Daily Papers

While large language models (LLMs) have enabled learning knowledge from the pre-training corpora, the acquired knowledge may be fundamentally incorrect or outdated over time, which necessitates rectifying the knowledge of the language model (LM) after the training. A promising approach involves employing a hyper-network to generate parameter shift, whereas existing hyper-networks suffer from inferior scalability in synchronous editing operation amount. To mitigate the problem, we propose the MAssive Language Model Editing Network (MALMEN), which formulates the parameter shift aggregation as the least square problem, subsequently updating the LM parameters using the normal equation. To accommodate editing multiple facts simultaneously with limited memory budgets, we separate the computation on the hyper-network and LM, enabling arbitrary batch size on both neural networks. Our method is evaluated by editing up to thousands of facts on LMs with different architectures, i.e., BERT-base, GPT-2, T5-XL (2.8B), and GPT-J (6B), across various knowledge-intensive NLP tasks, i.e., closed book fact-checking and question answering. Remarkably, MALMEN is capable of editing hundreds of times more facts than strong baselines with the identical hyper-network architecture and outperforms editor specifically designed for GPT. Our code is available at https://github.com/ChenmienTan/malmen.

According to the generally accepted phase diagram of QCD, at low temperature and high baryon number density the chiral phase transition of QCD is of first order and the co-existence of the Nambu-Goldstone phase and the Wigner phase should appear. This is in conflict with the usual claim that the quark gap equation has no Wigner solution in the case of nonzero current quark mass. In this paper we analyze the reason why the Wigner solution does not exist in the usual treatment and try to propose a new approach to discuss this question. As a first step, we adopt a modified Nambu-Jona-Lasinio (NJL) model to study the Wigner solution at finite current quark mass. We then generalize this approach to the case of finite chemical potential and discuss partial restoration of chiral symmetry at finite chemical potential and compare our results with those in the normal NJL model.

We address effects of spin-orbit coupling (SOC), phenomenologically added to a two-component Bose-Einstein condensate composed of particles moving by Levy flights, in one- and two-dimensional (1D and 2D) settings. The corresponding system of coupled Gross-Pitaevskii equations includes fractional kinetic-energy operators, characterized by the Levy index, \alpha < 2 (the normal kinetic energy corresponds to \alpha = 2). The SOC terms, with strength \lambda, produce strong effects in the 2D case: they create families of stable solitons of the semi-vortex (SV) and mixed-mode (MM) types in the interval of 1 < \alpha < 2, where the supercritical collapse does not admit the existence of stable solitons in the absence of the SOC. At \lambda --> 0, amplitudes of these solitons vanish as (\lambda)^{1/(\alpha - 1)}.

Gradient descent has proven to be a powerful and effective technique for optimization in numerous machine learning applications. Recent advances in computational neuroscience have shown that learning in standard gradient descent optimization formulation is not consistent with learning in biological systems. This has opened up interesting avenues for building biologically inspired learning techniques. One such approach is inspired by Dale's law, which states that inhibitory and excitatory synapses do not swap roles during the course of learning. The resulting exponential gradient descent optimization scheme leads to log-normally distributed synaptic weights. Interestingly, the density that satisfies the Fokker-Planck equation corresponding to the stochastic differential equation (SDE) with geometric Brownian motion (GBM) is the log-normal density. Leveraging this connection, we start with the SDE governing geometric Brownian motion, and show that discretizing the corresponding reverse-time SDE yields a multiplicative update rule, which surprisingly, coincides with the sampling equivalent of the exponential gradient descent update founded on Dale's law. Furthermore, we propose a new formalism for multiplicative denoising score-matching, subsuming the loss function proposed by Hyvaerinen for non-negative data. Indeed, log-normally distributed data is positive and the proposed score-matching formalism turns out to be a natural fit. This allows for training of score-based models for image data and results in a novel multiplicative update scheme for sample generation starting from a log-normal density. Experimental results on MNIST, Fashion MNIST, and Kuzushiji datasets demonstrate generative capability of the new scheme. To the best of our knowledge, this is the first instance of a biologically inspired generative model employing multiplicative updates, founded on geometric Brownian motion.

Since their discovery in the late 1960's the population of known neutron stars (NSs) has grown to ~2500. The last five decades of observations have yielded many surprises and demonstrated that the observational properties of NSs are remarkably diverse. The surveys that will be performed with SKA (the Square Kilometre Array) will produce a further tenfold increase in the number of Galactic NSs known. Moreover, the SKA's broad spectral coverage, sub-arraying and multi-beaming capabilities will allow us to characterise these sources with unprecedented efficiency, in turn enabling a giant leap in the understanding of their properties. Here we review the NS population and outline our strategies for studying each of the growing number of diverse classes that are populating the "NS zoo". Some of the main scientific questions that will be addressed by the much larger statistical samples and vastly improved timing efficiency provided by SKA include: (i) the spin period and spin-down rate distributions (and thus magnetic fields) at birth, and the associated information about the SNe wherein they are formed; (ii) the radio pulsar-magnetar connection; (iii) the link between normal radio pulsars, intermittent pulsars and rotating radio transients; (iv) the slowest possible spin period for a radio pulsar (revealing the conditions at the pulsar death-line); (v) proper motions of pulsars (revealing SN kick physics); (vi) the mass distribution of NSs (vii) the fastest possible spin period for a recycled pulsar (constraining magnetosphere-accretion disc interactions, gravitational wave radiation and the equation-of-state); (viii) the origin of high eccentricity millisecond pulsars (MSPs); (ix) the formation channels for recently identified triple systems; and finally (x) how isolated MSPs are formed. We expect that the SKA will break new ground unveiling exotic systems that will challenge... [abridged]

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

Mathematical equations have been unreasonably effective in describing complex natural phenomena across various scientific disciplines. However, discovering such insightful equations from data presents significant challenges due to the necessity of navigating extremely high-dimensional combinatorial and nonlinear hypothesis spaces. Traditional methods of equation discovery largely focus on extracting equations from data alone, often neglecting the rich domain-specific prior knowledge that scientists typically depend on. To bridge this gap, we introduce LLM-SR, a novel approach that leverages the extensive scientific knowledge and robust code generation capabilities of Large Language Models (LLMs) to discover scientific equations from data in an efficient manner. Specifically, LLM-SR treats equations as programs with mathematical operators and combines LLMs' scientific priors with evolutionary search over equation programs. The LLM iteratively proposes new equation skeletons, drawing from its physical understanding, which are then optimized against data to estimate skeleton parameters. We demonstrate LLM-SR's effectiveness across three diverse scientific domains, where it discovers physically accurate equations that provide significantly better fits to in-domain and out-of-domain data compared to the well-established equation discovery baselines

A common statistical task lies in showing asymptotic normality of certain statistics. In many of these situations, classical textbook results on weak convergence theory suffice for the problem at hand. However, there are quite some scenarios where stronger results are needed in order to establish an asymptotic normal approximation uniformly over a family of probability measures. In this note we collect some results in this direction. We restrict ourselves to weak convergence in mathbb R^d with continuous limit measures.

Recent text-to-3D methods employing diffusion models have made significant advancements in 3D human generation. However, these approaches face challenges due to the limitations of the text-to-image diffusion model, which lacks an understanding of 3D structures. Consequently, these methods struggle to achieve high-quality human generation, resulting in smooth geometry and cartoon-like appearances. In this paper, we observed that fine-tuning text-to-image diffusion models with normal maps enables their adaptation into text-to-normal diffusion models, which enhances the 2D perception of 3D geometry while preserving the priors learned from large-scale datasets. Therefore, we propose HumanNorm, a novel approach for high-quality and realistic 3D human generation by learning the normal diffusion model including a normal-adapted diffusion model and a normal-aligned diffusion model. The normal-adapted diffusion model can generate high-fidelity normal maps corresponding to prompts with view-dependent text. The normal-aligned diffusion model learns to generate color images aligned with the normal maps, thereby transforming physical geometry details into realistic appearance. Leveraging the proposed normal diffusion model, we devise a progressive geometry generation strategy and coarse-to-fine texture generation strategy to enhance the efficiency and robustness of 3D human generation. Comprehensive experiments substantiate our method's ability to generate 3D humans with intricate geometry and realistic appearances, significantly outperforming existing text-to-3D methods in both geometry and texture quality. The project page of HumanNorm is https://humannorm.github.io/.

Normalizing flows (NF) are a class of powerful generative models that have gained popularity in recent years due to their ability to model complex distributions with high flexibility and expressiveness. In this work, we introduce a new type of normalizing flow that is tailored for modeling positions and orientations of multiple objects in three-dimensional space, such as molecules in a crystal. Our approach is based on two key ideas: first, we define smooth and expressive flows on the group of unit quaternions, which allows us to capture the continuous rotational motion of rigid bodies; second, we use the double cover property of unit quaternions to define a proper density on the rotation group. This ensures that our model can be trained using standard likelihood-based methods or variational inference with respect to a thermodynamic target density. We evaluate the method by training Boltzmann generators for two molecular examples, namely the multi-modal density of a tetrahedral system in an external field and the ice XI phase in the TIP4P water model. Our flows can be combined with flows operating on the internal degrees of freedom of molecules and constitute an important step towards the modeling of distributions of many interacting molecules.

Monocular normal estimation for transparent objects is critical for laboratory automation, yet it remains challenging due to complex light refraction and reflection. These optical properties often lead to catastrophic failures in conventional depth and normal sensors, hindering the deployment of embodied AI in scientific environments. We propose TransNormal, a novel framework that adapts pre-trained diffusion priors for single-step normal regression. To handle the lack of texture in transparent surfaces, TransNormal integrates dense visual semantics from DINOv3 via a cross-attention mechanism, providing strong geometric cues. Furthermore, we employ a multi-task learning objective and wavelet-based regularization to ensure the preservation of fine-grained structural details. To support this task, we introduce TransNormal-Synthetic, a physics-based dataset with high-fidelity normal maps for transparent labware. Extensive experiments demonstrate that TransNormal significantly outperforms state-of-the-art methods: on the ClearGrasp benchmark, it reduces mean error by 24.4% and improves 11.25° accuracy by 22.8%; on ClearPose, it achieves a 15.2% reduction in mean error. The code and dataset will be made publicly available at https://longxiang-ai.github.io/TransNormal.

S. L. Woronowicz's theory of introducing C*-algebras generated by unbounded elements is applied to q-normal operators satisfying the defining relation of the quantum complex plane. The unique non-degenerate C*-algebra of bounded operators generated by a q-normal operator is computed and an abstract description is given by using crossed product algebras. If the spectrum of the modulus of the q-normal operator is the positive half line, this C*-algebra will be considered as the algebra of continuous functions on the quantum complex plane vanishing at infinity, and its unitization will be viewed as the algebra of continuous functions on a quantum 2-sphere.

Straight line equation y=mx with slope m, when singularly perturbed as ay^3+y=mx with a positive parameter a, results in S-shaped curves or S-curves on a real plane. As arightarrow 0, we get back y=mx which is a cumulative distribution function of a continuous uniform distribution that describes the occurrence of every event in an interval to be equally probable. As arightarrowinfty, the derivative of y has finite support only at y=0 resembling a degenerate distribution. Based on these arguments, in this work, we propose that these S-curves can represent maximum entropy uniform distribution to a zero entropy single value. We also argue that these S-curves are superposable as they are only parametrically nonlinear but fundamentally linear. So far, the superposed forms have been used to capture the patterns of natural systems such as nonlinear dynamics of biological growth and kinetics of enzyme reactions. Here, we attempt to use the S-curve and its superposed form as statistical models. We fit the models on a classical dataset containing flower measurements of iris plants and analyze their usefulness in pattern recognition. Based on these models, we claim that any non-uniform pattern can be represented as a singular perturbation to uniform distribution. However, our parametric estimation procedure have some limitations such as sensitivity to initial conditions depending on the data at hand.

Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies to compute forces and higher-order derivatives. Furthermore, such densities are often defined on non-trivial topologies. A recent example are Boltzmann Generators for generating 3D-structures of peptides and small proteins. These generative models leverage the space of internal coordinates (dihedrals, angles, and bonds), which is a product of hypertori and compact intervals. In this work, we introduce a class of smooth mixture transformations working on both compact intervals and hypertori. Mixture transformations employ root-finding methods to invert them in practice, which has so far prevented bi-directional flow training. To this end, we show that parameter gradients and forces of such inverses can be computed from forward evaluations via the inverse function theorem. We demonstrate two advantages of such smooth flows: they allow training by force matching to simulation data and can be used as potentials in molecular dynamics simulations.

Layer Normalization (LayerNorm) is an inherent component in all Transformer-based models. In this paper, we show that LayerNorm is crucial to the expressivity of the multi-head attention layer that follows it. This is in contrast to the common belief that LayerNorm's only role is to normalize the activations during the forward pass, and their gradients during the backward pass. We consider a geometric interpretation of LayerNorm and show that it consists of two components: (a) projection of the input vectors to a d-1 space that is orthogonal to the left[1,1,...,1right] vector, and (b) scaling of all vectors to the same norm of d. We show that each of these components is important for the attention layer that follows it in Transformers: (a) projection allows the attention mechanism to create an attention query that attends to all keys equally, offloading the need to learn this operation by the attention; and (b) scaling allows each key to potentially receive the highest attention, and prevents keys from being "un-select-able". We show empirically that Transformers do indeed benefit from these properties of LayeNorm in general language modeling and even in computing simple functions such as "majority". Our code is available at https://github.com/tech-srl/layer_norm_expressivity_role .

Inferring unbiased treatment effects has received widespread attention in the machine learning community. In recent years, our community has proposed numerous solutions in standard settings, high-dimensional treatment settings, and even longitudinal settings. While very diverse, the solution has mostly relied on neural networks for inference and simultaneous correction of assignment bias. New approaches typically build on top of previous approaches by proposing new (or refined) architectures and learning algorithms. However, the end result -- a neural-network-based inference machine -- remains unchallenged. In this paper, we introduce a different type of solution in the longitudinal setting: a closed-form ordinary differential equation (ODE). While we still rely on continuous optimization to learn an ODE, the resulting inference machine is no longer a neural network. Doing so yields several advantages such as interpretability, irregular sampling, and a different set of identification assumptions. Above all, we consider the introduction of a completely new type of solution to be our most important contribution as it may spark entirely new innovations in treatment effects in general. We facilitate this by formulating our contribution as a framework that can transform any ODE discovery method into a treatment effects method.

Approximating nonlinear differential equations using a neural network provides a robust and efficient tool for various scientific computing tasks, including real-time predictions, inverse problems, optimal controls, and surrogate modeling. Previous works have focused on embedding dynamical systems into networks through two approaches: learning a single solution operator (i.e., the mapping from input parametrized functions to solutions) or learning the governing system of equations (i.e., the constitutive model relative to the state variables). Both of these approaches yield different representations for the same underlying data or function. Additionally, observing that families of differential equations often share key characteristics, we seek one network representation across a wide range of equations. Our method, called Predicting Operators and Symbolic Expressions (PROSE), learns maps from multimodal inputs to multimodal outputs, capable of generating both numerical predictions and mathematical equations. By using a transformer structure and a feature fusion approach, our network can simultaneously embed sets of solution operators for various parametric differential equations using a single trained network. Detailed experiments demonstrate that the network benefits from its multimodal nature, resulting in improved prediction accuracy and better generalization. The network is shown to be able to handle noise in the data and errors in the symbolic representation, including noisy numerical values, model misspecification, and erroneous addition or deletion of terms. PROSE provides a new neural network framework for differential equations which allows for more flexibility and generality in learning operators and governing equations from data.

Recovering a 3D surface from its surface normal map, a problem known as normal integration, is a key component for photometric shape reconstruction techniques such as shape-from-shading and photometric stereo. The vast majority of existing approaches for normal integration handle only implicitly the presence of depth discontinuities and are limited to orthographic or ideal pinhole cameras. In this paper, we propose a novel formulation that allows modeling discontinuities explicitly and handling generic central cameras. Our key idea is based on a local planarity assumption, that we model through constraints between surface normals and ray directions. Compared to existing methods, our approach more accurately approximates the relation between depth and surface normals, achieves state-of-the-art results on the standard normal integration benchmark, and is the first to directly handle generic central camera models.

Continuous normalizing flows (CNFs) construct invertible mappings between an arbitrary complex distribution and an isotropic Gaussian distribution using Neural Ordinary Differential Equations (neural ODEs). It has not been tractable on large datasets due to the incremental complexity of the neural ODE training. Optimal Transport theory has been applied to regularize the dynamics of the ODE to speed up training in recent works. In this paper, a temporal optimization is proposed by optimizing the evolutionary time for forward propagation of the neural ODE training. In this appoach, we optimize the network weights of the CNF alternately with evolutionary time by coordinate descent. Further with temporal regularization, stability of the evolution is ensured. This approach can be used in conjunction with the original regularization approach. We have experimentally demonstrated that the proposed approach can significantly accelerate training without sacrifying performance over baseline models.

The paper shows that the application of the fixed-effect multiple linear regression model to an overparameterized dataset is equivalent to fitting the data with a hyper-curve parameterized by a single scalar parameter. This equivalence allows for a predictor-focused approach, where each predictor is described by a function of the chosen parameter. It is proven that a linear model will produce exact predictions even in the presence of nonlinear dependencies that violate the model assumptions. Parameterization in terms of the dependent variable and the monomial basis in the predictor function space are applied here to both synthetic and experimental data. The hyper-curve approach is especially suited for the regularization of problems with noise in predictor variables and can be used to remove noisy and "improper" predictors from the model.

One of the most common and universal problems in science is to investigate a function. The prediction can be made by an Artificial Neural Network (ANN) or a mathematical model. Both approaches have their advantages and disadvantages. Mathematical models were sought as more trustworthy as their prediction is based on the laws of physics expressed in the form of mathematical equations. However, the majority of existing mathematical models include different empirical parameters, and both approaches inherit inevitable experimental errors. At the same time, the approximation of neural networks can reproduce the solution extremely well if fed with a sufficient amount of data. The difference is that an ANN requires big data to build its accurate approximation whereas a typical mathematical model needs just several data points to estimate an empirical constant. Therefore, the common problem that developer meet is the inaccuracy of mathematical models and artificial neural network. An another common challenge is the computational complexity of the mathematical models, or lack of data for a sufficient precision of the Artificial Neural Networks. In the presented paper those problems are addressed using the integration of a mathematical model with an artificial neural network. In the presented analysis, an ANN predicts just a part of the mathematical model and its weights and biases are adjusted based on the output of the mathematical model. The performance of Integrated Mathematical modeling - Artificial Neural Network (IMANN) is compared to a Dense Neural Network (DNN) with the use of the benchmarking functions. The obtained calculation results indicate that such an approach could lead to an increase of precision as well as limiting the data-set required for learning.

Normalization layer constitutes an essential component in neural networks. In transformers, the predominantly used RMSNorm constrains vectors to a unit hypersphere, followed by dimension-wise rescaling through a learnable scaling coefficient γ to maintain the representational capacity of the model. However, RMSNorm discards the input norm information in forward pass and a static scaling factor γ may be insufficient to accommodate the wide variability of input data and distributional shifts, thereby limiting further performance improvements, particularly in zero-shot scenarios that large language models routinely encounter. To address this limitation, we propose SeeDNorm, which enhances the representational capability of the model by dynamically adjusting the scaling coefficient based on the current input, thereby preserving the input norm information and enabling data-dependent, self-rescaled dynamic normalization. During backpropagation, SeeDNorm retains the ability of RMSNorm to dynamically adjust gradient according to the input norm. We provide a detailed analysis of the training optimization for SeedNorm and proposed corresponding solutions to address potential instability issues that may arise when applying SeeDNorm. We validate the effectiveness of SeeDNorm across models of varying sizes in large language model pre-training as well as supervised and unsupervised computer vision tasks. By introducing a minimal number of parameters and with neglligible impact on model efficiency, SeeDNorm achieves consistently superior performance compared to previously commonly used normalization layers such as RMSNorm and LayerNorm, as well as element-wise activation alternatives to normalization layers like DyT.

Monocular normal estimation aims to estimate the normal map from a single RGB image of an object under arbitrary lights. Existing methods rely on deep models to directly predict normal maps. However, they often suffer from 3D misalignment: while the estimated normal maps may appear to have a correct appearance, the reconstructed surfaces often fail to align with the geometric details. We argue that this misalignment stems from the current paradigm: the model struggles to distinguish and reconstruct varying geometry represented in normal maps, as the differences in underlying geometry are reflected only through relatively subtle color variations. To address this issue, we propose a new paradigm that reformulates normal estimation as shading sequence estimation, where shading sequences are more sensitive to various geometric information. Building on this paradigm, we present RoSE, a method that leverages image-to-video generative models to predict shading sequences. The predicted shading sequences are then converted into normal maps by solving a simple ordinary least-squares problem. To enhance robustness and better handle complex objects, RoSE is trained on a synthetic dataset, MultiShade, with diverse shapes, materials, and light conditions. Experiments demonstrate that RoSE achieves state-of-the-art performance on real-world benchmark datasets for object-based monocular normal estimation.

In this work, we generalize the reaction-diffusion equation in statistical physics, Schr\"odinger equation in quantum mechanics, Helmholtz equation in paraxial optics into the neural partial differential equations (NPDE), which can be considered as the fundamental equations in the field of artificial intelligence research. We take finite difference method to discretize NPDE for finding numerical solution, and the basic building blocks of deep neural network architecture, including multi-layer perceptron, convolutional neural network and recurrent neural networks, are generated. The learning strategies, such as Adaptive moment estimation, L-BFGS, pseudoinverse learning algorithms and partial differential equation constrained optimization, are also presented. We believe it is of significance that presented clear physical image of interpretable deep neural networks, which makes it be possible for applying to analog computing device design, and pave the road to physical artificial intelligence.

While the use of combination therapy is increasing in prevalence for cancer treatment, it is often difficult to predict the exact interactions between different treatment forms, and their synergistic/antagonistic effects on patient health and therapy outcome. In this research, a system of ordinary differential equations is constructed to model nonlinear dynamics between tumor cells, immune cells, and three forms of therapy: chemotherapy, immunotherapy, and radiotherapy. This model is then used to generate optimized combination therapy plans using optimal control theory. In-silico experiments are conducted to simulate the response of the patient model to various treatment plans. This is the first mathematical model in current literature to introduce radiotherapy as an option alongside immuno- and chemotherapy, permitting more flexible and effective treatment plans that reflect modern therapeutic approaches.

Postprandial hyperglycemia, marked by the blood glucose level exceeding the normal range after meals, is a critical indicator of progression toward type 2 diabetes in prediabetic and healthy individuals. A key metric for understanding blood glucose dynamics after eating is the postprandial area under the curve (PAUC). Predicting PAUC in advance based on a person's diet and activity level and explaining what affects postprandial blood glucose could allow an individual to adjust their lifestyle accordingly to maintain normal glucose levels. In this paper, we propose GlucoLens, an explainable machine learning approach to predict PAUC and hyperglycemia from diet, activity, and recent glucose patterns. We conducted a five-week user study with 10 full-time working individuals to develop and evaluate the computational model. Our machine learning model takes multimodal data including fasting glucose, recent glucose, recent activity, and macronutrient amounts, and provides an interpretable prediction of the postprandial glucose pattern. Our extensive analyses of the collected data revealed that the trained model achieves a normalized root mean squared error (NRMSE) of 0.123. On average, GlucoLense with a Random Forest backbone provides a 16% better result than the baseline models. Additionally, GlucoLens predicts hyperglycemia with an accuracy of 74% and recommends different options to help avoid hyperglycemia through diverse counterfactual explanations. Code available: https://github.com/ab9mamun/GlucoLens.

Deep Equilibrium Models (DEQs) and Neural Ordinary Differential Equations (Neural ODEs) are two branches of implicit models that have achieved remarkable success owing to their superior performance and low memory consumption. While both are implicit models, DEQs and Neural ODEs are derived from different mathematical formulations. Inspired by homotopy continuation, we establish a connection between these two models and illustrate that they are actually two sides of the same coin. Homotopy continuation is a classical method of solving nonlinear equations based on a corresponding ODE. Given this connection, we proposed a new implicit model called HomoODE that inherits the property of high accuracy from DEQs and the property of stability from Neural ODEs. Unlike DEQs, which explicitly solve an equilibrium-point-finding problem via Newton's methods in the forward pass, HomoODE solves the equilibrium-point-finding problem implicitly using a modified Neural ODE via homotopy continuation. Further, we developed an acceleration method for HomoODE with a shared learnable initial point. It is worth noting that our model also provides a better understanding of why Augmented Neural ODEs work as long as the augmented part is regarded as the equilibrium point to find. Comprehensive experiments with several image classification tasks demonstrate that HomoODE surpasses existing implicit models in terms of both accuracy and memory consumption.