Daily Papers

This paper presents a modeling effort to explore the underlying physics of temperature evolution during additive friction stir deposition (AFSD) by a human-AI teaming approach. AFSD is an emerging solid-state additive manufacturing technology that deposits materials without melting. However, both process modeling and modeling of the AFSD tool are at an early stage. In this paper, a human-AI teaming approach is proposed to combine models based on first principles with AI. The resulting human-informed machine learning method, denoted as AFSD-Physics, can effectively learn the governing equations of temperature evolution at the tool and the build from in-process measurements. Experiments are designed and conducted to collect in-process measurements for the deposition of aluminum 7075 with a total of 30 layers. The acquired governing equations are physically interpretable models with low computational cost and high accuracy. Model predictions show good agreement with the measurements. Experimental validation with new process parameters demonstrates the model's generalizability and potential for use in tool temperature control and process optimization.

In this work, we present a novel physics-based data-driven framework for reduced-order modeling of laser ignition in a model rocket combustor based on parameterized neural ordinary differential equations (PNODE). Deep neural networks are embedded as functions of high-dimensional parameters of laser ignition to predict various terms in a 0D flow model including the heat source function, pre-exponential factors, and activation energy. Using the governing equations of a 0D flow model, our PNODE needs only a limited number of training samples and predicts trajectories of various quantities such as temperature, pressure, and mass fractions of species while satisfying physical constraints. We validate our physics-based PNODE on solution snapshots of high-fidelity Computational Fluid Dynamics (CFD) simulations of laser-induced ignition in a prototype rocket combustor. We compare the performance of our physics-based PNODE with that of kernel ridge regression and fully connected neural networks. Our results show that our physics-based PNODE provides solutions with lower mean absolute errors of average temperature over time, thus improving the prediction of successful laser ignition with high-dimensional parameters.

Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or boundary conditions or different input configurations. This study proposes the Variational Physics-Informed Neural Operator (VINO), a deep learning method designed for solving PDEs by minimizing the energy formulation of PDEs. This framework can be trained without any labeled data, resulting in improved performance and accuracy compared to existing deep learning methods and conventional PDE solvers. By discretizing the domain into elements, the variational format allows VINO to overcome the key challenge in physics-informed neural operators, namely the efficient evaluation of the governing equations for computing the loss. Comparative results demonstrate VINO's superior performance, especially as the mesh resolution increases. As a result, this study suggests a better way to incorporate physical laws into neural operators, opening a new approach for modeling and simulating nonlinear and complex processes in science and engineering.

Discovering governing equations from observational data remains a fundamental challenge in scientific modeling, particularly when the underlying mathematical structure is unknown. Traditional sparse identification methods like SINDy excel at discovering parsimonious models but require researchers to specify candidate basis functions a priori, a limitation that often leads to model failure when critical terms are omitted or when systems exhibit unconventional dynamics. Purely symbolic regression approaches offer unlimited flexibility but struggle with noise sensitivity and frequently produce overly complex, unstable equations. We present AutoSINDy, a hybrid Discovery-then-Solve framework that combines the exploratory power of symbolic regression with the robust sparsity-promoting capabilities of SINDy. Our method operates in three stages: (1) PySR-based symbolic regression discovers candidate functional forms from bootstrapped data chunks; (2) a curation pipeline decomposes, expands, and filters these expressions using collinearity analysis to construct a minimal yet comprehensive library; and (3) SINDy identifies sparse governing equations from this custom-tailored library. Extensive experiments across canonical nonlinear systems demonstrate that AutoSINDy consistently recovers ground-truth equations even under high observational noise, achieving a ground-truth recovery rate of 92.8% across all trials. Compared with standard SINDy using enriched libraries and standalone symbolic regression, AutoSINDy achieves higher predictive accuracy, superior generalization to unseen trajectories, and substantially lower symbolic complexity.

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.

Predicting the microstructural and morphological evolution of materials through phase-field modelling is computationally intensive, particularly for high-throughput parametric studies. While neural operators such as the Fourier neural operator (FNO) show promise in accelerating the solution of parametric partial differential equations (PDEs), the lack of explicit physical constraints, may limit generalisation and long-term accuracy for complex phase-field dynamics. Here, we develop a physics-informed neural operator framework to learn parametric phase-field PDEs, namely PF-PINO. By embedding the residuals of phase-field governing equations into the data-fidelity loss function, our framework effectively enforces physical constraints during training. We validate PF-PINO against benchmark phase-field problems, including electrochemical corrosion, dendritic crystal solidification, and spinodal decomposition. Our results demonstrate that PF-PINO significantly outperforms conventional FNO in accuracy, generalisation capability, and long-term stability. This work provides a robust and efficient computational tool for phase-field modelling and highlights the potential of physics-informed neural operators to advance scientific machine learning for complex interfacial evolution problems.

Many engineering problems involve phenomena whose governing equations are poorly characterized or only partially known. Surrogate modeling techniques such as neural networks can capture the behavior of these systems, but they typically demand large training datasets that are difficult to obtain in engineering contexts and yield models with limited physical interpretability. The Sparse Identification of Nonlinear Dynamics (SINDy) method addresses both limitations by performing sparse regression over libraries of candidate nonlinear terms, recovering interpretable governing equations from comparatively small datasets. Although SINDy has been demonstrated extensively on canonical benchmark systems, its application to practical engineering problems is less widely documented. This tutorial introduces the SINDy method and progressively builds toward its main extensions, from noise-robust weak-form and ensembling-based variants to constrained and parametrizable formulations. The paper and the accompanying tutorial (available at https://github.com/paullililili/SINDy4Engineers) is organized in three parts: the first introduces the standard SINDy algorithm and progressively extends it, inviting readers without prior knowledge to follow each step and adapt the methods to their own problems; the remaining two parts present detailed case studies on (1) the system identification of an unmanned aerial vehicle and (2) a chaotic thermosyphon heat exchanger. Through these examples, we aim to demonstrate that SINDy is simple to implement yet flexible enough to serve as a valuable identification tool for advanced engineering applications.

Given the existence of various forward and inverse problems in combustion studies and applications that necessitate distinct methods for resolution, a framework to solve them in a unified way is critically needed. A promising approach is the integration of machine learning methods with governing equations of combustion systems, which exhibits superior generality and few-shot learning ability compared to purely data-driven methods. In this work, the FlamePINN-1D framework is proposed to solve the forward and inverse problems of 1D laminar flames based on physics-informed neural networks. Three cases with increasing complexity have been tested: Case 1 are freely-propagating premixed (FPP) flames with simplified physical models, while Case 2 and Case 3 are FPP and counterflow premixed (CFP) flames with detailed models, respectively. For forward problems, FlamePINN-1D aims to solve the flame fields and infer the unknown eigenvalues (such as laminar flame speeds) under the constraints of governing equations and boundary conditions. For inverse problems, FlamePINN-1D aims to reconstruct the continuous fields and infer the unknown parameters (such as transport and chemical kinetics parameters) from noisy sparse observations of the flame. Our results strongly validate these capabilities of FlamePINN-1D across various flames and working conditions. Compared to traditional methods, FlamePINN-1D is differentiable and mesh-free, exhibits no discretization errors, and is easier to implement for inverse problems. The inverse problem results also indicate the possibility of optimizing chemical mechanisms from measurements of laboratory 1D flames. Furthermore, some proposed strategies, such as hard constraints and thin-layer normalization, are proven to be essential for the robust learning of FlamePINN-1D. The code for this paper is partially available at https://github.com/CAME-THU/FlamePINN-1D.

Computational Fluid Dynamics (CFD) simulation by the numerical solution of the Navier-Stokes equations is an essential tool in a wide range of applications from engineering design to climate modeling. However, the computational cost and memory demand required by CFD codes may become very high for flows of practical interest, such as in aerodynamic shape optimization. This expense is associated with the complexity of the fluid flow governing equations, which include non-linear partial derivative terms that are of difficult solution, leading to long computational times and limiting the number of hypotheses that can be tested during the process of iterative design. Therefore, we propose DeepCFD: a convolutional neural network (CNN) based model that efficiently approximates solutions for the problem of non-uniform steady laminar flows. The proposed model is able to learn complete solutions of the Navier-Stokes equations, for both velocity and pressure fields, directly from ground-truth data generated using a state-of-the-art CFD code. Using DeepCFD, we found a speedup of up to 3 orders of magnitude compared to the standard CFD approach at a cost of low error rates.

Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly matched to PDE solutions with strongly heterogeneous complexity across space and space--time, leaving capacity insufficient where the physics is difficult and redundant where it is simple. We introduce physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics. We instantiate this principle with Representation Capacity Adaptation for PDEs (ReCAP), a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging. ReCAP uses responsibility-weighted error indicators and the geometry of residual energy to determine where and how to refine. To limit the disturbance introduced by splitting, we introduce quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation. We further establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Across five challenging 3D and 4D PDE benchmarks against 11 physics-informed solvers, ReCAP achieves the lowest relative L^2 error on every problem, reducing error by 10.7%--27.5% relative to the strongest competing result. These results suggest that physics-informed solvers need not merely learn their parameters---they can learn how their representational capacity should be organized.

The accessibility of spatially distributed data, enabled by affordable sensors, field, and numerical experiments, has facilitated the development of data-driven solutions for scientific problems, including climate change, weather prediction, and urban planning. Neural Partial Differential Equations (Neural PDEs), which combine deep learning (DL) techniques with domain expertise (e.g., governing equations) for parameterization, have proven to be effective in capturing valuable correlations within spatiotemporal datasets. However, sparse and noisy measurements coupled with modeling approximation introduce aleatoric and epistemic uncertainties. Therefore, quantifying uncertainties propagated from model inputs to outputs remains a challenge and an essential goal for establishing the trustworthiness of Neural PDEs. This work evaluates various Uncertainty Quantification (UQ) approaches for both Forward and Inverse Problems in scientific applications. Specifically, we investigate the effectiveness of Bayesian methods, such as Hamiltonian Monte Carlo (HMC) and Monte-Carlo Dropout (MCD), and a more conventional approach, Deep Ensembles (DE). To illustrate their performance, we take two canonical PDEs: Burger's equation and the Navier-Stokes equation. Our results indicate that Neural PDEs can effectively reconstruct flow systems and predict the associated unknown parameters. However, it is noteworthy that the results derived from Bayesian methods, based on our observations, tend to display a higher degree of certainty in their predictions as compared to those obtained using the DE. This elevated certainty in predictions suggests that Bayesian techniques might underestimate the true underlying uncertainty, thereby appearing more confident in their predictions than the DE approach.

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.

In this paper, a mathematical model is developed to describe the evolution of the concentration of compounds through a gas chromatography column. The model couples mass balances and kinetic equations for all components. Both single and multiple-component cases are considered with constant or variable velocity. Non-dimensionalisation indicates the small effect of diffusion. The system where diffusion is neglected is analysed using Laplace transforms. In the multiple-component case, it is demonstrated that the competition between the compounds is negligible and the equations may be decoupled. This reduces the problem to solving a single integral equation to determine the concentration profile for all components (since they are scaled versions of each other). For a given analyte, we then only two parameters need to be fitted to the data. To verify this approach, the full governing equations are also solved numerically using the finite difference method and a global adaptive quadrature method to integrate the Laplace transformation. Comparison with the Laplace solution verifies the high degree of accuracy of the simpler Laplace form. The Laplace solution is then verified against experimental data from BTEX chromatography. This novel method, which involves solving a single equation and fitting parameters in pairs for individual components, is highly efficient. It is significantly faster and simpler than the full numerical solution and avoids the computationally expensive methods that would normally be used to fit all curves at the same time.

Unsupervised physical parameter estimation from video lacks a common benchmark: existing methods evaluate on non-overlapping synthetic data, the sole real-world dataset is restricted to single-body systems, and no established protocol addresses governing-equation identification. This work introduces IRIS, a high-fidelity benchmark comprising 240 real-world videos captured at 4K resolution and 60fps, spanning both single- and multi-body dynamics with independently measured ground-truth parameters and uncertainty estimates. Each dynamical system is recorded under controlled laboratory conditions and paired with its governing equations, enabling principled evaluation. A standardized evaluation protocol is defined encompassing parameter accuracy, identifiability, extrapolation, robustness, and governing-equation selection. Multiple baselines are evaluated, including a multi-step physics loss formulation and four complementary equation-identification strategies (VLM temporal reasoning, describe-then-classify prompting, CNN-based classification, and path-based labelling), establishing reference performance across all IRIS scenarios and exposing systematic failure modes that motivate future research. The dataset, annotations, evaluation toolkit, and all baseline implementations are publicly released.

Molecular clouds exhibit complex, multi-scale dynamics driven by the interplay of turbulence, self-gravity, and gas pressure. Modelling these systems with conventional grid-based methods becomes increasingly expensive as spatial resolution and dimensionality increase. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative by incorporating the governing equations directly into the training objective, but their ability to model realistic turbulent self-gravitating flows remains largely unexplored. We apply PINNs to isothermal molecular clouds initialized with astrophysically motivated turbulent velocity fields, considering subsonic and supersonic evolution in two and three dimensions. We benchmark the predicted density and velocity fields against the finite-volume code Athena++. PINNs accurately recover the global evolution of the flow over multiple free-fall times across turbulent realizations, Mach numbers, and dimensionalities. Their training cost also exhibits weak dependence on spatial resolution, increasing by less than 25% from 128^3 to 1024^3 in our experiments. These results demonstrate that PINNs can model substantially more realistic self-gravitating turbulent flows than previously explored and provide a promising complementary framework for studying these systems.

We introduce Poseidon, a foundation model for learning the solution operators of PDEs. It is based on a multiscale operator transformer, with time-conditioned layer norms that enable continuous-in-time evaluations. A novel training strategy leveraging the semi-group property of time-dependent PDEs to allow for significant scaling-up of the training data is also proposed. Poseidon is pretrained on a diverse, large scale dataset for the governing equations of fluid dynamics. It is then evaluated on a suite of 15 challenging downstream tasks that include a wide variety of PDE types and operators. We show that Poseidon exhibits excellent performance across the board by outperforming baselines significantly, both in terms of sample efficiency and accuracy. Poseidon also generalizes very well to new physics that is not seen during pretraining. Moreover, Poseidon scales with respect to model and data size, both for pretraining and for downstream tasks. Taken together, our results showcase the surprising ability of Poseidon to learn effective representations from a very small set of PDEs during pretraining in order to generalize well to unseen and unrelated PDEs downstream, demonstrating its potential as an effective, general purpose PDE foundation model. Finally, the Poseidon model as well as underlying pretraining and downstream datasets are open sourced, with code being available at https://github.com/camlab-ethz/poseidon and pretrained models and datasets at https://huggingface.co/camlab-ethz.

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.

Building models that generalize across physical systems without retraining remains a central challenge in computational science. Here we introduce In-Context Modeling (ICM), a retrain-free paradigm that infers physical relationships directly from observational fields. Rather than encoding system-specific behavior in fixed parameters, ICM assimilates measurements as physical context and performs inference through a single forward pass. Trained in a physics-informed, label-free manner using governing equations, a single model generalizes across unseen materials, geometries, and loading conditions. Demonstrated on hyperelasticity, ICM integrates with finite-element simulations and is validated using experimental full-field measurements. Moreover, performance improves with increasing data diversity and computational budget, exhibiting favorable scaling behavior analogous to foundation models. By recasting physical modeling as in-context inference, this work establishes a transferable paradigm for retrain-free scientific learning and a foundation for scalable modeling across computational science.

Understanding the physical world requires perceptual models grounded in physical laws rather than mere statistical correlations. However, existing multimodal learning frameworks, focused on vision and language, lack physical consistency and overlook the intrinsic causal relationships among an object's geometry, material, vibration modes, and the sounds it produces. We introduce VibraVerse, a large-scale geometry-acoustics alignment dataset that explicitly bridges the causal chain from 3D geometry -> physical attributes -> modal parameters -> acoustic signals. Each 3D model has explicit physical properties (density, Young's modulus, Poisson's ratio) and volumetric geometry, from which modal eigenfrequencies and eigenvectors are computed for impact sound synthesis under controlled excitations. To establish this coherence, we introduce CLASP, a contrastive learning framework for cross-modal alignment that preserves the causal correspondence between an object's physical structure and its acoustic response. This framework enforces physically consistent alignment across modalities, ensuring that every sample is coherent, traceable to the governing equations, and embedded within a unified representation space spanning shape, image, and sound. Built upon VibraVerse, we define a suite of benchmark tasks for geometry-to-sound prediction, sound-guided shape reconstruction, and cross-modal representation learning. Extensive validations on these tasks demonstrate that models trained on VibraVerse exhibit superior accuracy, interpretability, and generalization across modalities. These results establish VibraVerse as a benchmark for physically consistent and causally interpretable multimodal learning, providing a foundation for sound-guided embodied perception and a deeper understanding of the physical world. The dataset will be open-sourced.

Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

Modern power systems require fast and accurate dynamic simulations for stability assessment, digital twins, and real-time control, but classical ODE solvers are often too slow for large-scale or online applications. We propose a neural-operator framework for surrogate modeling of power system components, using Deep Operator Networks (DeepONets) to learn mappings from system states and time-varying inputs to full trajectories without step-by-step integration. To enhance generalization and data efficiency, we introduce Physics-Informed DeepONets (PI-DeepONets), which embed the residuals of governing equations into the training loss. Our results show that DeepONets, and especially PI-DeepONets, achieve accurate predictions under diverse scenarios, providing over 30 times speedup compared to high-order ODE solvers. Benchmarking against Physics-Informed Neural Networks (PINNs) highlights superior stability and scalability. Our results demonstrate neural operators as a promising path toward real-time, physics-aware simulation of power system dynamics.

Recovering governing Ordinary Differential Equations (ODEs) from data is a central challenge in modeling dynamical systems across scientific domains. Existing approaches cast discovery as a static inference problem over fixed datasets, assuming that the observed trajectories are sufficiently informative. However, dynamical systems evolve over large state spaces, and limited data can make multiple equations observationally indistinguishable, leading to identifiability gaps and the recovery of incorrect governing equations. To address this, we introduce LLM-ACES, or LLM-guided Active Closed-loop Equation Search, a closed-loop framework that jointly optimizes symbolic hypothesis construction and adaptive data acquisition. In LLM-ACES, a large language model (LLM) proposes operator priors that partition the large search space into distinct regions, within which candidate equations are fit to the observed data. The disagreement among these candidates guides the acquisition of informative trajectories, creating a feedback loop that iteratively refines both the hypothesis space and the discovered dynamics. On 122 ODE systems spanning ODEBench and ODEBase, LLM-ACES achieves the lowest median NMSE, outperforming state-of-the-art baselines by several orders of magnitude while achieving a high symbolic accuracy of 46.2% and 52.4%, respectively. Our analysis further shows that LLM-ACES is sample-efficient, achieving better performance with one-tenth the data. Furthermore, LLM-ACES's feedback-driven data acquisition makes it robust to noise and recovers the correct symbolic structure, while baselines introduce spurious terms that fit the data locally but obscure the true governing relationships.

Bayesian parameter inference is useful to improve Li-ion battery diagnostics and can help formulate battery aging models. However, it is computationally intensive and cannot be easily repeated for multiple cycles, multiple operating conditions, or multiple replicate cells. To reduce the computational cost of Bayesian calibration, numerical solvers for physics-based models can be replaced with faster surrogates. A physics-informed neural network (PINN) is developed as a surrogate for the pseudo-2D (P2D) battery model calibration. For the P2D surrogate, additional training regularization was needed as compared to the PINN single-particle model (SPM) developed in Part I. Both the PINN SPM and P2D surrogate models are exercised for parameter inference and compared to data obtained from a direct numerical solution of the governing equations. A parameter inference study highlights the ability to use these PINNs to calibrate scaling parameters for the cathode Li diffusion and the anode exchange current density. By realizing computational speed-ups of 2250x for the P2D model, as compared to using standard integrating methods, the PINN surrogates enable rapid state-of-health diagnostics. In the low-data availability scenario, the testing error was estimated to 2mV for the SPM surrogate and 10mV for the P2D surrogate which could be mitigated with additional data.

Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied data distribution are expected to adhere to specific governing equations. We present a framework that unifies generative modeling and partial differential equation fulfillment by introducing a first-principle-based loss term that enforces generated samples to fulfill the underlying physical constraints. Our approach reduces the residual error by up to two orders of magnitude compared to previous work in a fluid flow case study and outperforms task-specific frameworks in relevant metrics for structural topology optimization. We also present numerical evidence that our extended training objective acts as a natural regularization mechanism against overfitting. Our framework is simple to implement and versatile in its applicability for imposing equality and inequality constraints as well as auxiliary optimization objectives.

Can reinforcement learning with hard, verifiable rewards teach a compact language model to reason about physics, or does it primarily learn to pattern-match toward correct answers? We study this question by training a 1.5B-parameter reasoning model on beam statics, a classic engineering problem, using parameter-efficient RLVR with binary correctness rewards from symbolic solvers, without teacher-generated reasoning traces. The best BeamPERL checkpoint achieves a 66.7% improvement in Pass@1 over the base model. However, the learned competence is anisotropic: the model generalizes compositionally (more loads) but fails under topological shifts (moved supports) that require the same equilibrium equations. Intermediate checkpoints yield the strongest reasoning, while continued optimization degrades robustness while maintaining reward. These findings reveal a key limitation of outcome-level alignment: reinforcement learning with exact physics rewards induces procedural solution templates rather than internalization of governing equations. The precision of the reward signal - even when analytically exact - does not by itself guarantee transferable physical reasoning. Our results suggest that verifiable rewards may need to be paired with structured reasoning scaffolding to move beyond template matching toward robust scientific reasoning.

We explore the possibility to use physics-informed neural networks to drastically accelerate the solution of ordinary differential-algebraic equations that govern the power system dynamics. When it comes to transient stability assessment, the traditionally applied methods either carry a significant computational burden, require model simplifications, or use overly conservative surrogate models. Conventional neural networks can circumvent these limitations but are faced with high demand of high-quality training datasets, while they ignore the underlying governing equations. Physics-informed neural networks are different: they incorporate the power system differential algebraic equations directly into the neural network training and drastically reduce the need for training data. This paper takes a deep dive into the performance of physics-informed neural networks for power system transient stability assessment. Introducing a new neural network training procedure to facilitate a thorough comparison, we explore how physics-informed neural networks compare with conventional differential-algebraic solvers and classical neural networks in terms of computation time, requirements in data, and prediction accuracy. We illustrate the findings on the Kundur two-area system, and assess the opportunities and challenges of physics-informed neural networks to serve as a transient stability analysis tool, highlighting possible pathways to further develop this method.

Autoregressive models accumulate error over long rollouts, yet at deployment there is no ground truth to measure it against. We train a single conditional latent diffusion model that steps a dynamical system forward or backward in time via a direction flag, and show that this bidirectionality supplies a measurement-free test-time error signal: rolling forward i steps and then backward i steps must return the model to its start, so the round-trip discrepancy C_i is a self-supervised proxy for the unobservable rollout error: no ensembles, no held-out data, no governing equations, for one extra rollout. We validate on compressible magnetohydrodynamics (MHD), an astrophysical turbulent radiative mixing layer, and natural face videos (CelebV-HQ). On held-out MHD trajectories, C_i ranks rollout error (Spearman 0.91-0.98 at fixed depth; 0.69 pm 0.16 within trajectories), and a simple calibrator fit on training rollouts predicts its magnitude to within 1.14times (68%) and 1.29times (95%) with near-nominal coverage - one nat beyond a depth-only predictor, transferring to all six decoded physical fields. The same signal flags the out-of-distribution Orszag-Tang vortex (AUROC 0.98; 1.0 by depth 10) exactly where sampling-dispersion baselines invert, and it cuts incurred error by 15% at 80% coverage - three times the depth-only baseline. Bidirectional training comes at negative cost, beating direction specialists in both directions, and the backward direction doubles as a fast inverse solver. On LE-PDE-UQ's turbulent Navier-Stokes benchmark, a single bidirectional model reaches accuracy within 1.3times of their ten-model ensemble at a tenth of the training cost, with the best training-free pixel-level calibration. Round-trip consistency turns reversibility into a practical trust signal for generative models.

Random network models play a prominent role in modeling, analyzing and understanding complex phenomena on real-life networks. However, a key property of networks is often neglected: many real-world networks exhibit spatial structure, the tendency of a node to select neighbors with a probability depending on physical distance. Here, we introduce a class of random spatial networks (RSNs) which generalizes many existing random network models but adds spatial structure. In these networks, nodes are placed randomly in space and joined in edges with a probability depending on their distance and their individual expected degrees, in a manner that crucially remains analytically tractable. We use this network class to propose a new generalization of small-world networks, where the average shortest path lengths in the graph are small, as in classical Watts-Strogatz small-world networks, but with close spatial proximity of nodes that are neighbors in the network playing the role of large clustering. Small-world effects are demonstrated on these spatial small-world networks without clustering. We are able to derive partial integro-differential equations governing susceptible-infectious-recovered disease spreading through an RSN, and we demonstrate the existence of traveling wave solutions. If the distance kernel governing edge placement decays slower than exponential, the population-scale dynamics are dominated by long-range hops followed by local spread of traveling waves. This provides a theoretical modeling framework for recent observations of how epidemics like Ebola evolve in modern connected societies, with long-range connections seeding new focal points from which the epidemic locally spreads in a wavelike manner.

A fundamental challenge in the cognitive sciences is discovering the dynamics that govern behaviour. Take the example of spoken language, which is characterised by a highly variable and complex set of physical movements that map onto the small set of cognitive units that comprise language. What are the fundamental dynamical principles behind the movements that structure speech production? In this study, we discover models in the form of symbolic equations that govern articulatory gestures during speech. A sparse symbolic regression algorithm is used to discover models from kinematic data on the tongue and lips. We explore these candidate models using analytical techniques and numerical simulations, and find that a second-order linear model achieves high levels of accuracy, but a nonlinear force is required to properly model articulatory dynamics in approximately one third of cases. This supports the proposal that an autonomous, nonlinear, second-order differential equation is a viable dynamical law for articulatory gestures in speech. We conclude by identifying future opportunities and obstacles in data-driven model discovery and outline prospects for discovering the dynamical principles that govern language, brain and behaviour.

We introduce EMMA, a physics-informed multimodal framework that recovers all identifiable dynamical parameters of a system directly from raw video, audio, and image-based time-series observations. Unlike prior video-only approaches that struggle with occluded states, hidden actuation inputs, or assumptions about known initial conditions and coordinate frames, EMMA performs joint inference of explicit parameters, implicit dynamical components, and calibration invariants within a unified continuous-time model. EMMA leverages a Liquid Time-Constant (LTC) network to learn latent dynamics from heterogeneous modalities while a physics-constrained loss enforces consistency with the governing differential equations. A unified feature pipeline enables consistent alignment across video trajectories, acoustic signatures, and chart-derived measurements, allowing EMMA to estimate parameters under forced, implicit, and multivariate dynamics without requiring segmentation masks, differentiable rendering, or specialized sensors. Across 100+ scenarios including five standard dynamical benchmarks (75 Delfys videos), real-world rover and quadrotor systems with hidden inputs, and simulation-chart case studies spanning biological and chaotic systems, EMMA delivers robust multi-parameter recovery and significantly outperforms existing single-modality and equation-discovery baselines. Our results establish EMMA as a general, scalable solution for physics-consistent model extraction from opportunistic multimodal data. Code and data are available at: https://github.com/ImpactLabASU/EMMA-CVPR2026

Magnetic resonance imaging (MRI) super-resolution is vital for improving diagnostic accessibility, yet most methods treat it as a deterministic mapping from a fixed low-resolution input to a high-resolution target. This overlooks a key property of MRI acquisition physics: spatial resolution and signal-to-noise ratio (SNR) are inherently coupled, making any given low-resolution scan merely one of many possible realizations under varying acquisition trade-offs. We rethink MRI super-resolution as a physics-aware reconstruction problem, in which the goal is to identify the optimal resolution-SNR configuration and then super-resolve it to obtain high-quality MRI results. A key implication of this formulation is that MRI resolution becomes dynamic rather than fixed. To handle such resolution-heterogeneous inputs, we adapt 2D Gaussian Splatting (2D GS) to MRI by formulating reconstruction as a coordinate-based, resolution-agnostic rendering problem. To further enhance fidelity, we introduce three innovations: (1) a prior-aware Gaussian representation that combines an Anatomical Structure Prior for tissue-specific kernel initialization with an Imaging System Prior that captures hardware characteristics via a covariance dictionary; (2) a physics-constrained signal modeling scheme that predicts intrinsic tissue parameters (proton density rho and effective relaxation rate R2) and synthesizes intensities through governing physical equations, ensuring biophysically plausible contrast; and (3) a meta-learning framework that alleviates paired-data scarcity by pretraining on simulated data and adapting to real-world conditions. Extensive experiments on dynamic-resolution datasets and standard benchmarks demonstrate that our method achieves state-of-the-art performance, highlighting its strong potential for clinical deployment.

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.

Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters. We illustrate the performance (predictive power) of our framework on the heat equation, with and without noise and/or forcing, and compare our results to the Forward Euler method. Moreover, we show the advantages of using a Hessian-Free Trust Region method to train the network.

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to train approximations to multiple differential equations simultaneously and are thus a general purpose solver that can be adapted to downstream tasks. Current PDE foundation models focus on either learning general solution operators and/or the governing system of equations, and thus only handle numerical or symbolic modalities. However, real-world applications may require more flexible data modalities, e.g. text analysis or descriptive outputs. To address this gap, we propose a novel multimodal deep learning approach that leverages a transformer-based architecture to approximate solution operators for a wide variety of ODEs and PDEs. Our method integrates numerical inputs, such as equation parameters and initial conditions, with text descriptions of physical processes or system dynamics. This enables our model to handle settings where symbolic representations may be incomplete or unavailable. In addition to providing accurate numerical predictions, our approach generates interpretable scientific text descriptions, offering deeper insights into the underlying dynamics and solution properties. The numerical experiments show that our model provides accurate solutions for in-distribution data (with average relative error less than 3.3%) and out-of-distribution data (average relative error less than 7.8%) together with precise text descriptions (with correct descriptions generated 100% of times). In certain tests, the model is also shown to be capable of extrapolating solutions in time.

Accurate prediction of thermal runaway in lithium-ion batteries is essential for ensuring the safety, efficiency, and reliability of modern energy storage systems. Conventional data-driven approaches, such as Long Short-Term Memory (LSTM) networks, can capture complex temporal dependencies but often violate thermodynamic principles, resulting in physically inconsistent predictions. Conversely, physics-based thermal models provide interpretability but are computationally expensive and difficult to parameterize for real-time applications. To bridge this gap, this study proposes a Physics-Informed Long Short-Term Memory (PI-LSTM) framework that integrates governing heat transfer equations directly into the deep learning architecture through a physics-based regularization term in the loss function. The model leverages multi-feature input sequences, including state of charge, voltage, current, mechanical stress, and surface temperature, to forecast battery temperature evolution while enforcing thermal diffusion constraints. Extensive experiments conducted on thirteen lithium-ion battery datasets demonstrate that the proposed PI-LSTM achieves an 81.9% reduction in root mean square error (RMSE) and an 81.3% reduction in mean absolute error (MAE) compared to the standard LSTM baseline, while also outperforming CNN-LSTM and multilayer perceptron (MLP) models by wide margins. The inclusion of physical constraints enhances the model's generalization across diverse operating conditions and eliminates non-physical temperature oscillations. These results confirm that physics-informed deep learning offers a viable pathway toward interpretable, accurate, and real-time thermal management in next-generation battery systems.

We investigate solutions to a coupled system of partial differential equations that describe a multilayered filtration system. Namely, we study the interaction of a viscous incompressible flow with bulk poroelasticity, via a poroelastic interface. The configuration consists of two 3D toroidal subdomains connected via a plate interface, which permits elastic deformation and perfusive fluid dynamics. The governing dynamics comprise Stokes equations in the bulk fluid region, Biot's equations in the bulk poroelastic region, and the recent poroplate of Mikelić at the interface. Coupling occurs on the top and lower surfaces of the plate, and involves conservation of mass, stress balance, and a certain slip condition for the fluid free-flow. We seek strong (and mild) solutions in the Hilbert space framework via the Lumer-Phillips theorem. The resolvent analysis employs a nonstandard mixed variational formulation which captures the complex, multi-physics coupling at the interface. We explicitly characterize the infinitesimal generator associated to the linear Cauchy problem and establish the generation of a C_0-semigroup on a suitably chosen finite-energy space. With the semigroup in hand, we may treat elastic nonlinearities for plate displacements through perturbation theory. These result parallel those for Biot-Stokes filtration systems, and complement the recently established weak solution theory for multilayer filtrations. The agency of the semigroup straightforwardly admits structural (plate) nonlinearity into the dynamics. Future stability and regularity analyses for multilayer filtrations are also made possible by these results, as well as a comparison of spectral and regularity properties between filtration configurations, and the elucidation of the mitigating poroplate dynamics as possibly regularizing and stabilizing.

Deep generative models are used to solve partial differential equations and model distributions of physical system states, but ensuring that the generated samples satisfy the governing laws remains challenging. Projection-based flow-matching methods enforce physics by correcting the flow from an unconstrained noise distribution. These corrections shift the generated samples away from the distribution of target solutions, especially in high noise regions. To address this limitation, we propose Source Anchoring for Physical Consistency (SAPC), a Functional Flow Matching method that encodes the physical constraints into the source noise before generation begins. We evaluate SAPC on five systems governed by partial differential equations, covering six tasks with linear and non-linear dynamics, and compare results against five baselines and the unconstrained backbone. Anchoring the source reduces the need for large corrections that drive samples onto admissible but off-distribution states, and SAPC reproduces the target distributions most accurately on every evaluated task, while matching the constraint precision of the best projection-based baselines. Ablation experiments show that this gain arises from pairing source projection with a matched training objective that regresses toward the projected source. These results identify the source distribution as a key design choice for physically consistent generative modelling.

Foundation models have revolutionized natural language processing through a ``train once, deploy anywhere'' paradigm, where a single pre-trained model adapts to countless downstream tasks without retraining. Access to a Physics Foundation Model (PFM) would be transformative -- democratizing access to high-fidelity simulations, accelerating scientific discovery, and eliminating the need for specialized solver development. Yet current physics-aware machine learning approaches remain fundamentally limited to single, narrow domains and require retraining for each new system. We present the General Physics Transformer (GPhyT), trained on 1.8 TB of diverse simulation data, that demonstrates foundation model capabilities are achievable for physics. Our key insight is that transformers can learn to infer governing dynamics from context, enabling a single model to simulate fluid-solid interactions, shock waves, thermal convection, and multi-phase dynamics without being told the underlying equations. GPhyT achieves three critical breakthroughs: (1) superior performance across multiple physics domains, outperforming specialized architectures by up to 29x, (2) zero-shot generalization to entirely unseen physical systems through in-context learning, and (3) stable long-term predictions through 50-timestep rollouts. By establishing that a single model can learn generalizable physical principles from data alone, this work opens the path toward a universal PFM that could transform computational science and engineering.

Discovering the underlying physical behavior of complex systems is a crucial, but less well-understood topic in many engineering disciplines. This study proposes a finite-difference inspired convolutional neural network framework to learn hidden partial differential equations from given data and iteratively estimate future dynamical behavior. The methodology designs the filter sizes such that they mimic the finite difference between the neighboring points. By learning the governing equation, the network predicts the future evolution of the solution by using only a few trainable parameters. In this paper, we provide numerical results to compare the efficiency of the second-order Trust-Region Conjugate Gradient (TRCG) method with the first-order ADAM optimizer.

To plan and optimize energy storage demands that account for Li-ion battery aging dynamics, techniques need to be developed to diagnose battery internal states accurately and rapidly. This study seeks to reduce the computational resources needed to determine a battery's internal states by replacing physics-based Li-ion battery models -- such as the single-particle model (SPM) and the pseudo-2D (P2D) model -- with a physics-informed neural network (PINN) surrogate. The surrogate model makes high-throughput techniques, such as Bayesian calibration, tractable to determine battery internal parameters from voltage responses. This manuscript is the first of a two-part series that introduces PINN surrogates of Li-ion battery models for parameter inference (i.e., state-of-health diagnostics). In this first part, a method is presented for constructing a PINN surrogate of the SPM. A multi-fidelity hierarchical training, where several neural nets are trained with multiple physics-loss fidelities is shown to significantly improve the surrogate accuracy when only training on the governing equation residuals. The implementation is made available in a companion repository (https://github.com/NREL/pinnstripes). The techniques used to develop a PINN surrogate of the SPM are extended in Part II for the PINN surrogate for the P2D battery model, and explore the Bayesian calibration capabilities of both surrogates.

According to a scalar theory of gravity with a preferred frame, electromagnetism in the presence of a gravitational field implies that there is an additional energy tensor, which might contribute to dark matter. The expression of this tensor is determined by a mere scalar p, that depends on the EM field and (for a weak field) on the Newtonian gravitational field. We briefly recall why this tensor arises and how the EM field in a galaxy can be calculated. The data fields that enter the PDE for the scalar field p oscillate very quickly in space and time, as does the EM field. This prevents integration of that PDE at the relevant galactic scale. Therefore, a homogenization of that PDE has to be operated. We discuss in some detail three possible ways of applying the homogenization theory to that PDE: time, space, or spacetime homogenization. The second and third ways may lead to feasible, albeit heavy calculations.

Neural surrogates promise large speedups over classical solvers for physical dynamics but fail silently at sharp dynamical events such as shocks, fronts, and contact. We present hybrid neural world models for physical dynamics: a recipe for training and deploying multi-horizon surrogates in physical state space, where a single network with continuous horizon conditioning is trained with direct supervision against textbook reference solvers to predict any future state at horizon T in one forward pass. Although no part of the training data, loss function, or architecture supervises discontinuity location, the trained surrogate encodes it implicitly, recoverable from its forward passes alone as a per-trajectory error map that concentrates on shocks, fronts, and contacts, and stays small elsewhere. The map is competitive with or better than standard label-free baselines including deep ensembles, learned error heads, gradient-magnitude indicators, and locally-adaptive conformal prediction, while using only a single trained network and requiring no calibration set or governing-equation knowledge. The recipe supports two operating points. Mode 1 runs the surrogate alone for maximum throughput, with same-hardware CPU speedups of 26x to 72x against textbook solvers on the PDE environments. Mode 2 uses the error map to gate a reference-solver fallback, deferring uncertain trajectories and roughly halving the surrogate's residual error at the default operating point. The recipe applies without modification across reaction-diffusion, compressible Euler, and rigid-body collision dynamics.

We study the problem of the transformation of a given reactant species into an immiscible product species, as they flow through a chemically active porous medium. We derive the equation governing the evolution of the volume fraction of the species -- in a one-dimensional macroscopic description --, identify the relevant dimensionless numbers, and provide simple models for capillary pressure and relative permeabilities, which are quantities of crucial importance when tackling multiphase flows in porous media. We set the domain of validity of our models and discuss the importance of viscous coupling terms in the extended Darcy's law. We investigate numerically the steady regime and demonstrate that the spatial transformation rate of the species along the reactor is non-monotonous, as testified by the existence of an inflection point in the volume fraction profiles. We obtain the scaling of the location of this inflection point with the dimensionless lengths of the problem. Eventually, we provide key elements for optimization of the reactor.

Fluorescence LiDAR (FLiDAR), a Light Detection and Ranging (LiDAR) technology employed for distance and depth estimation across medical, automotive, and other fields, encounters significant computational challenges in scattering media. The complex nature of the acquired FLiDAR signal, particularly in such environments, makes isolating photon time-of-flight (related to target depth) and intrinsic fluorescence lifetime exceptionally difficult, thus limiting the effectiveness of current analytical and computational methodologies. To overcome this limitation, we present a Physics-Guided Mixture-of-Experts (MoE) framework tailored for specialized modeling of diverse temporal components. In contrast to the conventional MoE approaches our expert models are informed by underlying physics, such as the radiative transport equation governing photon propagation in scattering media. Central to our approach is EvidenceMoE, which integrates Evidence-Based Dirichlet Critics (EDCs). These critic models assess the reliability of each expert's output by providing per-expert quality scores and corrective feedback. A Decider Network then leverages this information to fuse expert predictions into a robust final estimate adaptively. We validate our method using realistically simulated Fluorescence LiDAR (FLiDAR) data for non-invasive cancer cell depth detection generated from photon transport models in tissue. Our framework demonstrates strong performance, achieving a normalized root mean squared error (NRMSE) of 0.030 for depth estimation and 0.074 for fluorescence lifetime.

Physics is a field of science that has traditionally used the scientific method to answer questions about why natural phenomena occur and to make testable models that explain the phenomena. Discovering equations, laws and principles that are invariant, robust and causal explanations of the world has been fundamental in physical sciences throughout the centuries. Discoveries emerge from observing the world and, when possible, performing interventional studies in the system under study. With the advent of big data and the use of data-driven methods, causal and equation discovery fields have grown and made progress in computer science, physics, statistics, philosophy, and many applied fields. All these domains are intertwined and can be used to discover causal relations, physical laws, and equations from observational data. This paper reviews the concepts, methods, and relevant works on causal and equation discovery in the broad field of Physics and outlines the most important challenges and promising future lines of research. We also provide a taxonomy for observational causal and equation discovery, point out connections, and showcase a complete set of case studies in Earth and climate sciences, fluid dynamics and mechanics, and the neurosciences. This review demonstrates that discovering fundamental laws and causal relations by observing natural phenomena is being revolutionised with the efficient exploitation of observational data, modern machine learning algorithms and the interaction with domain knowledge. Exciting times are ahead with many challenges and opportunities to improve our understanding of complex systems.

We present a numerical framework to treat infinite time horizon spatially distributed optimal control problems via the associated canonical system derived by Pontryagin's Maximum Principle. The basic idea is to consider the canonical system in two steps. First we perform a bifurcation analysis of canonical steady states using the continuation and bifurcation package pde2path, yielding a number of so called flat and patterned canonical steady states. In a second step we link pde2path to the two point boundary value problem solver TOM to study time dependent canonical paths to steady states having the so called saddle point property. As an example we consider a shallow lake model with diffusion.

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.

We study a discrete model for generalized exchange-driven growth in which the particle exchanged between two clusters is not limited to be of size one. This set of models include as special cases the usual exchange-driven growth system and the coagulation-fragmentation system with binary fragmentation. Under reasonable general condition on the rate coefficients we establish the existence of admissible solutions, meaning solutions that are obtained as appropriate limit of solutions to a finite-dimensional truncation of the infinite-dimensional ODE. For these solutions we prove that, in the class of models we call isolated both the total number of particles and the total mass are conserved, whereas in those models we can non-isolated only the mass is conserved. Additionally, under more restrictive growth conditions for the rate equations we obtain uniqueness of solutions to the initial value problems.

Diffusion models (DMs) have proven to be effective in modeling high-dimensional distributions, leading to their widespread adoption for representing complex priors in Bayesian inverse problems (BIPs). However, current DM-based posterior sampling methods proposed for solving common BIPs rely on heuristic approximations to the generative process. To exploit the generative capability of DMs and avoid the usage of such approximations, we propose an ensemble-based algorithm that performs posterior sampling without the use of heuristic approximations. Our algorithm is motivated by existing works that combine DM-based methods with the sequential Monte Carlo (SMC) method. By examining how the prior evolves through the diffusion process encoded by the pre-trained score function, we derive a modified partial differential equation (PDE) governing the evolution of the corresponding posterior distribution. This PDE includes a modified diffusion term and a reweighting term, which can be simulated via stochastic weighted particle methods. Theoretically, we prove that the error between the true posterior distribution can be bounded in terms of the training error of the pre-trained score function and the number of particles in the ensemble. Empirically, we validate our algorithm on several inverse problems in imaging to show that our method gives more accurate reconstructions compared to existing DM-based methods.

A scuba diver holding constant depth balances on an unstable equilibrium: the gas carried in the suit and buoyancy compensator compresses with depth, so the buoyant force falls as the diver sinks and rises as the diver ascends. We represent the diver as a proportional-derivative controller that regulates this compressible-buoyancy saddle after a finite reaction delay, and we derive the governing delay differential equation from the vertical force balance and the isothermal gas law, reducing it to a damping ratio, two control gains, and a dimensionless delay. The characteristic spectrum, obtained by pseudospectral collocation of the semigroup generator and checked against a direct Newton solution of the characteristic equation, locates the Hopf boundary that separates stable hovering from sustained porpoising; for the baseline diver the critical reaction delay is 3.36 s and the onset period is 28.8 s. The bifurcation is supercritical, and because the saturating force is the quadratic hydrodynamic drag, the limit-cycle amplitude grows in proportion to the delay excess rather than as its square root. The safe-operating envelope shows that runaway ascent is triggered by saturation of the compensator, not by loss of linear stability, so a stable and an unstable diver can share the same escape threshold. As onset is approached, the lag-one autocorrelation and variance rise while the fitted recovery rate falls and matches the spectral abscissa, giving an eigenvalue-exact early warning of the transition.

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this two part treatise, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct classes of algorithms, namely continuous time and discrete time models. The resulting neural networks form a new class of data-efficient universal function approximators that naturally encode any underlying physical laws as prior information. In this first part, we demonstrate how these networks can be used to infer solutions to partial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters.

We study the parameter dependence and transversality properties of trajectories of the controlled ODE $ U_x = P, qquad P_x = - f(U) - βP + tilde α(x)U, arising from traveling-wave solutions of a bistable reaction-diffusion equation with control. Optimal profiles minimizing the L^1-norm of the control \tilde α are determined by the unstable and stable manifolds Γ_u, Γ_s, the critical curve P^*(U) = Uf(U), and the associated function F(U) = dP^*{dU} + f(U){P^*(U)}. We establish generic transversality properties for these objects with respect to perturbations of the speed β and the nonlinear source f, together with quantitative estimates on the first- and second-order dependence of the invariant manifolds on these parameters. We also analyze the loss of regularity at the critical speed β^{**}, where the unstable manifold approaches the strongly stable manifold of the equilibrium (U^*,0)$. These results provide the ODE ingredients used in the analysis of the structure and regularity of optimal profiles in paper [S. Bianchini and C. Trifone. Traveling profiles and control cost for a PDE describing the evolution of invasive species. 2026. arXiv: 2606.23979 [math.OC]].

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.

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

Initial boundary value problems arise commonly in applications with engineering and natural systems governed by nonlinear partial differential equations (PDEs). Operator learning is an emerging field for solving these equations by using a neural network to learn a map between infinite dimensional input and output function spaces. These neural operators are trained using a combination of data (observations or simulations) and PDE-residuals (physics-loss). A major drawback of existing neural approaches is the requirement to retrain with new initial/boundary conditions, and the necessity for a large amount of simulation data for training. We develop a physics-informed transformer neural operator (named PINTO) that efficiently generalizes to unseen initial and boundary conditions, trained in a simulation-free setting using only physics loss. The main innovation lies in our new iterative kernel integral operator units, implemented using cross-attention, to transform the PDE solution's domain points into an initial/boundary condition-aware representation vector, enabling efficient learning of the solution function for new scenarios. The PINTO architecture is applied to simulate the solutions of important equations used in engineering applications: advection, Burgers, and steady and unsteady Navier-Stokes equations (three flow scenarios). For these five test cases, we show that the relative errors during testing under challenging conditions of unseen initial/boundary conditions are only one-fifth to one-third of other leading physics informed operator learning methods. Moreover, our PINTO model is able to accurately solve the advection and Burgers equations at time steps that are not included in the training collocation points. The code is available at https://github.com/quest-lab-iisc/PINTO

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering and mathematical problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, and more. While this has led to solving many complex phenomena, there are some limitations. Conventional approaches such as Finite Element Methods (FEMs) and Finite Differential Methods (FDMs) require considerable time and are computationally expensive. In contrast, data driven machine learning-based methods such as neural networks provide a faster, fairly accurate alternative, and have certain advantages such as discretization invariance and resolution invariance. This article aims to provide a comprehensive insight into how data-driven approaches can complement conventional techniques to solve engineering and physics problems, while also noting some of the major pitfalls of machine learning-based approaches. Furthermore, we highlight, a novel and fast machine learning-based approach (~1000x) to learning the solution operator of a PDE operator learning. We will note how these new computational approaches can bring immense advantages in tackling many problems in fundamental and applied physics.

We derive a thermodynamically-consistent model of fluid flow through a poroelastic medium. Starting from elastic and fluid free-energy densities, an energy-dissipation rate, and a kinematic constraint, the force-balance equations are derived using variational principles, with the pressure--density constitutive relation emerging as a direct consequence of the variational structure; the same kinematic constraint also supplies the total-flux transport structure. In the ideal-gas limit, the model linearization recovers the classical linear Biot equations. For power-law fluid energies, it yields isentropic pressure--density relations. A key advantage of the variational formulation is that extensions to richer physics, such as thermal effects, chemical reactions, or multi-component fluids, can be incorporated systematically by augmenting the energy and dissipation functionals without redesigning the force-balance or transport closure. We support the model with an energy-compatible two-field discretization and study consolidation under a surface load with three lateral-boundary treatments and three fluid-compressibility exponents.

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.

In this paper, we rigorously derive the fundamental PDEs of fluid mechanics, such as the compressible Euler and incompressible Navier-Stokes-Fourier equations, starting from the hard sphere particle systems undergoing elastic collisions. This resolves Hilbert's sixth problem, as it pertains to the program of deriving the fluid equations from Newton's laws by way of Boltzmann's kinetic theory. The proof relies on the derivation of Boltzmann's equation on 2D and 3D tori, which is an extension of our previous work (arXiv:2408.07818).

We investigate optimal control problems governed by the elliptic partial differential equation -Delta u=f subject to Dirichlet boundary conditions on a given domain Omega. The control variable in this setting is the right-hand side f, and the objective is to minimize a cost functional that depends simultaneously on the control f and on the associated state function u. We establish the existence of optimal controls and analyze their qualitative properties by deriving necessary conditions for optimality. In particular, when pointwise constraints of the form alphale flebeta are imposed a priori on the control, we examine situations where a {\it bang-bang} phenomenon arises, that is where the optimal control f assumes only the extremal values alpha and beta. More precisely, the control takes the form f=alpha1_E+beta1_{Omegasetminus E}, thereby placing the problem within the framework of shape optimization. Under suitable assumptions, we further establish certain regularity properties for the optimal sets E. Finally, in the last part of the paper, we present numerical simulations that illustrate our theoretical findings through a selection of representative examples.

Accurate and efficient simulations of physical phenomena governed by partial differential equations (PDEs) are important for scientific and engineering progress. While traditional numerical solvers are powerful, they are often computationally expensive. Recently, data-driven methods have emerged as alternatives, but they frequently suffer from error accumulation and limited physical consistency, especially in multiphysics and complex geometries. To address these challenges, we propose PEGNet, a Physics-Embedded Graph Network that incorporates PDE-guided message passing to redesign the graph neural network architecture. By embedding key PDE dynamics like convection, viscosity, and diffusion into distinct message functions, the model naturally integrates physical constraints into its forward propagation, producing more stable and physically consistent solutions. Additionally, a hierarchical architecture is employed to capture multi-scale features, and physical regularization is integrated into the loss function to further enforce adherence to governing physics. We evaluated PEGNet on benchmarks, including custom datasets for respiratory airflow and drug delivery, showing significant improvements in long-term prediction accuracy and physical consistency over existing methods. Our code is available at https://github.com/Yanghuoshan/PEGNet.

Modelling physical systems with partial differential equations (PDEs) is central to science and engineering, yet in most real applications the PDE model is incomplete: relationships such as constitutive or thermal laws are unknown. Existing surrogate approaches close this gap by learning the PDE solution from data, but remain tied to a specific configuration (geometry, boundary conditions, discretisation) and recover the solution rather than the missing physics itself. We introduce FEML, an end-to-end differentiable framework that couples the known PDE with a machine-learned operator for the missing physics. Embedding the PDE solver into training lets this operator be learned directly from the PDE solution, even when its own output cannot be measured - for example, stress in constitutive laws. Because the operator is independent of the system configuration, a law learned in one setting transfers zero-shot to new geometries, boundary conditions, and discretisations, and can be inspected by domain specialists. FEML represents the operator with structure-preserving operator networks (SPONs), which retain key continuous properties at the discrete level. We demonstrate FEML across solid mechanics and thermal transport. From synthetic data we progressively discover an elastoplastic law - the nonlinear elastic response, then the plastic hardening law - and compose them into a foundation constitutive model that transfers zero-shot to a 3D torsion problem. Moving to real data, we learn coupled plastic-hardening and ductile-damage laws from a benchmark shear-coupon test, reproducing the measured response, including post-peak softening, to within the experimental scatter. Finally, we recover a temperature-dependent conductivity from transient heat-flow data and apply symbolic regression to the learned operator to extract a closed-form law matching the ground truth.

The principle of detailed balance states that in equilibrium each elementary process is equilibrated by its reverse process. For many real physico-chemical complex systems (e.g. homogeneous combustion, heterogeneous catalytic oxidation, most enzyme reactions etc), detailed mechanisms include both reversible and irreversible reactions. In this case, the principle of detailed balance cannot be applied directly. We represent irreversible reactions as limits of reversible steps and obtain the principle of detailed balance for complex mechanisms with some irreversible elementary processes. We proved two consequences of the detailed balance for these mechanisms: the structural condition and the algebraic condition that form together the extended form of detailed balance. The algebraic condition is the principle of detailed balance for the reversible part. The structural condition is: the convex hull of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span of the stoichiometric vectors of the reversible reaction. Physically, this means that the irreversible reactions cannot be included in oriented pathways. The systems with the extended form of detailed balance are also the limits of the reversible systems with detailed balance when some of the equilibrium concentrations (or activities) tend to zero. Surprisingly, the structure of the limit reaction mechanism crucially depends on the relative speeds of this tendency to zero.

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 consider funnel control for linear infinite-dimensional systems that are impedance passive, meaning that they satisfy an energy balance in which the stored energy equals the squared norm of the state and the supplied power is the inner product of input and output. For the analysis we employ the system node approach, which offers a unified framework for infinite-dimensional systems with boundary and distributed control and observation. The resulting closed-loop dynamics are governed by a nonlinear evolution equation; we establish its solvability and hence the applicability of funnel control to this class. The applicability is illustrated by an Euler-Bernoulli beam, which is studied in two distinct scenarios: once with boundary control and once with distributed control.

We study the Cauchy problem for the one-dimensional isothermal compressible Navier--Stokes system with the van der Waals equation of state, and analyze the well-posedness and asymptotic stability of steady-state phase-transition solutions at subcritical temperatures. Working in Lagrangian coordinates, we use variational techniques to construct piecewise smooth steady-state solutions corresponding to two-phase equilibrium states under finite-mass perturbations. Introducing artificial-viscosity regularization, we first establish existence of the global minimizer of the regularized variational problem, unique within each fixed symmetry class; passing to the vanishing artificial-viscosity limit then yields a physically admissible steady-state solution with two phase interfaces. For initial perturbations that are small with respect to energy, piecewise-gradient, total-variation, and integral norms but may exhibit large pointwise jumps across phase boundaries, we derive uniform a priori bounds for weak solutions near the two-phase equilibrium profile. We further prove that these global weak solutions converge to the steady-state two-phase equilibrium in L^infty(R) as ttoinfty, thereby characterizing their large-time asymptotic behavior. Physically, finite mass perturbations break the uniqueness of the uniform steady state and generate infinitely many piecewise constant two-phase equilibria satisfying the Maxwell equal-area rule. Artificial viscosity acts as a selection principle that singles out the physically relevant two-interface steady-state configuration, and this two-phase equilibrium is asymptotically stable under small integral perturbations that allow large-amplitude discontinuities at phase interfaces.

Solving partial differential equations (PDEs) with machine learning has recently attracted great attention, as PDEs are fundamental tools for modeling real-world systems that range from fundamental physical science to advanced engineering disciplines. Most real-world physical systems across various disciplines are actually involved in multiple coupled physical fields rather than a single field. However, previous machine learning studies mainly focused on solving single-field problems, but overlooked the importance and characteristics of multiphysics problems in real world. Multiphysics PDEs typically entail multiple strongly coupled variables, thereby introducing additional complexity and challenges, such as inter-field coupling. Both benchmarking and solving multiphysics problems with machine learning remain largely unexamined. To identify and address the emerging challenges in multiphysics problems, we mainly made three contributions in this work. First, we collect the first general multiphysics dataset, the Multiphysics Bench, that focuses on multiphysics PDE solving with machine learning. Multiphysics Bench is also the most comprehensive PDE dataset to date, featuring the broadest range of coupling types, the greatest diversity of PDE formulations, and the largest dataset scale. Second, we conduct the first systematic investigation on multiple representative learning-based PDE solvers, such as PINNs, FNO, DeepONet, and DiffusionPDE solvers, on multiphysics problems. Unfortunately, naively applying these existing solvers usually show very poor performance for solving multiphysics. Third, through extensive experiments and discussions, we report multiple insights and a bag of useful tricks for solving multiphysics with machine learning, motivating future directions in the study and simulation of complex, coupled physical systems.

While recent AI-for-math has made strides in pure mathematics, areas of applied mathematics, particularly PDEs, remain underexplored despite their significant real-world applications. We present PDE-Controller, a framework that enables large language models (LLMs) to control systems governed by partial differential equations (PDEs). Our approach enables LLMs to transform informal natural language instructions into formal specifications, and then execute reasoning and planning steps to improve the utility of PDE control. We build a holistic solution comprising datasets (both human-written cases and 2 million synthetic samples), math-reasoning models, and novel evaluation metrics, all of which require significant effort. Our PDE-Controller significantly outperforms prompting the latest open-source and GPT models in reasoning, autoformalization, and program synthesis, achieving up to a 62% improvement in utility gain for PDE control. By bridging the gap between language generation and PDE systems, we demonstrate the potential of LLMs in addressing complex scientific and engineering challenges. We will release all data, model checkpoints, and code at https://pde-controller.github.io/.

Partial Differential Equations are precise in modelling the physical, biological and graphical phenomena. However, the numerical methods suffer from the curse of dimensionality, high computation costs and domain-specific discretization. We aim to explore pros and cons of different PDE solvers, and apply them to specific scientific simulation problems, including forwarding solution, inverse problems and equations discovery. In particular, we extend the recent CNF (NeurIPS 2023) framework solver to multi-dependent-variable and non-linear settings, together with down-stream applications. The outcomes include implementation of selected methods, self-tuning techniques, evaluation on benchmark problems and a comprehensive survey of neural PDE solvers and scientific simulation applications.

We propose a new method to ensure neural ordinary differential equations (ODEs) satisfy output specifications by using invariance set propagation. Our approach uses a class of control barrier functions to transform output specifications into constraints on the parameters and inputs of the learning system. This setup allows us to achieve output specification guarantees simply by changing the constrained parameters/inputs both during training and inference. Moreover, we demonstrate that our invariance set propagation through data-controlled neural ODEs not only maintains generalization performance but also creates an additional degree of robustness by enabling causal manipulation of the system's parameters/inputs. We test our method on a series of representation learning tasks, including modeling physical dynamics and convexity portraits, as well as safe collision avoidance for autonomous vehicles.

In this paper, a PDE-ODE model with discontinuity in the flux as well as a flux constraint is analyzed. A modified Riemann solution is proposed and the existence of a weak solution to the Cauchy problem is rigorously investigated using the wavefront tracking scheme.

In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." In this manuscript we introduce the SciML software ecosystem as a tool for mixing the information of physical laws and scientific models with data-driven machine learning approaches. We describe a mathematical object, which we denote universal differential equations (UDEs), as the unifying framework connecting the ecosystem. We show how a wide variety of applications, from automatically discovering biological mechanisms to solving high-dimensional Hamilton-Jacobi-Bellman equations, can be phrased and efficiently handled through the UDE formalism and its tooling. We demonstrate the generality of the software tooling to handle stochasticity, delays, and implicit constraints. This funnels the wide variety of SciML applications into a core set of training mechanisms which are highly optimized, stabilized for stiff equations, and compatible with distributed parallelism and GPU accelerators.

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively ``easy'' PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively ``hard'' PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating conservation constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticities. In each case, it achieves superior predictive performance on downstream tasks.

We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.

We find and analyse solutions of Einstein's equations in arbitrary d dimensions and in the presence of a scalar field with a Liouville potential coupled to a Maxwell field. We consider spacetimes of cylindrical symmetry or again subspaces of dimension d-2 with constant curvature and analyse in detail the field equations and manifest their symmetries. The field equations of the full system are shown to reduce to a single or couple of ODE's which can be used to solve analytically or numerically the theory for the symmetry at hand. Further solutions can also be generated by a solution generating technique akin to the EM duality in the absence of a cosmological constant. We then find and analyse explicit solutions including black holes and gravitating solitons for the case of four dimensional relativity and the higher-dimensional oxydised 5-dimensional spacetime. The general solution is obtained for a certain relation between couplings in the case of cylindrical symmetry.

Physics-Informed machine learning models have recently emerged with some interesting and unique features that can be applied to reservoir engineering. In particular, physics-informed neural networks (PINN) leverage the fact that neural networks are a type of universal function approximators that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can be described by partial differential equations. The transient diffusivity equation is a fundamental equation in reservoir engineering and the general solution to this equation forms the basis for Pressure Transient Analysis (PTA). The diffusivity equation is derived by combining three physical principles, the continuity equation, Darcy's equation, and the equation of state for a slightly compressible liquid. Obtaining general solutions to this equation is imperative to understand flow regimes in porous media. Analytical solutions of the transient diffusivity equation are usually hard to obtain due to the stiff nature of the equation caused by the steep gradients of the pressure near the well. In this work we apply physics-informed neural networks to the one and two dimensional diffusivity equation and demonstrate that decomposing the space domain into very few subdomains can overcome the stiffness problem of the equation. Additionally, we demonstrate that the inverse capabilities of PINNs can estimate missing physics such as permeability and distance from sealing boundary similar to buildup tests without shutting in the well.

We study the dynamics of an inextensible, closed interface subject to bending forces and immersed in a two-dimensional and incompressible Stokes fluid. We formulate the problem as a boundary integral equation in terms of the tangent angle and demonstrate the well-posedness in suitable time-weighted spaces of the resulting nonlinear and nonlocal system. The solution is furthermore shown to be smooth for positive times. Numerical computations are performed to initiate the study of the long-time behavior of the interface.

The classical theory of the wind-driven surface layer of a rotating ocean closes the momentum balance with a local flux-gradient law, in which the turbulent stress at a given depth is proportional to the shear at that depth. It predicts a surface current deflected forty-five degrees from the wind, which exceeds most direct measurements. This study asks what follows when locality is relaxed. Beginning from the exact integral relation between turbulent stress and mean shear, and requiring the memory kernel to carry no preferred vertical scale, we obtain a power-law kernel and therefore a stress law of fractional order. The resulting equation cannot be posed on the velocity, because the fractional derivative of a bounded profile vanishes at the surface, so the wind stress cannot be applied, while the alternative definition of the derivative leaves the surface current unbounded. Posed on the stress instead, the problem is solvable in closed form in Mittag-Leffler functions at every order between zero and one. The surface deflection then depends on the closure order alone and is smaller than forty-five degrees throughout, whereas the depth-integrated transport stays exactly normal to the wind, because that constraint follows from the momentum balance and not from the closure. The far field decays as a power law rather than exponentially, and its amplitude vanishes in the local limit, so that limit is singular. Under a suddenly applied stress the surface transient decays algebraically. Three independent algorithms agree closely, and no observational or model data are used.

In this paper, we consider a varying terminal time structure for the stochastic optimal control problem under state constraints, in which the terminal time varies with the mean value of the state. In this new stochastic optimal control system, the control domain does not need to be convex and the diffusion coefficient contains the control variable. To overcome the difficulty in the proof of the related Pontryagin's stochastic maximum principle, we develop asymptotic first- and second-order adjoint equations for the varying terminal time, and then establish its variational equation. In the end, two examples are given to verify the main results of this study.

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.

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.

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.

The growing body of research shows how to replace classical partial differential equation (PDE) integrators with neural networks. The popular strategy is to generate the input-output pairs with a PDE solver, train the neural network in the regression setting, and use the trained model as a cheap surrogate for the solver. The bottleneck in this scheme is the number of expensive queries of a PDE solver needed to generate the dataset. To alleviate the problem, we propose a computationally cheap augmentation strategy based on general covariance and simple random coordinate transformations. Our approach relies on the fact that physical laws are independent of the coordinate choice, so the change in the coordinate system preserves the type of a parametric PDE and only changes PDE's data (e.g., initial conditions, diffusion coefficient). For tried neural networks and partial differential equations, proposed augmentation improves test error by 23% on average. The worst observed result is a 17% increase in test error for multilayer perceptron, and the best case is a 80% decrease for dilated residual network.

Partial differential equations (PDEs) are foundational to modeling in science and engineering, but constructing reliable numerical solvers remains labor-intensive, demanding expert knowledge of discretization schemes, stability conditions, and boundary treatments. Recent work has begun to frame PDE solving as a code-generation task for large language models (LLMs), yet existing approaches operate primarily at inference time: relying on prompting, debugging, self-refinement, and test-time scaling rather than adapting the model itself. In parallel, reinforcement learning with verifiable rewards has emerged as a post-training paradigm for code and math reasoning, but its verifiers are typically binary: a compiler runs, or a test passes. Such signals discard the graded structure of scientific correctness, where two solvers may both execute and yet differ in solution accuracy by orders of magnitude. In this work, we introduce RLVP: Reinforcement Learning with Verifiable Physics, an RL post-training framework for multi-PDE solver code generation. RLVP addresses this verifiability gap with a hybrid verifier: hard program-validity checks ensure executability, while continuous physics rewards score function-space accuracy and PDE-residual consistency. A single policy is post-trained across diverse PDE families spanning hyperbolic, parabolic, elliptic, and incompressible-flow systems. RLVP improves over both pre-trained and supervised-only baselines on PDE benchmarks, and shows zero-shot improvement transfer to held-out PDEs. We show that a smaller LLM post-trained with RLVP can outperform prompting a frontier model on in-distribution PDE solver generation. The trained policy shows evidence of compositionality in numerical motifs: it recombines stencils, time-stepping schemes, and boundary-handling primitives learned from the PDEs used in training into generated solvers for unseen PDE problems.

Partial differential equations (PDEs) are fundamental to modeling physical systems, yet solving them remains a complex challenge. Traditional numerical solvers rely on expert knowledge to implement and are computationally expensive, while neural-network-based solvers require large training datasets and often lack interpretability. In this work, we frame PDE solving as a code generation task and introduce CodePDE, the first inference framework for generating PDE solvers using large language models (LLMs). Leveraging advanced inference-time algorithms and scaling strategies, CodePDE unlocks critical capacities of LLM for PDE solving: reasoning, debugging, selfrefinement, and test-time scaling -- all without task-specific tuning. CodePDE achieves superhuman performance across a range of representative PDE problems. We also present a systematic empirical analysis of LLM generated solvers, analyzing their accuracy, efficiency, and numerical scheme choices. Our findings highlight the promise and the current limitations of LLMs in PDE solving, offering a new perspective on solver design and opportunities for future model development. Our code is available at https://github.com/LithiumDA/CodePDE.

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.

This paper presents a mathematical framework for optimizing drug delivery in cancer treatment using a nonlocal model of solid tumor growth. We present a coupled system of partial differential equations that incorporate long-range cellular interactions through integral terms and drug-induced cell death. The model accounts for spatial heterogeneity in both tumor cell density and drug concentration while capturing the complex dynamics of drug resistance development. We first establish the well-posedness of the coupled system by proving the existence and uniqueness of a solution under appropriate regularity conditions. The optimal control problem is then formulated to minimize tumor size while accounting for drug toxicity constraints. Using variational methods, we derive the necessary optimality conditions and characterize the optimal control through an adjoint system. Theoretical results can help to design effective chemotherapy schedules that balance treatment efficacy with adverse effects.

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.

We investigate a coupled system of partial differential equations modeling the interaction between Brinkman flow, biofilm evolution, and nutrient transport in a porous medium. The model captures the mutual influence between the fluid velocity and the biofilm through drag and diffusion coefficients that depend on the local biofilm volume fraction. A hard constraint on the admissible range of the biofilm fraction is incorporated through the subdifferential of an indicator functional, which leads naturally to an evolution variational inequality formulation for the biofilm dynamics. Assuming standard coercivity, ellipticity, and growth conditions on the model coefficients and reaction terms, we prove the global-in-time existence of weak solutions. The analysis relies on a decomposition of the system into three interconnected subproblems: the Brinkman equation with a fixed biofilm profile, the constrained biofilm evolution treated through maximal monotone operator theory, and the nutrient equation viewed as a semilinear parabolic problem. These components are then coupled through a Leray--Schauder type fixed-point argument, with the passage to the limit justified by Aubin--Lions and Simon compactness results. We further establish the nonnegativity of the nutrient concentration under a natural quasi-positivity assumption on the reaction term. Finally, we provide conditional uniqueness results for weak solutions in two spatial dimensions under additional smallness assumptions.

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.

Recent advances in deep learning have inspired numerous works on data-driven solutions to partial differential equation (PDE) problems. These neural PDE solvers can often be much faster than their numerical counterparts; however, each presents its unique limitations and generally balances training cost, numerical accuracy, and ease of applicability to different problem setups. To address these limitations, we introduce several methods to apply latent diffusion models to physics simulation. Firstly, we introduce a mesh autoencoder to compress arbitrarily discretized PDE data, allowing for efficient diffusion training across various physics. Furthermore, we investigate full spatio-temporal solution generation to mitigate autoregressive error accumulation. Lastly, we investigate conditioning on initial physical quantities, as well as conditioning solely on a text prompt to introduce text2PDE generation. We show that language can be a compact, interpretable, and accurate modality for generating physics simulations, paving the way for more usable and accessible PDE solvers. Through experiments on both uniform and structured grids, we show that the proposed approach is competitive with current neural PDE solvers in both accuracy and efficiency, with promising scaling behavior up to sim3 billion parameters. By introducing a scalable, accurate, and usable physics simulator, we hope to bring neural PDE solvers closer to practical use.

We analyze the state constrained inverse Stefan type parabolic free boundary problem as an optimal control problem in the Sobolev-Besov spaces framework. Boundary heat flux, density of heat sources, and free boundary are components of the control vector. Cost functional is the sum of the L_2-norm declinations of the temperature measurement at the final moment, the phase transition temperature, the final position of the free boundary, and the penalty term, taking into account the state constraint on the temperature. We prove the existence of optimal control, Frechet differentiability, and optimality condition in the Besov spaces under minimal regularity assumptions on the data. We pursue space-time discretization through finite differences and prove that the sequence of discrete optimal control problems converges to the original problem both with respect to functional and control.

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.

Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated training data tailored to a given setting, one may instead wish to learn useful information from heterogeneous sources, or from real dynamical systems observations that are messy or incomplete. In this work, we learn general-purpose representations of PDEs from heterogeneous data by implementing joint embedding methods for self-supervised learning (SSL), a framework for unsupervised representation learning that has had notable success in computer vision. Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs.

Explaining observed phenomena through symbolic, interpretable formulas is a fundamental goal of science. Recently, large language models (LLMs) have emerged as promising tools for symbolic equation discovery, owing to their broad domain knowledge and strong reasoning capabilities. However, most existing LLM-based systems try to guess equations directly from data, without modeling the multi-step reasoning process that scientists often follow: first inferring physical properties such as symmetries, then using these as priors to restrict the space of candidate equations. We introduce KeplerAgent, an agentic framework that explicitly follows this scientific reasoning process. The agent coordinates physics-based tools to extract intermediate structure and uses these results to configure symbolic regression engines such as PySINDy and PySR, including their function libraries and structural constraints. Across a suite of physical equation benchmarks, KeplerAgent achieves substantially higher symbolic accuracy and greater robustness to noisy data than both LLM and traditional baselines.

Scientific equation discovery is a fundamental task in the history of scientific progress, enabling the derivation of laws governing natural phenomena. Recently, Large Language Models (LLMs) have gained interest for this task due to their potential to leverage embedded scientific knowledge for hypothesis generation. However, evaluating the true discovery capabilities of these methods remains challenging, as existing benchmarks often rely on common equations that are susceptible to memorization by LLMs, leading to inflated performance metrics that do not reflect discovery. In this paper, we introduce LLM-SRBench, a comprehensive benchmark with 239 challenging problems across four scientific domains specifically designed to evaluate LLM-based scientific equation discovery methods while preventing trivial memorization. Our benchmark comprises two main categories: LSR-Transform, which transforms common physical models into less common mathematical representations to test reasoning beyond memorized forms, and LSR-Synth, which introduces synthetic, discovery-driven problems requiring data-driven reasoning. Through extensive evaluation of several state-of-the-art methods, using both open and closed LLMs, we find that the best-performing system so far achieves only 31.5% symbolic accuracy. These findings highlight the challenges of scientific equation discovery, positioning LLM-SRBench as a valuable resource for future research.

Scientific observations are frequently distributed across locations, time periods, and institutions. Combining such observations into a continuous, differentiable field enables recovering governing physical parameters from its derivatives. This paper makes two contributions in this setting. First, the established additive structure of fixed-basis ridge-regression statistics is applied to tensor-product spline fields: each data holder computes a local Gram matrix and moment vector, and the merged solution is mathematically identical to centralized fitting, with no raw data shared and no iterative synchronization. This property is specific to the fixed-feature squared-error setting; the present derivation does not establish an analogous guarantee for general jointly trained multilayer networks. Second, a complete pipeline connects distributed observations to physical parameter inference through field reconstruction, derivative extraction, and linear regression. The pipeline is validated on four PDEs: diffusion, wave, heat-with-source, and the nonlinear viscous Burgers equation, recovering governing parameters to sub-percent accuracy in the linear cases and 5\% for Burgers. In all cases, distributed merging introduces zero degradation relative to centralized fitting. Synthetic experiments validate parameter recovery; application to 41 years of NOAA sea-surface temperature data validates field reconstruction and aggregation equivalence on real spatiotemporal observations. Source code to reproduce all experiments is available at https://github.com/NAVEENMN/splinemerge.

Partial differential equations (PDEs) are important tools to model physical systems, and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works like a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDE, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

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.

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.

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

We introduce Tadpole, a novel foundation model for three-dimensional partial differential equations (PDEs) that addresses key challenges in transferability, scalability to high dimensionality, and multi-functionality. Tadpole is pre-trained as an autoencoder on synthetic 3D PDE data generated by an efficient online data-generation framework. This enables large-scale, diverse training without storage or I/O overhead, demonstrated by scaling to an equivalent of hundreds of terabytes of training data. By autoencoding single-channel spatial crops, Tadpole learns rich and transferable representations across heterogeneous physical systems with varying numbers of state variables and spatial resolutions. Although pre-trained solely as an autoencoder, Tadpole can be efficiently applied for multiple downstream tasks beyond reconstruction, including dynamics learning and generative modeling. For dynamics learning, we propose a novel parameter-efficient fine-tuning strategy that integrates low-rank adaptation, latent-space transformations, and reintroduced skip connections, achieving accurate temporal modeling with a minimal number of trainable parameters. Tadpole demonstrates strong fine-tuning performance across various downstream tasks, highlighting its versatility and effectiveness as a foundation model for 3D PDE learning. Source code and pre-trained weights of Tadpole are available at https://github.com/tum-pbs/tadpole

This paper studies the existence and asymptotic stability of traveling shock waves for the one-dimensional full compressible Navier--Stokes--Poisson equations, which model viscous heat-conducting ions in a self-consistent electrostatic field. We first prove the existence and uniqueness of a smooth shock profile of small amplitude by the center manifold theorem, and then establish its large-time asymptotic stability under suitably small smooth perturbations. The proof is based on an energy estimate for a specially constructed integrated system derived from the full compressible Navier--Stokes--Poisson equations. To carry out this estimate, we introduce suitable anti-derivative variables that incorporate the effects of the self-consistent electrostatic potential and heat conduction, thereby reformulating the original system as a dissipative integrated one. The subsequent energy estimates are completed by carefully handling the induced couplings.

We present a method for parametrizing sub-grid processes in the Shallow Water equations. We define coarse variables and local spatial averages and use a feed-forward neural network to learn sub-grid fluxes. Our method results in a local parametrization that uses a four-point computational stencil, which has several advantages over globally coupled parametrizations. We demonstrate numerically that our method improves energy balance in long-term turbulent simulations and also accurately reproduces individual solutions. The neural network parametrization can be easily combined with flux limiting to reduce oscillations near shocks. More importantly, our method provides reliable parametrizations, even in dynamical regimes that are not included in the training data.

We establish an existence of equilibrium result for a class of non-Markovian mean-field games with unbounded control space in weak formulation. Our result is based on new existence and stability results for quadratic-growth generalized McKean-Vlasov BSDEs. Unlike earlier approaches, our approach does not require boundedness assumptions on the model parameters or time horizons and allows for running costs that are quadratic in the control variable.

Solving the two-dimensional shallow water equations is a fundamental problem in flood simulation technology. In recent years, physics-informed neural networks (PINNs) have emerged as a novel methodology for addressing this problem. Given their advantages in parallel computing, the potential for data assimilation and parameter calibration, and the rapid advancement of artificial intelligence, it is crucial to investigate both the capabilities and limitations of PINNs. While current research has demonstrated the significant potential of PINNs, many aspects of this new approach remain to be explored. In this study, we employ PINNs enhanced by dimensional transformation and N-LAAF techniques to validate their effectiveness in solving two-dimensional free surface flow with rainfall on terrain topography. The shallow water equations primarily exist in two forms: the variables form and the conservative form. Through theoretical analysis and experimental validation, we demonstrate that a hybrid variable-conservation form offers superior performance. Additionally, we find that incorporating the energy conservation law, specifically the entropy condition, does not yield substantial improvements and may even lead to training failure. Furthermore, we have developed an open-source module on the PINNacle platform for solving shallow water equations using PINNs, which includes over ten case studies and various equation forms, to promote research and application in this field.

Deep generative models such as flow matching and diffusion models have shown great potential in learning complex distributions and dynamical systems, but often act as black-boxes, neglecting underlying physics. In contrast, physics-based simulation models described by ODEs/PDEs remain interpretable, but may have missing or unknown terms, unable to fully describe real-world observations. We bridge this gap with a novel grey-box method that integrates incomplete physics models directly into generative models. Our approach learns dynamics from observational trajectories alone, without ground-truth physics parameters, in a simulation-free manner that avoids scalability and stability issues of Neural ODEs. The core of our method lies in modelling a structured variational distribution within the flow matching framework, by using two latent encodings: one to model the missing stochasticity and multi-modal velocity, and a second to encode physics parameters as a latent variable with a physics-informed prior. Furthermore, we present an adaptation of the framework to handle second-order dynamics. Our experiments on representative ODE/PDE problems show that our method performs on par with or superior to fully data-driven approaches and previous grey-box baselines, while preserving the interpretability of the physics model. Our code is available at https://github.com/DMML-Geneva/VGB-DM.

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill-conditioning of the optimization problem caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics-aware models. Furthermore, while traditional physics-informed methods solve for a single PDE instance, our approach extends to parametric PDEs. Specifically, we integrate the physical loss gradient with PDE parameters, allowing our method to solve over a distribution of PDE parameters, including coefficients, initial conditions, and boundary conditions. We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets, comparing both training and test-time optimization performance. The code is available at https://github.com/2ailesB/neural-parametric-solver.

I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for this system given bounded initial data. In particular, this shows that the model does not allow for thermal blow-up. If the initial temperature and fuel density also satisfy certain integrability conditions, the L^2-norms of these global solutions are uniformly bounded in time. Additionally, I use a bootstrap argument to show that small initial temperatures give rise to solutions that decay to zero as time goes to infinity, proving the existence of initial states that do not develop into travelling combustion waves.

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 derive a set of recursion relations for the perturbative evaluation of the scattering of two black holes in Einstein gravity. To illustrate, we solve the equations up to third post-Minkowskian order in the gravitational coupling constant G, recovering the correct result, including the back-reaction from gravitational radiation.

Inverse scattering problem for an operator, which is a sum of the operator of the third derivative and of an operator of multiplication by a real function, is solved. The main closed system of equations of inverse problem is obtained. This system contains scattering coefficients and bound state elements as independent parameters. Form of the simplest reflectionless potential (analogous to soliton) is found.

Synthesizing optimal controllers for dynamical systems often involves solving optimization problems with hard real-time constraints. These constraints determine the class of numerical methods that can be applied: computationally expensive but accurate numerical routines are replaced by fast and inaccurate methods, trading inference time for solution accuracy. This paper provides techniques to improve the quality of optimized control policies given a fixed computational budget. We achieve the above via a hypersolvers approach, which hybridizes a differential equation solver and a neural network. The performance is evaluated in direct and receding-horizon optimal control tasks in both low and high dimensions, where the proposed approach shows consistent Pareto improvements in solution accuracy and control performance.

We investigate singularities of radial solutions of the classical two-phase and one-phase Stefan problem. In a remarkable result, M.A. Herrero and J.J.L. Velazquez show existence of a solution such that the free boundary described by g(t) satisfies in dimension ngeq 3 $g(t)=c_n t |log t|^{-1{n - 2}} (1 + o(1)), tto 0,$ and they conjecture that these asymptotics hold in general. Here we prove the conjecture for all radial solutions, which means complete asymptotic rigidity in that class. In order to obtain these precise asymptotics for the two-phase Stefan problem, for which frequency formulas and monotonicity formulas are unknown, we apply the Potential Reduction to ODE method introduced in arXiv:2608.18913 for the Neumann Bernoulli problem. The philosophy of that method is that the asymptotic shape of the free surface is governed by a precise ordinary differential equation. The fact that we can apply it in this paper to an equation of different type with quasilinear principal part, which is in our regime not the perturbation of a linear operator, confirms the versatility of the method.

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.

This project focuses on a method to extract a frequency comb in mechanical means, for general interest and numerous practical applications in MEMS. The method of execution is the implementation of a beam that is exhibiting non-linear dynamics that is perturbed and analyzed for its transverse vibrations. The perturbation is an external harmonic driver with a chosen small amplitude and frequency (which is slightly detuned from the beam eigenfrequency), that when engaged with the unperturbed beam oscillations, causes it reach a state of "injection pulling" - an effect that occurs when one harmonic oscillator is coupled with a second one and causes it to oscillate in a frequency near its own. This causes the beam to reach SNIC bifurcation, rendering a frequency comb as desired. Theoretical analysis showed that the problem can be modelled using a non-linear equation of the beam, that translates to a form of the non-linear Duffing equation. While a solution to the dynamics function of the beam is hard to obtain in practice due to mathematical difficulties, a slow evolution model is suggested that is composed of functions of a amplitude and phase. Using several additional mathematical assumptions, the amplitude is seen to be related to the phase, while the phase equation solution is seen to be of the form of Adler's equation. These assumptions ultimately reduce the entire behaviour of the beam to a relatively simple solution to the Adler equation, which has a known analytical solution. Computerized numerical simulations are run on it to check the results and compare them to the theory and desired outcome. The results agreed with the theory and produce the expected frequency comb, showing the assumptions to be valid in extracting the comb.

This work investigates the use of smooth neural networks for modeling dynamic variations of implicit surfaces under the level set equation (LSE). For this, it extends the representation of neural implicit surfaces to the space-time R^3times R, which opens up mechanisms for continuous geometric transformations. Examples include evolving an initial surface towards general vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of initial conditions. The network training considers two constraints. A data term is responsible for fitting the initial condition to the corresponding time instant, usually R^3 times {0}. Then, a LSE term forces the network to approximate the underlying geometric evolution given by the LSE, without any supervision. The network can also be initialized based on previously trained initial conditions, resulting in faster convergence compared to the standard approach.

Autoregressive learning of time-stepping operators provides an effective approach to data-driven partial differential equation (PDE) simulation, yet for conservation laws, they face a fundamental challenge: learned updates may violate global conservation over long rollouts. For the important subclass of mass-conservation-type equations, the problem is compounded by inherent physical bounds (e.g., nonnegativity or concentrations in [0,1]) whose violation further destabilizes predictions. We introduce FluxNet, which learns cumulative transport amounts representing the total conserved quantity redistributed between each cell and a configurable neighborhood over the full surrogate interval. A conservative update guarantees exact discrete conservation by construction; modular capacity-constrained transport heads (L, U, and D) enforce lower bounds, upper bounds, or near-zero dual-bound violations through architectural design. Unlike flux-rate surrogates that require temporal integration and thus inherit CFL constraints, FluxNet involves no such integration; configurable transport neighborhoods enable large-timestep prediction at full spatial resolution. Ghost cells extend the framework to non-periodic boundaries. Experiments on four benchmarks (1D convection--diffusion, 2D shallow water, 1D traffic flow, 2D Cahn--Hilliard) demonstrate exact conservation, structural bound preservation, architecture modularity, and superior stability over flux-rate surrogates at large temporal strides. The code is publicly available at: https://github.com/Lan-zs/FluxNet.

The statistical mechanics of Gibbs is a juxtaposition of subjective, probabilistic ideas on the one hand and objective, mechanical ideas on the other. In this paper, we follow the path set out by Jaynes, including elements added subsequently to that original work, to explore the consequences of the purely statistical point of view. We show how standard methods in the equilibrium theory could have been derived simply from a description of the available problem information. In addition, our presentation leads to novel insights into questions associated with symmetry and non-equilibrium statistical mechanics. Two surprising consequences to be explored in further work are that (in)distinguishability factors are automatically predicted from the problem formulation and that a quantity related to the thermodynamic entropy production is found by considering information loss in non-equilibrium processes. Using the problem of ion channel thermodynamics as an example, we illustrate the idea of building up complexity by successively adding information to create progressively more complex descriptions of a physical system. Our result is that such statistical mechanical descriptions can be used to create transparent, computable, experimentally-relevant models that may be informed by more detailed atomistic simulations. We also derive a theory for the kinetic behavior of this system, identifying the nonequilibrium `process' free energy functional. The Gibbs relation for this functional is a fluctuation-dissipation theorem applicable arbitrarily far from equilibrium, that captures the effect of non-local and time-dependent behavior from transient driving forces. Based on this work, it is clear that statistical mechanics is a general tool for constructing the relationships between constraints on system information.

In aerodynamics, accurately modeling subsonic compressible flow over airfoils is critical for aircraft design. However, solving the governing nonlinear perturbation velocity potential equation presents computational challenges. Traditional approaches often rely on linearized equations or finite, truncated domains, which introduce non-negligible errors and limit applicability in real-world scenarios. In this study, we propose a novel framework utilizing Physics-Informed Neural Networks (PINNs) to solve the full nonlinear compressible potential equation in an unbounded (infinite) domain. We address the unbounded-domain and convergence challenges inherent in standard PINNs by incorporating a coordinate transformation and embedding physical asymptotic constraints directly into the network architecture. Furthermore, we employ a Multi-Stage PINN (MS-PINN) approach to iteratively minimize residuals, achieving solution accuracy approaching machine precision. We validate this framework by simulating flow over circular and elliptical geometries, comparing our results against traditional finite-domain and linearized solutions. Our findings quantify the noticeable discrepancies introduced by domain truncation and linearization, particularly at higher Mach numbers, and demonstrate that this new framework is a robust, high-fidelity tool for computational fluid dynamics.