consistent dynamic models, such as Deep Lagrangian Networks (DeLaN), can improve the generalizability and interpretability of the resulting behavior. However, in complex environments, the number of objects to potentially interact with is vast, and their physical properties are often uncertain. This complexity makes it infeasible to employ a single global model. Therefore, we need to resort to online system identification of context-aware models that capture only the currently relevant aspects of the environment. While physical principles such as the conservation of energy may not hold across varying contexts, ensuring physical plausibility for any individual context-aware model can still be highly desirable, particularly when using it for receding horizon control methods such as model predictive control (MPC). Hence, in this work, we extend DeLaN to make it context-aware, combine it with a recurrent network for online system identification, and integrate it with an MPC for adaptive, physics-consistent control. We also combine DeLaN with a residual dynamics model to leverage the fact that a nominal model of the robot is typically available. We evaluate our method on a 7-DOF robot arm for trajectory tracking under varying loads. Our method reduces the end-effector tracking error by 39%, compared to a 21% improvement achieved by a baseline that uses an extended Kalman filter.

Reasoning system dynamics is one of the most important analytical approaches for many scientific studies. With the initial state of a system as input, the recent graph neural networks (GNNs)-based methods are capable of predicting the future state distant in time with high accuracy. Although these methods have diverse designs in modeling the coordinates and interacting forces of the system, we show that they actually share a common paradigm that learns the integration of the velocity over the interval between the initial and terminal coordinates. However, their integrand is constant w.r.t. time. Inspired by this observation, we propose a new approach to predict the integration based on several velocity estimations with Newton-Cotes formulas and prove its effectiveness theoretically. Extensive experiments on several benchmarks empirically demonstrate consistent and significant improvement compared with the state-of-the-art methods.

This work presents a probabilistic digital twin framework for response prediction in dynamical systems governed by misspecified physics. The approach integrates Gaussian Process Latent Force Models (GPLFM) and Bayesian Neural Networks (BNNs) to enable end-to-end uncertainty-aware inference and prediction. In the diagnosis phase, model-form errors (MFEs) are treated as latent input forces to a nominal linear dynamical system and jointly estimated with system states using GPLFM from sensor measurements. A BNN is then trained on posterior samples to learn a probabilistic nonlinear mapping from system states to MFEs, while capturing diagnostic uncertainty. For prognosis, this mapping is used to generate pseudo-measurements, enabling state prediction via Kalman filtering. The framework allows for systematic propagation of uncertainty from diagnosis to prediction, a key capability for trustworthy digital twins. The framework is demonstrated using four nonlinear examples: a single degree of freedom (DOF) oscillator, a multi-DOF system, and two established benchmarks -- the Bouc-Wen hysteretic system and the Silverbox experimental dataset -- highlighting its predictive accuracy and robustness to model misspecification.

In recent years, Physics-Informed Neural Networks (PINNs) have emerged as a powerful and robust framework for solving nonlinear differential equations across a wide range of scientific and engineering disciplines, including biology, geophysics, astrophysics and fluid dynamics. In the PINN framework, the governing partial differential equations, along with initial and boundary conditions, are encoded directly into the loss function, enabling the network to learn solutions that are consistent with the underlying physics. In this work, we employ the PINN framework to solve the dimensionless Navier-Stokes equations for three two-dimensional incompressible, steady, laminar flow problems without using any labeled data. The boundary and initial conditions are enforced in a hard manner, ensuring they are satisfied exactly rather than penalized during training. We validate the PINN predicted velocity profiles, drag coefficients and pressure profiles against the conventional computational fluid dynamics (CFD) simulations for moderate to high values of Reynolds number (Re). It is observed that the PINN predictions show good agreement with the CFD results at lower Re. We also extend our analysis to a transient condition and find that our method is equally capable of simulating complex time-dependent flow dynamics. To quantitatively assess the accuracy, we compute the L_2 normalized error, which lies in the range O(10^{-4}) - O(10^{-1}) for our chosen case studies.

Let (M omega) be a two dimensional symplectic manifold, phi: M to M a symplectomorphism with hyperbolic fixed point x and transversely intersecting stable and unstable manifolds W^s(phi, x) cap W^u(phi, x)=:H(phi, x). The intersection points are called homoclinic points, and the stable and unstable manifold are in this situation Lagrangian submanifolds. For this Lagrangian intersection problem with its infinite number of intersection points and wild oscillation behavior, we first define a Floer homology generated by finite sets of so-called contractible homoclinic points. This generalizes very significantly the Floer homologies generated by (semi)primary points defined by us in earlier works. Nevertheless these Floer homologies only consider quite `local' aspects of W^s(phi, x) cap W^u(phi, x) since their generator sets are finite, but the number of all contractible homoclinic points is infinite. To overcome this issue, we construct a direct limit of these `local' homoclinic Floer homologies over suitable index sets. These direct limits thus accumulate the information gathered by the finitely generated local' homoclinic Floer homologies.

Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations (PDEs) at high resolution are computationally too expensive for many-query scenarios, e.g., uncertainty quantification, remeshing applications, topology optimization, and so forth. This limitation has motivated the application of data-driven surrogate models, where the microscale computations are substituted with a surrogate, usually acting as a black-box mapping between macroscale quantities. These models offer significant speedups but struggle with incorporating microscale physical constraints, such as the balance of linear momentum and constitutive models. In this contribution, we propose Equilibrium Neural Operator (EquiNO) as a complementary physics-informed PDE surrogate for predicting microscale physics and compare it with variational physics-informed neural and operator networks. Our framework, applicable to the so-called multiscale FE^{,2}, computations, introduces the FE-OL approach by integrating the finite element (FE) method with operator learning (OL). We apply the proposed FE-OL approach to quasi-static problems of solid mechanics. The results demonstrate that FE-OL can yield accurate solutions even when confronted with a restricted dataset during model development. Our results show that EquiNO achieves speedup factors exceeding 8000-fold compared to traditional methods and offers an optimal balance between data-driven and physics-based strategies.

We propose a hybrid neural network (NN) and PDE approach for learning generalizable PDE dynamics from motion observations. Many NN approaches learn an end-to-end model that implicitly models both the governing PDE and constitutive models (or material models). Without explicit PDE knowledge, these approaches cannot guarantee physical correctness and have limited generalizability. We argue that the governing PDEs are often well-known and should be explicitly enforced rather than learned. Instead, constitutive models are particularly suitable for learning due to their data-fitting nature. To this end, we introduce a new framework termed "Neural Constitutive Laws" (NCLaw), which utilizes a network architecture that strictly guarantees standard constitutive priors, including rotation equivariance and undeformed state equilibrium. We embed this network inside a differentiable simulation and train the model by minimizing a loss function based on the difference between the simulation and the motion observation. We validate NCLaw on various large-deformation dynamical systems, ranging from solids to fluids. After training on a single motion trajectory, our method generalizes to new geometries, initial/boundary conditions, temporal ranges, and even multi-physics systems. On these extremely out-of-distribution generalization tasks, NCLaw is orders-of-magnitude more accurate than previous NN approaches. Real-world experiments demonstrate our method's ability to learn constitutive laws from videos.

Neural networks have shown great potential in accelerating the solution of partial differential equations (PDEs). Recently, there has been a growing interest in introducing physics constraints into training neural PDE solvers to reduce the use of costly data and improve the generalization ability. However, these physics constraints, based on certain finite dimensional approximations over the function space, must resolve the smallest scaled physics to ensure the accuracy and stability of the simulation, resulting in high computational costs from large input, output, and neural networks. This paper proposes a general acceleration methodology called NeuralStagger by spatially and temporally decomposing the original learning tasks into several coarser-resolution subtasks. We define a coarse-resolution neural solver for each subtask, which requires fewer computational resources, and jointly train them with the vanilla physics-constrained loss by simply arranging their outputs to reconstruct the original solution. Due to the perfect parallelism between them, the solution is achieved as fast as a coarse-resolution neural solver. In addition, the trained solvers bring the flexibility of simulating with multiple levels of resolution. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations, which leads to an additional 10sim100times speed-up. Moreover, the experiment also shows that the learned model could be well used for optimal control.

Tobasco et al. [Physics Letters A, 382:382-386, 2018; see https://doi.org/10.1016/j.physleta.2017.12.023] recently suggested that trajectories of ODE systems that optimize the infinite-time average of a certain observable can be localized using sublevel sets of a function that arise when bounding such averages using so-called auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimization is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localize the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed efficiently through direct nonlinear optimization. Such points provide good initial conditions for UPO computations. As a proof of concept, we then combine these methods with a single-shooting Newton-Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow. We discover three previously unknown families of UPOs, one of which simultaneously minimizes the mean energy dissipation rate and maximizes the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125.

Scientific machine learning and the advent of the Physics-Informed Neural Network (PINN) show considerable potential in their capacity to identify solutions to complex differential equations. Over the past two years, much work has gone into the development of PINNs capable of solving the gravity field modeling problem -- i.e.\ learning a differentiable form of the gravitational potential from position and acceleration estimates. While the past PINN gravity models (PINN-GMs) have demonstrated advantages in model compactness, robustness to noise, and sample efficiency; there remain key modeling challenges which this paper aims to address. Specifically, this paper introduces the third generation of the Physics-Informed Neural Network Gravity Model (PINN-GM-III) which solves the problems of extrapolation error, bias towards low-altitude samples, numerical instability at high-altitudes, and compliant boundary conditions through numerous modifications to the model's design. The PINN-GM-III is tested by modeling a known heterogeneous density asteroid, and its performance is evaluated using seven core metrics which showcases its strengths against its predecessors and other analytic and numerical gravity models.

We analyze Newton's method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the method. This significantly reduces the overall arithmetical complexity of second-order optimization schemes. By using the cubic regularization technique, we establish fast global convergence of our method to a second-order stationary point, while the Hessian does not need to be updated each iteration. For convex problems, we justify global and local superlinear rates for lazy Newton steps with quadratic regularization, which is easier to compute. The optimal frequency for updating the Hessian is once every d iterations, where d is the dimension of the problem. This provably improves the total arithmetical complexity of second-order algorithms by a factor d.

A major challenge of AI + Science lies in their inherent incompatibility: today's AI is primarily based on connectionism, while science depends on symbolism. To bridge the two worlds, we propose a framework to seamlessly synergize Kolmogorov-Arnold Networks (KANs) and science. The framework highlights KANs' usage for three aspects of scientific discovery: identifying relevant features, revealing modular structures, and discovering symbolic formulas. The synergy is bidirectional: science to KAN (incorporating scientific knowledge into KANs), and KAN to science (extracting scientific insights from KANs). We highlight major new functionalities in the pykan package: (1) MultKAN: KANs with multiplication nodes. (2) kanpiler: a KAN compiler that compiles symbolic formulas into KANs. (3) tree converter: convert KANs (or any neural networks) to tree graphs. Based on these tools, we demonstrate KANs' capability to discover various types of physical laws, including conserved quantities, Lagrangians, symmetries, and constitutive laws.

We introduce a new predator-prey model by replacing the growth and predation constant by a square matrix, and the population density as a population vector. The classical Lotka-Volterra model describes a population that either modulates or converges. Stability analysis of such models have been extensively studied by the works of Merdan (https://doi.org/10.1016/j.chaos.2007.06.062). The new model adds complexity by introducing an age group structure where the population of each age group evolves as prescribed by the Leslie matrix. The added complexity changes the behavior of the model such that the population either displays roughly an exponential growth or decay. We first provide an exact equation that describes a time evolution and use analytic techniques to obtain an approximate growth factor. We also discuss the variants of the Leslie model, i.e., the complex value predator-prey model and the competitive model. We then prove the Last Species Standing theorem that determines the dominant population in the large time limit. The recursive structure of the model denies the application of simple regression. We discuss a machine learning scheme that allows an admissible fit for the population evolution of Paramecium Aurelia and Paramecium Caudatum. Another potential avenue to simplify the computation is to use the machinery of quantum operators. We demonstrate the potential of this approach by computing the Hamiltonian of a simple Leslie system.

The prediction-based nonlinear reference governor (PRG) is an add-on algorithm to enforce constraints on pre-stabilized nonlinear systems by modifying, whenever necessary, the reference signal. The implementation of PRG carries a heavy computational burden, as it may require multiple numerical simulations of the plant model at each sample time. To this end, this paper proposes an alternative approach based on machine learning, where we first use a regression neural network (NN) to approximate the input-output map of the PRG from a set of training data. During the real-time operation, at each sample time, we use the trained NN to compute a nominal reference command, which may not be constraint admissible due to training errors and limited data. We adopt a novel sensitivity-based approach to minimally adjust the nominal reference while ensuring constraint enforcement. We thus refer to the resulting control strategy as the modified neural network reference governor (MNN-RG), which is significantly more computationally efficient than the PRG. The computational and theoretical properties of MNN-RG are presented. Finally, the effectiveness and limitations of the proposed method are studied by applying it as a load governor for constraint management in automotive fuel cell systems through simulation-based case studies.

We rigorously state the connection between the EHZ-capacity of convex Lagrangian products Ktimes TsubsetR^ntimesR^n and the minimal length of closed (K,T)-Minkowski billiard trajectories. This connection was made explicit for the first time by Artstein-Avidan and Ostrover under the assumption of smoothness and strict convexity of both K and T. We prove this connection in its full generality, i.e., without requiring any conditions on the convex bodies K and T. This prepares the computation of the EHZ-capacity of convex Lagrangian products of two convex polytopes by using discrete computational methods.

Many problems in astrophysics cover multiple orders of magnitude in spatial and temporal scales. While simulating systems that experience rapid changes in these conditions, it is essential to adapt the (time-) step size to capture the behavior of the system during those rapid changes and use a less accurate time step at other, less demanding, moments. We encounter three problems with traditional methods. Firstly, making such changes requires expert knowledge of the astrophysics as well as of the details of the numerical implementation. Secondly, some parameters that determine the time-step size are fixed throughout the simulation, which means that they do not adapt to the rapidly changing conditions of the problem. Lastly, we would like the choice of time-step size to balance accuracy and computation effort. We address these challenges with Reinforcement Learning by training it to select the time-step size dynamically. We use the integration of a system of three equal-mass bodies that move due to their mutual gravity as an example of its application. With our method, the selected integration parameter adapts to the specific requirements of the problem, both in terms of computation time and accuracy while eliminating the expert knowledge needed to set up these simulations. Our method produces results competitive to existing methods and improve the results found with the most commonly-used values of time-step parameter. This method can be applied to other integrators without further retraining. We show that this extrapolation works for variable time-step integrators but does not perform to the desired accuracy for fixed time-step integrators.

A primary challenge in using neural networks to approximate nonlinear dynamical systems governed by partial differential equations (PDEs) is transforming these systems into a suitable format, especially when dealing with non-linearizable dynamics or the need for infinite-dimensional spaces for linearization. This paper introduces DyMixOp, a novel neural operator framework for PDEs that integrates insights from complex dynamical systems to address this challenge. Grounded in inertial manifold theory, DyMixOp transforms infinite-dimensional nonlinear PDE dynamics into a finite-dimensional latent space, establishing a structured foundation that maintains essential nonlinear interactions and enhances physical interpretability. A key innovation is the Local-Global-Mixing (LGM) transformation, inspired by convection dynamics in turbulence. This transformation effectively captures both fine-scale details and nonlinear interactions, while mitigating spectral bias commonly found in existing neural operators. The framework is further strengthened by a dynamics-informed architecture that connects multiple LGM layers to approximate linear and nonlinear dynamics, reflecting the temporal evolution of dynamical systems. Experimental results across diverse PDE benchmarks demonstrate that DyMixOp achieves state-of-the-art performance, significantly reducing prediction errors, particularly in convection-dominated scenarios reaching up to 86.7\%, while maintaining computational efficiency and scalability.

The enduring challenge in the field of artificial intelligence has been the control of systems to achieve desired behaviours. While for systems governed by straightforward dynamics equations, methods like Linear Quadratic Regulation (LQR) have historically proven highly effective, most real-world tasks, which require a general problem-solver, demand world models with dynamics that cannot be easily described by simple equations. Consequently, these models must be learned from data using neural networks. Most model predictive control (MPC) algorithms designed for visual world models have traditionally explored gradient-free population-based optimisation methods, such as Cross Entropy and Model Predictive Path Integral (MPPI) for planning. However, we present an exploration of a gradient-based alternative that fully leverages the differentiability of the world model. In our study, we conduct a comparative analysis between our method and other MPC-based alternatives, as well as policy-based algorithms. In a sample-efficient setting, our method achieves on par or superior performance compared to the alternative approaches in most tasks. Additionally, we introduce a hybrid model that combines policy networks and gradient-based MPC, which outperforms pure policy based methods thereby holding promise for Gradient-based planning with world models in complex real-world tasks.

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the L^2 difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

Even for known nonlinear dynamical systems, feedback controller synthesis is a difficult problem that often requires leveraging the particular structure of the dynamics to induce a stable closed-loop system. For general nonlinear models, including those fit to data, there may not be enough known structure to reliably synthesize a stabilizing feedback controller. In this paper, we discuss a state-dependent nonlinear tracking controller formulation based on a state-dependent Riccati equation for general nonlinear control-affine systems. This formulation depends on a nonlinear factorization of the system of vector fields defining the control-affine dynamics, which always exists under mild smoothness assumptions. We propose a method for learning this factorization from a finite set of data. On a variety of simulated nonlinear dynamical systems, we empirically demonstrate the efficacy of learned versions of this controller in stable trajectory tracking. Alongside our learning method, we evaluate recent ideas in jointly learning a controller and stabilizability certificate for known dynamical systems; we show experimentally that such methods can be frail in comparison.

Equilibrium propagation (EP) is a compelling alternative to the backpropagation of error algorithm (BP) for computing gradients of neural networks on biological or analog neuromorphic substrates. Still, the algorithm requires weight symmetry and infinitesimal equilibrium perturbations, i.e., nudges, to estimate unbiased gradients efficiently. Both requirements are challenging to implement in physical systems. Yet, whether and how weight asymmetry affects its applicability is unknown because, in practice, it may be masked by biases introduced through the finite nudge. To address this question, we study generalized EP, which can be formulated without weight symmetry, and analytically isolate the two sources of bias. For complex-differentiable non-symmetric networks, we show that the finite nudge does not pose a problem, as exact derivatives can still be estimated via a Cauchy integral. In contrast, weight asymmetry introduces bias resulting in low task performance due to poor alignment of EP's neuronal error vectors compared to BP. To mitigate this issue, we present a new homeostatic objective that directly penalizes functional asymmetries of the Jacobian at the network's fixed point. This homeostatic objective dramatically improves the network's ability to solve complex tasks such as ImageNet 32x32. Our results lay the theoretical groundwork for studying and mitigating the adverse effects of imperfections of physical networks on learning algorithms that rely on the substrate's relaxation dynamics.

This study introduces amangkurat, an open-source Python library designed for the robust numerical simulation of relativistic scalar field dynamics governed by the nonlinear Klein-Gordon equation in (1+1)D spacetime. The software implements a hybrid computational strategy that couples Fourier pseudo-spectral spatial discretization with a symplectic Størmer-Verlet temporal integrator, ensuring both exponential spatial convergence for smooth solutions and long-term preservation of Hamiltonian structure. To optimize performance, the solver incorporates adaptive timestepping based on Courant-Friedrichs-Lewy (CFL) stability criteria and utilizes Just-In-Time (JIT) compilation for parallelized force computation. The library's capabilities are validated across four canonical physical regimes: dispersive linear wave propagation, static topological kink preservation in phi-fourth theory, integrable breather dynamics in the sine-Gordon model, and non-integrable kink-antikink collisions. Beyond standard numerical validation, this work establishes a multi-faceted analysis framework employing information-theoretic entropy metrics (Shannon, Rényi, and Tsallis), kernel density estimation, and phase space reconstruction to quantify the distinct phenomenological signatures of these regimes. Statistical hypothesis testing confirms that these scenarios represent statistically distinguishable dynamical populations. Benchmarks on standard workstation hardware demonstrate that the implementation achieves high computational efficiency, making it a viable platform for exploratory research and education in nonlinear field theory.

Reinforcement learning provides a framework for learning control policies that can reproduce diverse motions for simulated characters. However, such policies often exploit unnatural high-frequency signals that are unachievable by humans or physical robots, making them poor representations of real-world behaviors. Existing work addresses this issue by adding a reward term that penalizes a large change in actions over time. This term often requires substantial tuning efforts. We propose to use the action Jacobian penalty, which penalizes changes in action with respect to the changes in simulated state directly through auto differentiation. This effectively eliminates unrealistic high-frequency control signals without task specific tuning. While effective, the action Jacobian penalty introduces significant computational overhead when used with traditional fully connected neural network architectures. To mitigate this, we introduce a new architecture called a Linear Policy Net (LPN) that significantly reduces the computational burden for calculating the action Jacobian penalty during training. In addition, a LPN requires no parameter tuning, exhibits faster learning convergence compared to baseline methods, and can be more efficiently queried during inference time compared to a fully connected neural network. We demonstrate that a Linear Policy Net, combined with the action Jacobian penalty, is able to learn policies that generate smooth signals while solving a number of motion imitation tasks with different characteristics, including dynamic motions such as a backflip and various challenging parkour skills. Finally, we apply this approach to create policies for dynamic motions on a physical quadrupedal robot equipped with an arm.

A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most prior theoretical analyses have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as nonlinear elliptic variational PDEs, whose solutions minimize an Euler-Lagrange energy functional E(u) = int_Omega L(x, u(x), nabla u(x)) - f(x) u(x)dx. We show that if composing a function with Barron norm b with partial derivatives of L produces a function of Barron norm at most B_L b^p, the solution to the PDE can be epsilon-approximated in the L^2 sense by a function with Barron norm Oleft(left(dB_Lright)^{max{p log(1/ epsilon), p^{log(1/epsilon)}}}right). By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating p, epsilon, B_L as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.

We propose a physics-informed neural particle method (PINN--PM) for the spatially homogeneous Landau equation. The method adopts a Lagrangian interacting-particle formulation and jointly parameterizes the time-dependent score and the characteristic flow map with neural networks. Instead of advancing particles through explicit time stepping, the Landau dynamics is enforced via a continuous-time residual defined along particle trajectories. This design removes time-discretization error and yields a mesh-free solver that can be queried at arbitrary times without sequential integration. We establish a rigorous stability analysis in an L^2_v framework. The deviation between learned and exact characteristics is controlled by three interpretable sources: (i) score approximation error, (ii) empirical particle approximation error, and (iii) the physics residual of the neural flow. This trajectory estimate propagates to density reconstruction, where we derive an L^2_v error bound for kernel density estimators combining classical bias--variance terms with a trajectory-induced contribution. Using Hyvarinen's identity, we further relate the oracle score-matching gap to the L^2_v score error and show that the empirical loss concentrates at the Monte Carlo rate, yielding computable a posteriori accuracy certificates. Numerical experiments on analytical benchmarks, including the two- and three-dimensional BKW solutions, as well as reference-free configurations, demonstrate stable transport, preservation of macroscopic invariants, and competitive or improved accuracy compared with time-stepping score-based particle and blob methods while using significantly fewer particles.

We present Kinodynamic RRT*, an incremental sampling-based approach for asymptotically optimal motion planning for robots with linear differential constraints. Our approach extends RRT*, which was introduced for holonomic robots (Karaman et al. 2011), by using a fixed-final-state-free-final-time controller that exactly and optimally connects any pair of states, where the cost function is expressed as a trade-off between the duration of a trajectory and the expended control effort. Our approach generalizes earlier work on extending RRT* to kinodynamic systems, as it guarantees asymptotic optimality for any system with controllable linear dynamics, in state spaces of any dimension. Our approach can be applied to non-linear dynamics as well by using their first-order Taylor approximations. In addition, we show that for the rich subclass of systems with a nilpotent dynamics matrix, closed-form solutions for optimal trajectories can be derived, which keeps the computational overhead of our algorithm compared to traditional RRT* at a minimum. We demonstrate the potential of our approach by computing asymptotically optimal trajectories in three challenging motion planning scenarios: (i) a planar robot with a 4-D state space and double integrator dynamics, (ii) an aerial vehicle with a 10-D state space and linearized quadrotor dynamics, and (iii) a car-like robot with a 5-D state space and non-linear dynamics.

We describe and analyse Levenberg-Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under Hölder metric subregularity of the function defining the equation and Hölder continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the Łojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the Łojasiewicz gradient inequality assumption.

A common pipeline in learning-based control is to iteratively estimate a model of system dynamics, and apply a trajectory optimization algorithm - e.g.~iLQR - on the learned model to minimize a target cost. This paper conducts a rigorous analysis of a simplified variant of this strategy for general nonlinear systems. We analyze an algorithm which iterates between estimating local linear models of nonlinear system dynamics and performing iLQR-like policy updates. We demonstrate that this algorithm attains sample complexity polynomial in relevant problem parameters, and, by synthesizing locally stabilizing gains, overcomes exponential dependence in problem horizon. Experimental results validate the performance of our algorithm, and compare to natural deep-learning baselines.

Partial Differential Equations are foundational in modeling science and natural systems such as fluid dynamics and weather forecasting. The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers by proposing a scalable and efficient alternative. To enhance the efficiency and accuracy of LE-PDE, we incorporate the Mamba model, an advanced machine learning model known for its predictive efficiency and robustness in handling complex dynamic systems with a progressive learning strategy. The LE-PDE was tested on several benchmark problems. The method demonstrated a marked reduction in computational time compared to traditional solvers and standalone deep learning models while maintaining high accuracy in predicting system behavior over time. Our method doubles the inference speed compared to the LE-PDE while retaining the same level of parameter efficiency, making it well-suited for scenarios requiring long-term predictions.

Systems of interacting objects often evolve under the influence of field effects that govern their dynamics, yet previous works have abstracted away from such effects, and assume that systems evolve in a vacuum. In this work, we focus on discovering these fields, and infer them from the observed dynamics alone, without directly observing them. We theorize the presence of latent force fields, and propose neural fields to learn them. Since the observed dynamics constitute the net effect of local object interactions and global field effects, recently popularized equivariant networks are inapplicable, as they fail to capture global information. To address this, we propose to disentangle local object interactions -- which are SE(n) equivariant and depend on relative states -- from external global field effects -- which depend on absolute states. We model interactions with equivariant graph networks, and combine them with neural fields in a novel graph network that integrates field forces. Our experiments show that we can accurately discover the underlying fields in charged particles settings, traffic scenes, and gravitational n-body problems, and effectively use them to learn the system and forecast future trajectories.

Large language model (LLM)-based agentic frameworks increasingly adopt the paradigm of dynamically generating task-specific agents. We suggest that not only agents but also specialized software modules for scientific and engineering tasks can be generated on demand. We demonstrate this concept in the field of solid mechanics. There, so-called constitutive models are required to describe the relationship between mechanical stress and body deformation. Constitutive models are essential for both the scientific understanding and industrial application of materials. However, even recent data-driven methods of constitutive modeling, such as constitutive artificial neural networks (CANNs), still require substantial expert knowledge and human labor. We present a framework in which an LLM generates a CANN on demand, tailored to a given material class and dataset provided by the user. The framework covers LLM-based architecture selection, integration of physical constraints, and complete code generation. Evaluation on three benchmark problems demonstrates that LLM-generated CANNs achieve accuracy comparable to or greater than manually engineered counterparts, while also exhibiting reliable generalization to unseen loading scenarios and extrapolation to large deformations. These findings indicate that LLM-based generation of physics-constrained neural networks can substantially reduce the expertise required for constitutive modeling and represent a step toward practical end-to-end automation.

Bundle adjustment is the common way to solve localization and mapping. It is an iterative process in which a system of non-linear equations is solved using two optimization methods, weighted by a damping factor. In the classic approach, the latter is chosen heuristically by the Levenberg-Marquardt algorithm on each iteration. This might take many iterations, making the process computationally expensive, which might be harmful to real-time applications. We propose to replace this heuristic by viewing the problem in a holistic manner, as a game, and formulating it as a reinforcement-learning task. We set an environment which solves the non-linear equations and train an agent to choose the damping factor in a learned manner. We demonstrate that our approach considerably reduces the number of iterations required to reach the bundle adjustment's convergence, on both synthetic and real-life scenarios. We show that this reduction benefits the classic approach and can be integrated with other bundle adjustment acceleration methods.

Fluid-structure interaction is common in engineering and natural systems, where floating-body motion is governed by added mass, drag, and background flows. Modeling these dissipative dynamics is difficult: black-box neural models regress state derivatives with limited interpretability and unstable long-horizon predictions. We propose Floating-Body Hydrodynamic Neural Networks (FHNN), a physics-structured framework that predicts interpretable hydrodynamic parameters such as directional added masses, drag coefficients, and a streamfunction-based flow, and couples them with analytic equations of motion. This design constrains the hypothesis space, enhances interpretability, and stabilizes integration. On synthetic vortex datasets, FHNN achieves up to an order-of-magnitude lower error than Neural ODEs, recovers physically consistent flow fields. Compared with Hamiltonian and Lagrangian neural networks, FHNN more effectively handles dissipative dynamics while preserving interpretability, which bridges the gap between black-box learning and transparent system identification.

With the widespread adoption of machine learning systems, the need to curtail their behavior has become increasingly apparent. This is evidenced by recent advancements towards developing models that satisfy robustness, safety, and fairness requirements. These requirements can be imposed (with generalization guarantees) by formulating constrained learning problems that can then be tackled by dual ascent algorithms. Yet, though these algorithms converge in objective value, even in non-convex settings, they cannot guarantee that their outcome is feasible. Doing so requires randomizing over all iterates, which is impractical in virtually any modern applications. Still, final iterates have been observed to perform well in practice. In this work, we address this gap between theory and practice by characterizing the constraint violation of Lagrangian minimizers associated with optimal dual variables, despite lack of convexity. To do this, we leverage the fact that non-convex, finite-dimensional constrained learning problems can be seen as parametrizations of convex, functional problems. Our results show that rich parametrizations effectively mitigate the issue of feasibility in dual methods, shedding light on prior empirical successes of dual learning. We illustrate our findings in fair learning tasks.

A new class of affine scaling matrices for the interior point Newton-type methods is considered to solve the nonlinear systems with simple bounds. We review the essential properties of a scaling matrix and consider several well-known scaling matrices proposed in the literature. We define a new scaling matrix that is the convex combination of these matrices. The proposed scaling matrix inherits those interesting properties of the individual matrices and satisfies additional desired requirements. The numerical experiments demonstrate the superiority of the new scaling matrix in solving several important test problems.

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity.

As LLMs advance their reasoning capabilities about the physical world, the absence of rigorous benchmarks for evaluating their ability to generate scientifically valid physical models has become a critical gap. Computational mechanics, which develops and applies mathematical models and numerical methods to predict the behavior of physical systems under forces, deformation, and constraints, provides an ideal foundation for structured scientific reasoning evaluation. Problems follow clear mathematical structure, enforce strict physical and numerical constraints, and support objective verification. The discipline requires constructing explicit models of physical systems and reasoning about geometry, spatial relationships, and material behavior, connecting directly to emerging AI goals in physical reasoning and world modeling. We introduce FEM-Bench, a computational mechanics benchmark designed to evaluate the ability of LLMs to generate correct finite element method (FEM) and related code. FEM-Bench 2025 contains a suite of introductory but nontrivial tasks aligned with material from a first graduate course on computational mechanics. These tasks capture essential numerical and physical modeling challenges while representing only a small fraction of the complexity present in the discipline. Despite their simplicity, state-of-the-art LLMs do not reliably solve all of them. In a five attempt run, the best performing model at function writing, Gemini 3 Pro, completed 30/33 tasks at least once and 26/33 tasks all five times. The best performing model at unit test writing, GPT-5, had an Average Joint Success Rate of 73.8%. Other popular models showed broad performance variation. FEM-Bench establishes a structured foundation for evaluating AI-generated scientific code, and future iterations will incorporate increasingly sophisticated tasks to track progress as models evolve.

This paper investigates the global convergence of stepsized Newton methods for convex functions. We propose several simple stepsize schedules with fast global convergence guarantees, up to O (k^{-3}), nearly matching lower complexity bounds Omega (k^{-3.5}) of second-order methods. For cases with multiple plausible smoothness parameterizations or an unknown smoothness constant, we introduce a stepsize backtracking procedure that ensures convergence as if the optimal smoothness parameters were known.

Modeling of conservative systems with neural networks is an area of active research. A popular approach is to use Hamiltonian neural networks (HNNs) which rely on the assumptions that a conservative system is described with Hamilton's equations of motion. Many recent works focus on improving the integration schemes used when training HNNs. In this work, we propose to enhance HNNs with an estimation of a continuous-time trajectory of the modeled system using an additional neural network, called a deep hidden physics model in the literature. We demonstrate that the proposed integration scheme works well for HNNs, especially with low sampling rates, noisy and irregular observations.

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.

Reduced-order modeling (ROM) of time-dependent and parameterized differential equations aims to accelerate the simulation of complex high-dimensional systems by learning a compact latent manifold representation that captures the characteristics of the solution fields and their time-dependent dynamics. Although high-fidelity numerical solvers generate the training datasets, they have thus far been excluded from the training process, causing the learned latent dynamics to drift away from the discretized governing physics. This mismatch often limits generalization and forecasting capabilities. In this work, we propose Physics-informed ROM (Φ-ROM) by incorporating differentiable PDE solvers into the training procedure. Specifically, the latent space dynamics and its dependence on PDE parameters are shaped directly by the governing physics encoded in the solver, ensuring a strong correspondence between the full and reduced systems. Our model outperforms state-of-the-art data-driven ROMs and other physics-informed strategies by accurately generalizing to new dynamics arising from unseen parameters, enabling long-term forecasting beyond the training horizon, maintaining continuity in both time and space, and reducing the data cost. Furthermore, Φ-ROM learns to recover and forecast the solution fields even when trained or evaluated with sparse and irregular observations of the fields, providing a flexible framework for field reconstruction and data assimilation. We demonstrate the framework's robustness across various PDE solvers and highlight its broad applicability by providing an open-source JAX implementation that is readily extensible to other PDE systems and differentiable solvers, available at https://phi-rom.github.io.

Probabilistic completeness is an important property in motion planning. Although it has been established with clear assumptions for geometric planners, the panorama of completeness results for kinodynamic planners is still incomplete, as most existing proofs rely on strong assumptions that are difficult, if not impossible, to verify on practical systems. In this paper, we focus on an important class of kinodynamic planners, namely those that interpolate trajectories in the state space. We provide a proof of probabilistic completeness for these planners under assumptions that can be readily verified from the system's equations of motion and the user-defined interpolation function. Our proof relies crucially on a property of interpolated trajectories, termed second-order continuity (SOC), which we show is tightly related to the ability of a planner to benefit from denser sampling. We analyze the impact of this property in simulations on a low-torque pendulum. Our results show that a simple RRT using a second-order continuous interpolation swiftly finds solution, while it is impossible for the same planner using standard Bezier curves (which are not SOC) to find any solution.

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f =(f_1, ldots, f_N)in C[x_1, ldots, x_n]^N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $alpha$-theory test to certify the quadratic convergence, and togive bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

An obstacle to artificial general intelligence is set by continual learning of multiple tasks of different nature. Recently, various heuristic tricks, both from machine learning and from neuroscience angles, were proposed, but they lack a unified theory ground. Here, we focus on continual learning in single-layered and multi-layered neural networks of binary weights. A variational Bayesian learning setting is thus proposed, where the neural networks are trained in a field-space, rather than gradient-ill-defined discrete-weight space, and furthermore, weight uncertainty is naturally incorporated, and modulates synaptic resources among tasks. From a physics perspective, we translate the variational continual learning into Franz-Parisi thermodynamic potential framework, where previous task knowledge acts as a prior and a reference as well. We thus interpret the continual learning of the binary perceptron in a teacher-student setting as a Franz-Parisi potential computation. The learning performance can then be analytically studied with mean-field order parameters, whose predictions coincide with numerical experiments using stochastic gradient descent methods. Based on the variational principle and Gaussian field approximation of internal preactivations in hidden layers, we also derive the learning algorithm considering weight uncertainty, which solves the continual learning with binary weights using multi-layered neural networks, and performs better than the currently available metaplasticity algorithm. Our proposed principled frameworks also connect to elastic weight consolidation, weight-uncertainty modulated learning, and neuroscience inspired metaplasticity, providing a theory-grounded method for the real-world multi-task learning with deep networks.

A faster than real time forecast system for solar wind and interplanetary magnetic field transients that is driven by hourly updated solar magnetograms is proposed to provide a continuous nowcast of the solar corona (<0.1AU) and 24-hours forecast of the solar wind at 1 AU by solving a full 3-D MHD model. This new model has been inspired by the concept of relativity of simultaneity used in the theory of special relativity. It is based on time transformation between two coordinate systems: the solar rest frame and a boosted system in which the current observations of the solar magnetic field and tomorrow's measurement of the solar wind at 1 AU are simultaneous. In this paper we derive the modified governing equations for both hydrodynamics (HD) and magnetohydrodynamics (MHD) and present a new numerical algorithm that only modifies the conserved quantities but preserves the original HD/MHD numerical flux. The proposed method enables an efficient numerical implementation, and thus a significantly longer forecast time than the traditional method.

This paper presents a new approach and algorithm for solving a class of constrained Bi-Level Optimization (BLO) problems in which the lower-level problem involves constraints coupling both upper-level and lower-level variables. Such problems have recently gained significant attention due to their broad applicability in machine learning. However, conventional gradient-based methods unavoidably rely on computationally intensive calculations related to the Hessian matrix. To address this challenge, we begin by devising a smooth proximal Lagrangian value function to handle the constrained lower-level problem. Utilizing this construct, we introduce a single-level reformulation for constrained BLOs that transforms the original BLO problem into an equivalent optimization problem with smooth constraints. Enabled by this reformulation, we develop a Hessian-free gradient-based algorithm-termed proximal Lagrangian Value function-based Hessian-free Bi-level Algorithm (LV-HBA)-that is straightforward to implement in a single loop manner. Consequently, LV-HBA is especially well-suited for machine learning applications. Furthermore, we offer non-asymptotic convergence analysis for LV-HBA, eliminating the need for traditional strong convexity assumptions for the lower-level problem while also being capable of accommodating non-singleton scenarios. Empirical results substantiate the algorithm's superior practical performance.

Solving mechanics problems using numerical methods requires comprehensive intelligent capability of retrieving relevant knowledge and theory, constructing and executing codes, analyzing the results, a task that has thus far mainly been reserved for humans. While emerging AI methods can provide effective approaches to solve end-to-end problems, for instance via the use of deep surrogate models or various data analytics strategies, they often lack physical intuition since knowledge is baked into the parametric complement through training, offering less flexibility when it comes to incorporating mathematical or physical insights. By leveraging diverse capabilities of multiple dynamically interacting large language models (LLMs), we can overcome the limitations of conventional approaches and develop a new class of physics-inspired generative machine learning platform, here referred to as MechAgents. A set of AI agents can solve mechanics tasks, here demonstrated for elasticity problems, via autonomous collaborations. A two-agent team can effectively write, execute and self-correct code, in order to apply finite element methods to solve classical elasticity problems in various flavors (different boundary conditions, domain geometries, meshes, small/finite deformation and linear/hyper-elastic constitutive laws, and others). For more complex tasks, we construct a larger group of agents with enhanced division of labor among planning, formulating, coding, executing and criticizing the process and results. The agents mutually correct each other to improve the overall team-work performance in understanding, formulating and validating the solution. Our framework shows the potential of synergizing the intelligence of language models, the reliability of physics-based modeling, and the dynamic collaborations among diverse agents, opening novel avenues for automation of solving engineering problems.

Neural simulators promise efficient surrogates for physics simulation, but scaling them is bottlenecked by the prohibitive cost of generating high-fidelity training data. Pre-training on abundant off-the-shelf geometries offers a natural alternative, yet faces a fundamental gap: supervision on static geometry alone ignores dynamics and can lead to negative transfer on physics tasks. We present GeoPT, a unified pre-trained model for general physics simulation based on lifted geometric pre-training. The core idea is to augment geometry with synthetic dynamics, enabling dynamics-aware self-supervision without physics labels. Pre-trained on over one million samples, GeoPT consistently improves industrial-fidelity benchmarks spanning fluid mechanics for cars, aircraft, and ships, and solid mechanics in crash simulation, reducing labeled data requirements by 20-60% and accelerating convergence by 2times. These results show that lifting with synthetic dynamics bridges the geometry-physics gap, unlocking a scalable path for neural simulation and potentially beyond. Code is available at https://github.com/Physics-Scaling/GeoPT.

Since batch algorithms suffer from lack of proficiency in confronting model mismatches and disturbances, this contribution proposes an adaptive scheme based on continuous Lyapunov function for online robot dynamic identification. This paper suggests stable updating rules to drive neural networks inspiring from model reference adaptive paradigm. Network structure consists of three parallel self-driving neural networks which aim to estimate robot dynamic terms individually. Lyapunov candidate is selected to construct energy surface for a convex optimization framework. Learning rules are driven directly from Lyapunov functions to make the derivative negative. Finally, experimental results on 3-DOF Phantom Omni Haptic device demonstrate efficiency of the proposed method.

Computing the optimal transport distance between statistical distributions is a fundamental task in machine learning. One remarkable recent advancement is entropic regularization and the Sinkhorn algorithm, which utilizes only matrix scaling and guarantees an approximated solution with near-linear runtime. Despite the success of the Sinkhorn algorithm, its runtime may still be slow due to the potentially large number of iterations needed for convergence. To achieve possibly super-exponential convergence, we present Sinkhorn-Newton-Sparse (SNS), an extension to the Sinkhorn algorithm, by introducing early stopping for the matrix scaling steps and a second stage featuring a Newton-type subroutine. Adopting the variational viewpoint that the Sinkhorn algorithm maximizes a concave Lyapunov potential, we offer the insight that the Hessian matrix of the potential function is approximately sparse. Sparsification of the Hessian results in a fast O(n^2) per-iteration complexity, the same as the Sinkhorn algorithm. In terms of total iteration count, we observe that the SNS algorithm converges orders of magnitude faster across a wide range of practical cases, including optimal transportation between empirical distributions and calculating the Wasserstein W_1, W_2 distance of discretized densities. The empirical performance is corroborated by a rigorous bound on the approximate sparsity of the Hessian matrix.

We introduce PhysGaussian, a new method that seamlessly integrates physically grounded Newtonian dynamics within 3D Gaussians to achieve high-quality novel motion synthesis. Employing a custom Material Point Method (MPM), our approach enriches 3D Gaussian kernels with physically meaningful kinematic deformation and mechanical stress attributes, all evolved in line with continuum mechanics principles. A defining characteristic of our method is the seamless integration between physical simulation and visual rendering: both components utilize the same 3D Gaussian kernels as their discrete representations. This negates the necessity for triangle/tetrahedron meshing, marching cubes, "cage meshes," or any other geometry embedding, highlighting the principle of "what you see is what you simulate (WS^2)." Our method demonstrates exceptional versatility across a wide variety of materials--including elastic entities, metals, non-Newtonian fluids, and granular materials--showcasing its strong capabilities in creating diverse visual content with novel viewpoints and movements. Our project page is at: https://xpandora.github.io/PhysGaussian/

Partial differential equations (PDEs) are instrumental for modeling dynamical systems in science and engineering. The advent of neural networks has initiated a significant shift in tackling these complexities though challenges in accuracy persist, especially for initial value problems. In this paper, we introduce the Time-Evolving Natural Gradient (TENG), generalizing time-dependent variational principles and optimization-based time integration, leveraging natural gradient optimization to obtain high accuracy in neural-network-based PDE solutions. Our comprehensive development includes algorithms like TENG-Euler and its high-order variants, such as TENG-Heun, tailored for enhanced precision and efficiency. TENG's effectiveness is further validated through its performance, surpassing current leading methods and achieving machine precision in step-by-step optimizations across a spectrum of PDEs, including the heat equation, Allen-Cahn equation, and Burgers' equation.

Diffusion models have demonstrated remarkable generation quality but at the cost of numerous function evaluations. Recently, advanced ODE-based solvers have been developed to mitigate the substantial computational demands of reverse-diffusion solving under limited sampling steps. However, these solvers, heavily inspired by Adams-like multistep methods, rely solely on t-related Lagrange interpolation. We show that t-related Lagrange interpolation is suboptimal for diffusion model and reveal a compact search space comprised of time steps and solver coefficients. Building on our analysis, we propose a novel differentiable solver search algorithm to identify more optimal solver. Equipped with the searched solver, rectified-flow models, e.g., SiT-XL/2 and FlowDCN-XL/2, achieve FID scores of 2.40 and 2.35, respectively, on ImageNet256 with only 10 steps. Meanwhile, DDPM model, DiT-XL/2, reaches a FID score of 2.33 with only 10 steps. Notably, our searched solver outperforms traditional solvers by a significant margin. Moreover, our searched solver demonstrates generality across various model architectures, resolutions, and model sizes.

We present a novel structure-preserving framework for solving the Vlasov-Poisson-Landau system of equations using a particle in cell (PIC) discretization combined with discrete gradient time integrators. The Vlasov-Poisson-Landau system is an accurate model for studying hot plasma dynamics at a kinetic scale where small-angle Coulomb collisions dominate. Our scheme guarantees conservation of mass, momentum and energy as well as preservation of the monotonicity of entropy production in both the time-continuous and discrete systems. We employ the conservative integrator for both the Hamiltonian Vlasov-Poisson equations and the dissipative Landau equation using the PETSc library (www.mcs.anl.gov/petsc) to showcase structure-preserving properties.

To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.

Hybrid systems are prevalent in robotics. However, ensuring the stability of hybrid systems is challenging due to sophisticated continuous and discrete dynamics. A system with all its system modes stable can still be unstable. Hence special treatments are required at mode switchings to stabilize the system. In this work, we propose a hierarchical, neural network (NN)-based method to control general hybrid systems. For each system mode, we first learn an NN Lyapunov function and an NN controller to ensure the states within the region of attraction (RoA) can be stabilized. Then an RoA NN estimator is learned across different modes. Upon mode switching, we propose a differentiable planner to ensure the states after switching can land in next mode's RoA, hence stabilizing the hybrid system. We provide novel theoretical stability guarantees and conduct experiments in car tracking control, pogobot navigation, and bipedal walker locomotion. Our method only requires 0.25X of the training time as needed by other learning-based methods. With low running time (10-50X faster than model predictive control (MPC)), our controller achieves a higher stability/success rate over other baselines such as MPC, reinforcement learning (RL), common Lyapunov methods (CLF), linear quadratic regulator (LQR), quadratic programming (QP) and Hamilton-Jacobian-based methods (HJB). The project page is on https://mit-realm.github.io/hybrid-clf.

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

In this paper, we propose a novel neural network framework, the Legendre-Kolmogorov-Arnold Network (Legendre-KAN) method, designed to solve fully nonlinear Monge-Ampère equations with Dirichlet boundary conditions. The architecture leverages the orthogonality of Legendre polynomials as basis functions, significantly enhancing both convergence speed and solution accuracy compared to traditional methods. Furthermore, the Kolmogorov-Arnold representation theorem provides a strong theoretical foundation for the interpretability and optimization of the network. We demonstrate the effectiveness of the proposed method through numerical examples, involving both smooth and singular solutions in various dimensions. This work not only addresses the challenges of solving high-dimensional and singular Monge-Ampère equations but also highlights the potential of neural network-based approaches for complex partial differential equations. Additionally, the method is applied to the optimal transport problem in image mapping, showcasing its practical utility in geometric image transformation. This approach is expected to pave the way for further enhancement of KAN-based applications and numerical solutions of PDEs across a wide range of scientific and engineering fields.

Physics-informed neural networks (PINNs) are a newly emerging research frontier in machine learning, which incorporate certain physical laws that govern a given data set, e.g., those described by partial differential equations (PDEs), into the training of the neural network (NN) based on such a data set. In PINNs, the NN acts as the solution approximator for the PDE while the PDE acts as the prior knowledge to guide the NN training, leading to the desired generalization performance of the NN when facing the limited availability of training data. However, training PINNs is a non-trivial task largely due to the complexity of the loss composed of both NN and physical law parts. In this work, we propose a new PINN training framework based on the multi-task optimization (MTO) paradigm. Under this framework, multiple auxiliary tasks are created and solved together with the given (main) task, where the useful knowledge from solving one task is transferred in an adaptive mode to assist in solving some other tasks, aiming to uplift the performance of solving the main task. We implement the proposed framework and apply it to train the PINN for addressing the traffic density prediction problem. Experimental results demonstrate that our proposed training framework leads to significant performance improvement in comparison to the traditional way of training the PINN.

The dynamical realisation of the equation of state p +rho =0 is studied. A non-pathological dynamics for the perturbations of such a system mimicking a dynamical cosmological constant (DCC) requires to go beyond the perfect fluid paradigm. It is shown that an anisotropic stress must be always present. The Hamiltonian of the system in isolation resembles the one of a Pais-Uhlenbeck oscillator and linear stability requires that it cannot be positive definite. The dynamics of linear cosmological perturbations in a DCC dominated Universe is studied in detail showing that when DCC is minimally coupled to gravity no dramatic instability is present. In contrast to what happens in a cosmological constant dominated Universe, the non-relativistic matter contrast is no longer constant and exhibits an oscillator behaviour at small scales while it grows weakly at large scales. In the gravitational waves sector, at small scales, the amplitude is still suppressed as the inverse power of the scale factor while it grows logarithmically at large scales. Also the vector modes propagate, though no growing mode is found.

This paper develops a policy learning method for tuning a pre-trained policy to adapt to additional tasks without altering the original task. A method named Adaptive Policy Gradient (APG) is proposed in this paper, which combines Bellman's principle of optimality with the policy gradient approach to improve the convergence rate. This paper provides theoretical analysis which guarantees the convergence rate and sample complexity of O(1/T) and O(1/epsilon), respectively, where T denotes the number of iterations and epsilon denotes the accuracy of the resulting stationary policy. Furthermore, several challenging numerical simulations, including cartpole, lunar lander, and robot arm, are provided to show that APG obtains similar performance compared to existing deterministic policy gradient methods while utilizing much less data and converging at a faster rate.

The gradients of convex functions are expressive models of non-trivial vector fields. For example, Brenier's theorem yields that the optimal transport map between any two measures on Euclidean space under the squared distance is realized as a convex gradient, which is a key insight used in recent generative flow models. In this paper, we study how to model convex gradients by integrating a Jacobian-vector product parameterized by a neural network, which we call the Input Convex Gradient Network (ICGN). We theoretically study ICGNs and compare them to taking the gradient of an Input-Convex Neural Network (ICNN), empirically demonstrating that a single layer ICGN can fit a toy example better than a single layer ICNN. Lastly, we explore extensions to deeper networks and connections to constructions from Riemannian geometry.

It is well-known that inverse dynamics models can improve tracking performance in robot control. These models need to precisely capture the robot dynamics, which consist of well-understood components, e.g., rigid body dynamics, and effects that remain challenging to capture, e.g., stick-slip friction and mechanical flexibilities. Such effects exhibit hysteresis and partial observability, rendering them, particularly challenging to model. Hence, hybrid models, which combine a physical prior with data-driven approaches are especially well-suited in this setting. We present a novel hybrid model formulation that enables us to identify fully physically consistent inertial parameters of a rigid body dynamics model which is paired with a recurrent neural network architecture, allowing us to capture unmodeled partially observable effects using the network memory. We compare our approach against state-of-the-art inverse dynamics models on a 7 degree of freedom manipulator. Using data sets obtained through an optimal experiment design approach, we study the accuracy of offline torque prediction and generalization capabilities of joint learning methods. In control experiments on the real system, we evaluate the model as a feed-forward term for impedance control and show the feedback gains can be drastically reduced to achieve a given tracking accuracy.

This paper introduces an innovative approach to analyzing and improving multi-hop reasoning in AI systems by drawing inspiration from Hamiltonian mechanics. We propose a novel framework that maps reasoning chains in embedding spaces to Hamiltonian systems, allowing us to leverage powerful analytical tools from classical physics. Our method defines a Hamiltonian function that balances the progression of reasoning (kinetic energy) against the relevance to the question at hand (potential energy). Using this framework, we analyze a large dataset of reasoning chains from a multi-hop question-answering task, revealing intriguing patterns that distinguish valid from invalid reasoning. We show that valid reasoning chains have lower Hamiltonian energy and move in ways that make the best trade-off between getting more information and answering the right question. Furthermore, we demonstrate the application of this framework to steer the creation of more efficient reasoning algorithms within AI systems. Our results not only provide new insights into the nature of valid reasoning but also open up exciting possibilities for physics-inspired approaches to understanding and improving artificial intelligence.

Iterative generative policies, such as diffusion models and flow matching, offer superior expressivity for continuous control but complicate Maximum Entropy Reinforcement Learning because their action log-densities are not directly accessible. To address this, we propose Field Least-Energy Actor-Critic (FLAC), a likelihood-free framework that regulates policy stochasticity by penalizing the kinetic energy of the velocity field. Our key insight is to formulate policy optimization as a Generalized Schrödinger Bridge (GSB) problem relative to a high-entropy reference process (e.g., uniform). Under this view, the maximum-entropy principle emerges naturally as staying close to a high-entropy reference while optimizing return, without requiring explicit action densities. In this framework, kinetic energy serves as a physically grounded proxy for divergence from the reference: minimizing path-space energy bounds the deviation of the induced terminal action distribution. Building on this view, we derive an energy-regularized policy iteration scheme and a practical off-policy algorithm that automatically tunes the kinetic energy via a Lagrangian dual mechanism. Empirically, FLAC achieves superior or comparable performance on high-dimensional benchmarks relative to strong baselines, while avoiding explicit density estimation.

Tagged magnetic resonance imaging (tMRI) has been employed for decades to measure the motion of tissue undergoing deformation. However, registration-based motion estimation from tMRI is difficult due to the periodic patterns in these images, particularly when the motion is large. With a larger motion the registration approach gets trapped in a local optima, leading to motion estimation errors. We introduce a novel "momenta, shooting, and correction" framework for Lagrangian motion estimation in the presence of repetitive patterns and large motion. This framework, grounded in Lie algebra and Lie group principles, accumulates momenta in the tangent vector space and employs exponential mapping in the diffeomorphic space for rapid approximation towards true optima, circumventing local optima. A subsequent correction step ensures convergence to true optima. The results on a 2D synthetic dataset and a real 3D tMRI dataset demonstrate our method's efficiency in estimating accurate, dense, and diffeomorphic 2D/3D motion fields amidst large motion and repetitive patterns.

We present the new public version of the KETJU supermassive black hole (SMBH) dynamics module, as implemented into GADGET-4. KETJU adds a small region around each SMBH where the dynamics of the SMBHs and stellar particles are integrated using an algorithmically regularised integrator instead of the leapfrog integrator with gravitational softening used by GADGET-4. This enables modelling SMBHs as point particles even during close interactions with stellar particles or other SMBHs, effectively removing the spatial resolution limitation caused by gravitational softening. KETJU also includes post-Newtonian corrections, which allows following the dynamics of SMBH binaries to sub-parsec scales and down to tens of Schwarzschild radii. Systems with multiple SMBHs are also supported, with the code also including the leading non-linear cross terms that appear in the post-Newtonian equations for such systems. We present tests of the code showing that it correctly captures, at sufficient mass resolution, the sinking driven by dynamical friction and binary hardening driven by stellar scattering. We also present an example application demonstrating how the code can be applied to study the dynamics of SMBHs in mergers of multiple galaxies and the effect they have on the properties of the surrounding galaxy. We expect that the presented KETJU SMBH dynamics module can also be straightforwardly incorporated into other codes similar to GADGET-4, which would allow coupling small-scale SMBH dynamics to the rich variety of galactic physics models that exist in the literature.

The Information Bottleneck (IB) principle has emerged as a promising approach for enhancing the generalization, robustness, and interpretability of deep neural networks, demonstrating efficacy across image segmentation, document clustering, and semantic communication. Among IB implementations, the IB Lagrangian method, employing Lagrangian multipliers, is widely adopted. While numerous methods for the optimizations of IB Lagrangian based on variational bounds and neural estimators are feasible, their performance is highly dependent on the quality of their design, which is inherently prone to errors. To address this limitation, we introduce Structured IB, a framework for investigating potential structured features. By incorporating auxiliary encoders to extract missing informative features, we generate more informative representations. Our experiments demonstrate superior prediction accuracy and task-relevant information preservation compared to the original IB Lagrangian method, even with reduced network size.

The Muon optimizer has received considerable attention for its strong performance in training large language models, yet the design principle behind its matrix-gradient orthogonalization remains largely elusive. In this paper, we introduce a surrogate model that not only sheds new light on the design of Muon, but more importantly leads to a new optimizer. In the same spirit as the derivation of Newton's method, the surrogate approximates the loss as a quadratic function of the perturbation to a weight matrix W using only three matrices: the gradient G, an output-space curvature matrix H, and the data matrix Z that stacks the layer inputs. By minimizing this surrogate in one step and adopting a certain isotropic assumption on the weights, we obtain the closed-form update rule (up to momentum and weight decay) W leftarrow W - ηcdot msgn(G(ZZ^top)^{-1}), where η is the learning rate and msgn(X)=UV^top if X=USV^top is a compact singular value decomposition. This new optimization method, which we refer to as Newton-Muon, shows that standard Muon can be interpreted as an implicit Newton-type method that neglects the right preconditioning induced by the input second moment. Empirically, on a reproduction of the earliest publicly released Modded-NanoGPT speedrun configuration using Muon for GPT-2 pretraining, Newton-Muon reaches the target validation loss in 6\% fewer iteration steps and reduces wall-clock training time by about 4\%.

Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics-informed neural network (PINN) involving challenging optimization landscapes due to higher-order derivatives. To tackle this issue, we propose an alternative approach using Monte Carlo approaches to estimate the solution to the PDE as a stochastic process for weak supervision during training. Leveraging the Walk-on-Spheres method, we introduce a learning scheme called Walk-on-Spheres Neural Operator (WoS-NO) which uses weak supervision from WoS to train any given neural operator. We propose to amortize the cost of Monte Carlo walks across the distribution of PDE instances using stochastic representations from the WoS algorithm to generate cheap, noisy, estimates of the PDE solution during training. This is formulated into a data-free physics-informed objective where a neural operator is trained to regress against these weak supervisions, allowing the operator to learn a generalized solution map for an entire family of PDEs. This strategy does not require expensive pre-computed datasets, avoids computing higher-order derivatives for loss functions that are memory-intensive and unstable, and demonstrates zero-shot generalization to novel PDE parameters and domains. Experiments show that for the same number of training steps, our method exhibits up to 8.75times improvement in L_2-error compared to standard physics-informed training schemes, up to 6.31times improvement in training speed, and reductions of up to 2.97times in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO

Path-velocity decomposition is an intuitive yet powerful approach to address the complexity of kinodynamic motion planning. The difficult trajectory planning problem is solved in two separate, simpler, steps: first, find a path in the configuration space that satisfies the geometric constraints (path planning), and second, find a time-parameterization of that path satisfying the kinodynamic constraints. A fundamental requirement is that the path found in the first step should be time-parameterizable. Most existing works fulfill this requirement by enforcing quasi-static constraints in the path planning step, resulting in an important loss in completeness. We propose a method that enables path-velocity decomposition to discover truly dynamic motions, i.e. motions that are not quasi-statically executable. At the heart of the proposed method is a new algorithm -- Admissible Velocity Propagation -- which, given a path and an interval of reachable velocities at the beginning of that path, computes exactly and efficiently the interval of all the velocities the system can reach after traversing the path while respecting the system kinodynamic constraints. Combining this algorithm with usual sampling-based planners then gives rise to a family of new trajectory planners that can appropriately handle kinodynamic constraints while retaining the advantages associated with path-velocity decomposition. We demonstrate the efficiency of the proposed method on some difficult kinodynamic planning problems, where, in particular, quasi-static methods are guaranteed to fail.

Stability guarantees are crucial when ensuring a fully autonomous robot does not take undesirable or potentially harmful actions. Unfortunately, global stability guarantees are hard to provide in dynamical systems learned from data, especially when the learned dynamics are governed by neural networks. We propose a novel methodology to learn neural contractive dynamical systems, where our neural architecture ensures contraction, and hence, global stability. To efficiently scale the method to high-dimensional dynamical systems, we develop a variant of the variational autoencoder that learns dynamics in a low-dimensional latent representation space while retaining contractive stability after decoding. We further extend our approach to learning contractive systems on the Lie group of rotations to account for full-pose end-effector dynamic motions. The result is the first highly flexible learning architecture that provides contractive stability guarantees with capability to perform obstacle avoidance. Empirically, we demonstrate that our approach encodes the desired dynamics more accurately than the current state-of-the-art, which provides less strong stability guarantees.

The two-body problem of variational electrodynamics possesses differential-delay equations of motion with state-dependent delays of neutral type and solutions that can have velocity discontinuities on countable sets. From a periodic orbit possessing some mild properties at breaking points, we define a synchronization function in R x R3, which is further used to construct two bounded oscillatory functions vanishing at breaking points and whose first derivatives are continuous and defined everywhere but at breaking points. The oscillatory functions are associated with a PDE identity in H2(R3), and we postulate ordering conditions for the PDE identity to define a Fredholm-Schroedinger operator with an O(1/r2) spin-orbit forcing term belonging to L2(R3). As an application, we introduce the Chemical Principle criterion to select orbits with asymptotically vanishing far-fields and estimate the Bohr radius parameter of the Fredholm-Schroedinger PDE using the boundary-layer of orbits chosen according to the Chemical Principle criterion. Last, working backward, we derive a Chemical Principle-like condition from the ordering conditions.

We propose a neural physics system for real-time, interactive fluid simulations. Traditional physics-based methods, while accurate, are computationally intensive and suffer from latency issues. Recent machine-learning methods reduce computational costs while preserving fidelity; yet most still fail to satisfy the latency constraints for real-time use and lack support for interactive applications. To bridge this gap, we introduce a novel hybrid method that integrates numerical simulation, neural physics, and generative control. Our neural physics jointly pursues low-latency simulation and high physical fidelity by employing a fallback safeguard to classical numerical solvers. Furthermore, we develop a diffusion-based controller that is trained using a reverse modeling strategy to generate external dynamic force fields for fluid manipulation. Our system demonstrates robust performance across diverse 2D/3D scenarios, material types, and obstacle interactions, achieving real-time simulations at high frame rates (11~29% latency) while enabling fluid control guided by user-friendly freehand sketches. We present a significant step towards practical, controllable, and physically plausible fluid simulations for real-time interactive applications. We promise to release both models and data upon acceptance.

We present Dojo, a differentiable physics engine for robotics that prioritizes stable simulation, accurate contact physics, and differentiability with respect to states, actions, and system parameters. Dojo models hard contact and friction with a nonlinear complementarity problem with second-order cone constraints. We introduce a custom primal-dual interior-point method to solve the second order cone program for stable forward simulation over a broad range of sample rates. We obtain smooth gradient approximations with this solver through the implicit function theorem, giving gradients that are useful for downstream trajectory optimization, policy optimization, and system identification applications. Specifically, we propose to use the central path parameter threshold in the interior point solver as a user-tunable design parameter. A high value gives a smooth approximation to contact dynamics with smooth gradients for optimization and learning, while a low value gives precise simulation rollouts with hard contact. We demonstrate Dojo's differentiability in trajectory optimization, policy learning, and system identification examples. We also benchmark Dojo against MuJoCo, PyBullet, Drake, and Brax on a variety of robot models, and study the stability and simulation quality over a range of sample frequencies and accuracy tolerances. Finally, we evaluate the sim-to-real gap in hardware experiments with a Ufactory xArm 6 robot. Dojo is an open source project implemented in Julia with Python bindings, with code available at https://github.com/dojo-sim/Dojo.jl.

Offline goal-conditioned reinforcement learning (GCRL) learns goal-conditioned policies from static pre-collected datasets. However, accurate value estimation remains a challenge due to the limited coverage of the state-action space. Recent physics-informed approaches have sought to address this by imposing physical and geometric constraints on the value function through regularization defined over first-order partial differential equations (PDEs), such as the Eikonal equation. However, these formulations can often be ill-posed in complex, high-dimensional environments. In this work, we propose a physics-informed regularization derived from the viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation. By providing a physics-based inductive bias, our approach grounds the learning process in optimal control theory, explicitly regularizing and bounding updates during value iterations. Furthermore, we leverage the Feynman-Kac theorem to recast the PDE solution as an expectation, enabling a tractable Monte Carlo estimation of the objective that avoids numerical instability in higher-order gradients. Experiments demonstrate that our method improves geometric consistency, making it broadly applicable to navigation and high-dimensional, complex manipulation tasks. Open-source codes are available at https://github.com/HrishikeshVish/phys-fk-value-GCRL.

Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While existing approaches focus primarily on learning simulations from the target PDE, they often overlook more fundamental physical principles underlying these equations. Inspired by how numerical solvers are compatible with simulations of different settings of PDEs, we propose a multiphysics training framework that jointly learns from both the original PDEs and their simplified basic forms. Our framework enhances data efficiency, reduces predictive errors, and improves out-of-distribution (OOD) generalization, particularly in scenarios involving shifts of physical parameters and synthetic-to-real transfer. Our method is architecture-agnostic and demonstrates consistent improvements in normalized root mean square error (nRMSE) across a wide range of 1D/2D/3D PDE problems. Through extensive experiments, we show that explicit incorporation of fundamental physics knowledge significantly strengthens the generalization ability of neural operators. We will release models and codes at https://sites.google.com/view/sciml-fundemental-pde.

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for the ability of these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, and neural network theory, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new algorithm, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep neural network training, and provide preliminary numerical evidence for its superior performance.

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.

We study a version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance varepsilon, using tools from variational analysis. In particular, we describe necessary conditions associated with the optimal classifier subject to such an adversary. Using the necessary conditions, we derive a geometric evolution equation which can be used to track the change in classification boundaries as varepsilon varies. This evolution equation may be described as an uncoupled system of differential equations in one dimension, or as a mean curvature type equation in higher dimension. In one dimension, and under mild assumptions on the data distribution, we rigorously prove that one can use the initial value problem starting from varepsilon=0, which is simply the Bayes classifier, in order to solve for the global minimizer of the adversarial problem for small values of varepsilon. In higher dimensions we provide a similar result, albeit conditional to the existence of regular solutions of the initial value problem. In the process of proving our main results we obtain a result of independent interest connecting the original adversarial problem with an optimal transport problem under no assumptions on whether classes are balanced or not. Numerical examples illustrating these ideas are also presented.

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

We present a flow-based approach to the optimal transport (OT) problem between two continuous distributions pi_0,pi_1 on R^d, of minimizing a transport cost E[c(X_1-X_0)] in the set of couplings (X_0,X_1) whose marginal distributions on X_0,X_1 equals pi_0,pi_1, respectively, where c is a cost function. Our method iteratively constructs a sequence of neural ordinary differentiable equations (ODE), each learned by solving a simple unconstrained regression problem, which monotonically reduce the transport cost while automatically preserving the marginal constraints. This yields a monotonic interior approach that traverses inside the set of valid couplings to decrease the transport cost, which distinguishes itself from most existing approaches that enforce the coupling constraints from the outside. The main idea of the method draws from rectified flow, a recent approach that simultaneously decreases the whole family of transport costs induced by convex functions c (and is hence multi-objective in nature), but is not tailored to minimize a specific transport cost. Our method is a single-object variant of rectified flow that guarantees to solve the OT problem for a fixed, user-specified convex cost function c.

We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an epsilon-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.