Partial Differential Equations

Coarse-grid numerical solvers can substantially reduce the computational cost of time-dependent PDE simulation, but under-resolution often degrades both the trajectory and the spatial fidelity of the solution. We introduce RECAST (Recurrent Error Correction And Super-resolution of coarse-grid Trajectories), a machine-learning framework designed to restore this lost accuracy while retaining coarse-grid evolution. RECAST combines learned correction within the numerical time-stepping loop with reconstruction of the corresponding fine-grid state from the corrected coarse history. We evaluate the framework on six one-dimensional PDE systems spanning transport, diffusion, dispersion, reaction, and wave dynamics, using spatial grids coarsened by factors of 8-16 and 1000-step closed-loop rollouts from unseen initial conditions. Across the test cases, RECAST remains closely aligned with the fine-grid reference solutions and reduces time-averaged relative error by approximately 50-92% compared with the corresponding uncorrected coarse-grid solvers. Additional tests show generalization to unseen PDE parameter values, while comparison with a contemporary coarse-correction architecture shows that RECAST achieves lower error and better long-horizon agreement with the fine-grid reference over 5000-step rollouts. These results demonstrate that the learned correction and reconstruction capabilities of RECAST can enable substantially coarser PDE evolution without the corresponding loss of solution fidelity, providing a proof-of-concept route toward machine-learning acceleration of higher-dimensional numerical simulations across science and engineering.

We systematically investigate finite-difference (FD) derivative computation in Physics-Informed Neural Networks (PINNs) as an alternative to automatic differentiation (AD). On three benchmark PDEs we show that, with a properly calibrated step size, FD matches AD in accuracy on every problem while running faster across the full tested batch-size range and using substantially less GPU memory, and that a stochastic variant we propose outperforms AD on a stationary problem. We further show that for neural architectures with inter-sample dependencies (e.g. BatchNorm, self-attention) the standard PyTorch autograd idiom is silently incorrect; the correct per-sample alternative is computationally infeasible at PINN-relevant batch sizes, while FD provides a forward-only approximation that is empirically an order of magnitude closer to the true per-sample derivative.

Quantifying myocardial perfusion from cardiac magnetic resonance (CMR) can be achieved by fitting tracer-kinetic models to the dynamic contrast-enhanced MR data. However, fitting the observed data with multi-compartment exchange models, which describe the evolution of the contrast agent in the tissue, to estimate perfusion parameters is a challenging inverse problem that is sensitive to noise and acquisition variability. Previously, physics-informed neural networks (PINNs) have been proposed as an alternative to conventional non-linear least squares fitting methods with promising results for quantitative perfusion CMR. In this work, we extend the previously proposed PINN framework with spatiotemporal implicit neural representations (INRs) to represent the MR signal as a continuous spatiotemporal function and to improve the accuracy, smoothness, and physical consistency of the PINN model. In realistic simulated CMR datasets, our proposed PINN with INRs demonstrates improved robustness and parameter estimation accuracy over the previously established methods. The code is available at https://github.com/q-cardIA/pinn-inr.

In recent years, neural networks have significantly advanced numerical solutions of partial differential equations (PDEs). However, solving PDEs with discontinuous solutions, such as hyperbolic conservation laws, remains challenging for neural network-based methods such as physics-informed neural networks (PINNs). Existing methods often rely on strong prior assumptions such as knowledge of discontinuity locations, or they introduce artificial smoothing terms that degrade accuracy. However, accurately solving these conservation laws and predicting the formation and propagation of discontinuities in solutions is crucial in many practical applications, including gas dynamics and traffic flow modeling. In this paper, we introduce a novel Weak-Entropy PINN (WEPINN) framework for hyperbolic conservation laws with discontinuous solutions. The method enforces the governing equations in their weak (integral) formulation and incorporates the entropy condition to select the physically admissible solution, while employing the discrete fast Fourier transform (DFFT) for efficient numerical integration. Our method is tested through extensive numerical experiments on a variety of scalar conservation laws and systems of conservation laws in one and two dimensional spaces. These experiments demonstrate that our method can accurately resolve sharp discontinuities while effectively capturing interactions between multiple shock and rarefaction waves.

Hamiltonian Neural Networks (HNNs) parameterize conservative dynamics through a learned scalar Hamiltonian, providing an architectural prior that is absent from generic vector-field neural networks. We evaluate this prior under a controlled protocol in which an HNN and a parameter-matched feedforward baseline are trained on the same RK4-generated trajectories, use the same central-difference derivative targets and optimization settings, and are integrated at inference with the same RK4 scheme. Results are reported over five independent training seeds. On the nonlinear pendulum, the HNN reduces mean energy drift by 42-fold and mean trajectory MSE by 15.8-fold at T = 100, approximately 16 pendulum periods. Its energy drift also remains bounded and exhibits substantially lower seed-to-seed variability than the standard-network baseline. An energy-stratified analysis shows that the difference becomes more pronounced as trajectories explore more nonlinear regions of phase space. As an additional diagnostic, we examine an explicit Störmer--Verlet-style rollout of the learned HNN. Because the learned Hamiltonian is not constrained to the separable form H(q,p) = T(p) + V(q), the standard symplecticity guarantee of velocity Verlet does not directly apply. We further apply the same matched-integrator protocol to the three-dimensional Kepler two-body problem. The HNN again exhibits lower trajectory, energy, and angular-momentum drift than the parameter-matched baseline. These experiments provide a controlled study of how Hamiltonian parameterization affects long-horizon prediction and physical consistency across two conservative dynamical systems.

Operator learning is a rapidly advancing area of computational science. It is particularly well suited to problems where a partial differential equation (PDE) must be solved repeatedly under varying physical configurations. Most existing architectures represent the solution operator in a fixed basis. While this assumption is well aligned with global structures, it is less suitable for phenomena governed by local interactions in physical space. We explore an alternative perspective motivated by the observation that the continuum limit of coupled oscillator systems can describe a broad class of PDEs. Building on this idea, we introduce the Kuramoto Neural Operator (KNO), which represents the solution through the evolution of a latent field of interacting oscillators. Across a diverse collection of PDE benchmarks, KNO achieves strong predictive performance, with improvements over competing approaches. Our experimental evaluation also includes an extensive ablation study that quantifies the contribution of each architectural component incorporated into KNO. Furthermore, we show that the model's prediction error is closely linked to the collective dynamics of the latent oscillators. It varies systematically with their degree of synchronization, providing insights into the underlying mechanisms.

Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces. However, we reveal that existing learnable projection mechanisms cannot ensure stable and balanced assignments from observation points to latent tokens, causing some latent tokens to be over-assigned while others remain underutilized. This limitation further restricts the design of hierarchical architectures, as assignment imbalance is continuously inherited and amplified across latent spaces, eventually causing severe token collapse in deeper spaces. To address these issues, we propose MoNo (Multiscale Optimal Transport Neural Operator), a progressive multiscale neural operator that efficiently solves PDEs on general geometries through stable latent-space construction. At its core is CoTAP (Cross-scale Optimal Transport Assignment and Projection), a novel latent-space construction method that formulates cross-space assignment between adjacent spaces as an entropy-regularized optimal transport problem, thereby constructing balanced bidirectional projections and stable latent spaces. CoTAP also ensures stable information transfer across multiple latent spaces, further enabling multiscale architectures on general geometries, which in turn support more efficient learning of long-range physical interactions. Extensive experiments demonstrate that MoNo outperforms existing state-of-the-art neural operators in both prediction performance and computational efficiency. Code is available at https://github.com/ZijiangY1116/MoNo.

In modern industry, keeping complex systems reliable, safe, and efficient hinges on Prognostics and Health Management (PHM). Machine Learning (ML) has largely driven advancements in diagnostics and prognostics, yet purely data-driven models face inherent limitations, such as poor generalization, an inability to infer causal relationships, and a lack of interpretability. Physics-Informed Machine Learning (PIML) helps mitigate these limitations by incorporating prior physical knowledge directly into the ML pipeline, thereby fostering growing interest in its application to PHM. This work investigates how PIML is being leveraged in the context of PHM through a systematic literature review of 212 studies. The review introduces a four-class classification scheme, consisting of observational bias, inductive bias, learning bias, and hybrid approaches, and further categorizes studies by PHM task. Across all four classes, the reviewed studies consistently demonstrate improved predictive performance over conventional baselines across a broad range of assets, although the literature is heavily skewed toward lithium-ion batteries and bearings, and dominated by problem-specific solutions. Overall, the review indicates that physics-informed approaches already provide tangible benefits, whereas claims of improvements concerning some of the aforementioned limitations lack sufficient supporting evidence. Future research should prioritize transferable design patterns, benchmarks comparing integration strategies, and uncertainty-aware models that are lightweight and robust enough for online deployment in real-world settings.

In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing. Building on the modified Walk-on-Spheres algorithm of Beznea et al. (arXiv:2209.01432), we introduce Monte Carlo estimators that explicitly incorporate sampled random times arising in the analyzed stochastic representations. We establish uniform error bounds for these estimators and show that, under suitable assumptions, a prescribed approximation accuracy is achieved with sample complexities growing at most polynomially in both the inverse accuracy and the problem dimension. Furthermore, we prove a deep neural network approximation result for the stochastic representations. Assuming suitable neural network representations of the boundary data and the distance function to the boundary, we use the constructed Monte Carlo to design deep neural networks that approximate the representation uniformly with a number of parameters growing at most polynomially in the inverse accuracy and the problem dimension. These results extend previous complexity analyses to a broader class of elliptic equations involving drift and killing.

Recently, tensor-based physics-informed neural networks (T-PINNs) have received increasing attention. However, existing T-PINNs still face a fundamental challenge: they mainly rely on pre-specified low-rank tensor decompositions with manually tuned ranks, which limits their ability to capture the underlying structures of multivariate solution functions and hinders their practical deployment. To address this challenge, we propose a hierarchical rank-evolving (abbreviated as HRE) representation for multivariate functions, which endows us to faithfully capture the underlying structure of the targeted multivariate function accompanying with automatic rank determination. Concretely, in the hierarchical design of HRE representation, the target multivariate function is decomposed as a small-scale inner tensor with a set of univariate functions along each mode, where a customized tensor network decomposition can be readily deployed to capture the underlying structure of the small-scale inner tensor. In HRE representation, the crucial hyperparameters, ranks, can be adaptively revealed during the decomposition, freeing us from manual rank tuning and making HRE practically applicable to real-world problems. Besides, we build the HRE-PINNs correspondingly. Extensive numerical experiments, including high-dimensional static problems (Helmholtz equation and Poisson equation), nonlinear time-dependent problems (Klein-Gordon equation), and complex fluid-dynamics problems (flow mixing equation and Navier-Stokes equation), demonstrate that HRE-PINNs consistently outperform existing state-of-the-art approaches in terms of accuracy.

Electromagnetic inverse scattering is a nonlinear and ill-posed computational imaging problem, where accurate reconstruction is challenging due to measurement limitations, noise, and high computational costs, especially for 3-D imaging. Although physics-driven neural networks (PDNNs) reduce the dependence on labeled training data, existing accelerated PDNN frameworks often rely on preliminary reconstruction-based region selection, which may introduce instability when the selected region is inaccurate. In this paper, a coordinate-residual physics-driven neural network (CRPDNN) is proposed for 3-D electromagnetic inverse scattering. CRPDNN represents the unknown complex contrast distribution using normalized spatial coordinates and a residual convolutional network, whose parameters are optimized by enforcing consistency between the measured and model-predicted scattered fields. Unlike existing subregion-accelerated PDNN approaches, CRPDNN does not require a preliminary reconstruction, thereby avoiding dependence on its accuracy. For the reported noise-free 3-D synthetic cases, CRPDNN achieves an average relative error of 2.10%, compared with 7.97% for CSI and 3.99% for $L_{2/3}$-FBE-WCIE, while providing approximately 5.5- and 12.1-fold speedups over the two baselines, respectively. Additional 2-D comparisons further demonstrate its stability and computational efficiency relative to existing PDNN frameworks. CRPDNN also maintains reliable reconstruction performance under noisy measurements, and the 3-D Fresnel experiments further indicate its potential for practical imaging applications.

Recent advances in Passive Acoustic Monitoring (PAM) offer an opportunity to obtain ecological spatial point-process data at unprecedented scale. However, realizing this opportunity necessitates the development of accurate and scalable localization methods. In real-world outdoor soundscapes, however, the assumptions underlying classical localization methods such as hyperbolic and score-based localization are routinely violated by multipath dominance, near-field effects, and complex propagation. Under these conditions, classical localization methods become brittle, with extreme errors possible even in small detection arrays. Rather than statistically replacing the underlying physics, we propose a method to refine it and increase robustness outside of ideal operating conditions: a learned model operating on physics-informed acoustic features corrects a fast hyperbolic solver where it produces implausible solutions, substantially reducing catastrophic worst-case errors while matching its median accuracy on field data. We further provide calibrated, geometry-aware uncertainty estimates suitable for propagation into downstream spatial models. Evaluating on distributed microphone arrays in real and simulated outdoor environments, we demonstrate that the proposed method yields robust, uncertainty-aware localization, providing a step toward scalable automated wildlife monitoring in complex acoustic environments.

Fourier neural operators (FNOs) provide efficient nonlocal spectral learning, but varying geometries and independently chosen discretizations remain difficult to accommodate. We introduce the ambient-domain extension Fourier neural operator (ADEx-FNO), a deterministic framework that incorporates geometry without modifying the defining Fourier-operator layers. Each physical domain is embedded in a fixed ambient hypercube and represented by a signed distance function. Inputs and solution fields are deterministically extended to the ambient domain, transferred to a common, potentially nonuniform rectilinear latent grid, processed by the FNO, then interpolated to an independently chosen target discretization and restricted to the physical domain. All geometry-transfer operations lie outside the optimization procedure and require no trainable graph, point-cloud, deformation, or geometry-decoding modules. ADEx-FNO achieves relative l2 errors of 0.32%-0.77% on held-out smooth-domain nonlinear Poisson and advection-reaction-diffusion problems in 2D and 3D, and is also evaluated on unseen nonsmooth geometries. A single ADEx-FNO inference is then used to initialize conventional CFD solvers. For all 29 converged 2D and 3D RANS cases, pseudo-time iterations decrease, with mean reductions of 44.17% and 43.03%, respectively, with comparable gains across three mesh resolutions. URANS cases reduce post-window physical-time advances by 18.52%-27.51%. In transfer from 2D URANS training data to DNS at different Mach and Reynolds numbers, the bootstrap interval decreases by 23.47%-48.21%, depending on the target statistic. In all CFD tests, ADEx-FNO provides only the initial field; the governing-equation solver controls the subsequent solution, while physical or statistical consistency is assessed separately from computational savings.

Accurate models of underwater vehicle motion are needed for autonomous execution of marine tasks like infrastructure inspection and scientific sampling. However, such motion is challenging to characterize using traditional physics-based methods. This paper presents a novel data-driven framework for learning stochastic underwater vehicle dynamics. Using Euler-Poincaré dynamics and the geometry of Lie groups, we develop a stochastic physics-informed neural network architecture that respects the physical and geometric constraints of underwater vehicles. Our approach leverages structure-preserving stochastic integration and builds upon moment matching and finite dimensional matching to ensure geometrically-consistent training. We evaluate our approach in simulation and on an underwater vehicle navigating dock pylons in a harbor environment. The results demonstrate that our method learns accurate and robust dynamics models, enabling safe model-based control in challenging marine environments.

Physics-informed neural networks applied to the level-set formulation of interface advection commonly augment the residual and initial-condition losses with an eikonal regulariser, penalising the deviation of $\|\nablaφ\|$ from unity. A previous two-dimensional study identified this weight as the dominant hyperparameter and found its optimum shifts by four orders of magnitude between rigid-body and deforming flows, but left open whether these principles transfer to three dimensions and whether single-seed results survive run-to-run variability. We answer both by repeating the weight selection across four 3D benchmarks (translating sphere, rotating sphere, slotted sphere, reversed vortex), sweeping six weights with three seeds at full training budget under a pre-registered selection rule. The ordering transfers: the selected weight tracks how far the exact solution departs from the signed-distance property, spanning four decades from $10^{-1}$ where it holds exactly to $10^{-5}$ where the interface is stretched. Values transfer only benchmark by benchmark; two of four carry over unchanged and two do not, so inheritance must be verified. The multi-seed protocol reveals that at small weights the seed-to-seed standard deviation equals the error itself, and the regulariser reduces it by more than an order of magnitude, buying reproducibility as well as accuracy. We benchmark against a fifth-order WENO solver on identical grids and error measures; the classical scheme is more accurate on all four problems, by two orders of magnitude on smooth rigid advection, with a margin that narrows with geometric difficulty and is smaller in volume conservation than in the field norm. Finally, we show that the relative $L_2$ error cannot certify the preservation of thin features, and report a feature-restricted measure that can.

In this work, viscous fluid flow governed by the Stokes equations in highly perforated domains is studied using physics-informed neural networks (PINNs). Perforated microstructures induce complex boundary conditions and fine-scale flow features that are difficult for standard neural networks to resolve. Conventional PINNs, even when combined with advanced training techniques, can suffer from a loss of accuracy and efficiency as the number of perforations increases. One important source of this difficulty is the soft enforcement of boundary conditions through penalty terms, which can lead to stiffness, gradient conflicts, and poor resolution of near-boundary flow structures. Hard constraints provide an alternative by encoding boundary conditions exactly into the network ansatz, but may introduce undesirable non-local effects due to the global nature of the approximation. To address these challenges, finite basis PINNs (FBPINNs), which are based on domain decomposition and localisation principles, are used together with hard boundary constraints that efficiently encode perforation-related boundary conditions. This approach helps mitigate spectral bias, improves overall accuracy, and exhibits convergence that is only weakly affected by the number of perforations, thereby providing an efficient and highly parallelisable neural network framework. The proposed approach is further supported with theoretical arguments, specifically focusing on the localisation and approximation properties of FBPINNs.

Pretrained partial differential equation (PDE) foundation models can generalize across different equations, but adapting them to unseen PDE systems typically requires dense solution data, which is often expensive or unavailable. To address this limitation, we propose an unsupervised PDE-based finetuning framework that eliminates the need for ground-truth solutions. We first pretrain a neighborhood attention Transformer on diverse time-dependent PDEs spanning varying spatial scales, yielding transferable representations across heterogeneous equations. In the adaptation stage, we construct a physics-based objective using the PDE residual and boundary conditions, and finetune the model on unseen equations via low-rank adaptation (LoRA). To address the uneven learning across physical quantities in standard LoRA, we introduce NSLoRA, a Newton-Schulz orthogonalized variant that rebalances adaptation. Our method achieves performance comparable to supervised LoRA finetuning without requiring any ground-truth solutions, while consistently outperforming competitive neural operator baselines and recent PDE foundation models across heterogeneous PDE benchmarks spanning multiple spatial dimensions.

Neural operators have become a central tool for solving partial differential equations (PDEs), with spectral operators offering efficient global mixing across spatial locations. However, many PDEs contain physics-sensitive local structures that are critical to the underlying physical behavior. For example, in Darcy flow, local material interfaces are often reflected by sharp changes in the permeability field and can strongly influence the solution. Existing spectral operators primarily adapt modal mixing based on center-point representations, making them insufficiently responsive to such localized structural variations. We propose the Edge-Conditioned Spectral Operator (ESO), a novel spectral operator framework that modulates global spectral mixing using local edge-wise variations. By incorporating the Pairwise-Variation Modal Mixer (PVMM) to inject local edge information into spectral mode selection, ESO preserves the global approximation capability of spectral neural operators while enabling the learned kernel to adapt to physics-sensitive local structures. Furthermore, we introduce a task-adaptive Physics-Aware Reweighting (PAR) that emphasizes physically important regions, identified by taskspecific physical quantities. Across nine PDE benchmarks, ESO consistently achieves state-of-the-art performance. Visual and region-wise analyses further demonstrate that ESO reduces solution errors near coefficient jumps, high-gradient flow structures, and other physically sensitive regions. The code is available at https://github.com/Tanpig-X/ESO.

In this paper, we extend a weak adversary neural network pushforward method for solving time dependent Boltzmann equation and a weak formulation of the collision operator is proposed where an invertible neural pushforward mapping is used to generating samples given by the distribution governed by the Boltzmann equation. The training of the pushforward mapping is learnt by enforcing the weak form of the Boltzmann equation. Numerical results have demonstrated the effectiveness of the proposed method.

We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations $\mathfrak L_βu=f$ using linearized ReLU$^k$ neural networks on the sphere, where $\mathfrak L_β$ is a positive elliptic spectral multiplier of order $β$. Given a parameter set $Θ_n=\{θ_{j}^*\}_{j=1}^n\subset\mathbb S^d$, we approximate $u$ in the linearized network space $L_n^k(Θ_n)$ by the discrete residual on the collocation points $\{η_i^*\}_{i=1}^m$ \begin{equation*} u_{n,m}\in\arg\min_{v_n\in L_n^k(Θ_n)}\frac1m\sum_{i=1}^m\left(f(η_i^)-\mathfrak L_βv_n(η_i^)\right)^2. \end{equation*} With $k>\frac{d-1}{2}+β$, for antipodally quasi-uniform network parameter sets and any quasi-uniform collocation points with $m\gtrsim n$, we prove that \begin{equation*} |u-u_{n,m}|{\mathcal H^β(\mathbb S^d)}\eqsim|f-\mathfrak L_βu{n,m}|{\mathcal L^2(\mathbb S^d)}\lesssim n^{-\frac{r}{d}} \begin{cases} |f|{\mathcal W^{r,p}(\mathbb S^d)},&\frac{d}{p}<r\leq \frac{d}{2},~p>2,\ |f|{\mathcal H^r(\mathbb S^d)},&r>\frac{d}{2}. \end{cases} \end{equation*} We also establish a high-probability residual estimate, up to a logarithmic factor and an arbitrarily small smoothness loss, for i.i.d.\ uniformly distributed collocation points. The key analytical ingredient is a Bernstein inequality for linearized ReLU$^k$ network spaces. If $\underline h$ denotes the antipodal separation distance of the network parameters, then \begin{equation*} |v_n|{\mathcal H^r(\mathbb S^d)}\lesssim\underline h^{-(r-s)}|v_n|_{\mathcal H^s(\mathbb S^d)},\qquad 0\leq s<r<k+\tfrac12. \end{equation*}

Machine learning offers a promising avenue to accelerate physical simulations by replacing computationally expensive traditional Partial Differential Equation (PDE) solvers with fast, differentiable surrogate models. However, standard auto-regressive ML emulators often suffer from error accumulation over long horizons and struggle to capture the stochasticity of complex physical systems. In this paper, we propose Kastor, a comprehensive methodology to adapt a deterministic physics foundation model into a highly efficient and accurate generative surrogate. First, we introduce a two-stage inference scheme that combines a large-stride causal auto-regressive model with a non-causal temporal super-resolution network, significantly reducing error accumulation while minimizing computational cost. Second, we present Mean prediction regularization (MPR), a novel training objective that constrains the generative model to predict the deterministic distribution mean under null noise conditioning. This regularization dramatically improves the performance and stability of both Functional Generative Networks (FGN) and diffusion-based emulators. Finally, we demonstrate that incorporating spatial gradient matching improves the accuracy and physical fidelity of the simulations as measured by power spectrum density. Extensive evaluations on diverse simulation datasets of the benchmark The Well show that with these components, our model outperforms competing methods in forecasting accuracy, spectral consistency, and computational efficiency. Our model achieves a 42.9% average reduction in forecasting compared to our reference based on the Walrus finetuning methodology, and outperforms Walrus for 8 out of 10 datasets on variance-normalized RMSE (VRMSE).

Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-frequency features; and representation-coefficient coupling, the entanglement of representation learning and coefficient fitting within a single nonconvex optimization objective. In this work, we propose the Fourier-enhanced alternating Levenberg--Marquardt PINN (FALM-PINN), an optimization framework that decouples representation learning from coefficient fitting. The upper-level problem learns a Fourier-enhanced basis that enriches the latent space with high-frequency components, while the lower-level problem resolves the coupling by fitting the projection coefficients on this basis, solving a nonlinear least-squares problem with the Levenberg--Marquardt algorithm. The framework applies to general nonlinear and coupled PDE systems, and reduces to a single-step convex optimization problem for linear PDEs. We prove global convergence of the alternating training scheme in both cases. Numerical examples on multiple challenging high-frequency and nonlinear PDEs show that FALM-PINN achieves relative $L^2$ errors up to two orders of magnitude lower than state-of-the-art baselines.

Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets. Parameterized PINNs (ParamPINNs) further take physical parameters as inputs, allowing a single model to represent a family of PDE solutions over a parameter domain. Existing ParamPINNs, however, still face inefficient training, uneven accuracy across parameters, and overfitting to a limited set of sampled parameter tasks, which can impair generalization to unsampled parameters. To address these issues, we propose a continual-learning physics-informed neural network (CL-PINN), which treats PDE instances at different parameter values as related tasks and learns them sequentially. CL-PINN combines Bayesian-optimization-based active parameter selection, task-wise dynamic loss weighting, sparse physics-constrained replay, and an optional parameter subnetwork to improve task allocation and knowledge retention under bounded active-task capacity. It requires no observational data and is designed to solve parameterized PDEs over relatively broad parameter domains under limited computational resources. Multi-seed evaluations on five benchmarks, including one continuous function and four parameterized PDEs, show that Bayesian selection substantially reduces objective-loss queries relative to grid-greedy search, while sparse replay mitigates forgetting of earlier tasks. Under the prescribed within-case resource protocols, CL-PINN generally provides higher and more balanced solution accuracy than fixed-sampling and grid-greedy baselines. CL-PINN offers a practical route toward learning PDE solutions that generalize across physical parameters and has the potential to support reusable physics-informed surrogates for large-scale engineering parameter studies.

Functional flow matching is posed on distributions of functions but implemented from finitely many coefficients or point values. Under scattered or adaptive refinement, the resulting conditioning sigma-algebras need not be nested, so martingale convergence does not justify the sensor limit. We prove strong $L^2$ convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions, with quantitative bounds for orthogonal projections and a point-sensor extension through a regularity space. For learned flows, coupling directly to a population superposition path yields an end-to-end Wasserstein bound without assuming uniqueness of the population finite-dimensional ODE. We verify sensor-independent constants for a normalized quadrature neural operator, including globally Lipschitz activations through an explicit magnitude recurrence. A noncommuting trace-class Gaussian example gives boundary multiplier $0$ under projected restriction and $0.72$ under exact conditioning. A spatial regularity--cubature certificate closes the operator-realization term, a Bernstein argument gives a $\widetilde{O}(n^{-1})$ excess-risk term for fixed model dimension and envelopes, and an exactly realizable clipped Gaussian scaling specialization yields an explicit end-to-end rate.

Neural PDE solver auto-design is fundamentally a search-space representation problem. In the space of unrestricted Python programs, valid solvers form an extremely sparse subset: most candidate programs are syntactically incorrect, semantically incompatible, or numerically unstable. Direct code generation therefore forces an LLM to spend most of its search capacity navigating implementation failures rather than reasoning about solver quality. ADSL-PDE addresses this challenge by introducing a structured search state between solver concepts and executable code. It represents the functional decisions that determine a neural PDE solver (architecture, physical constraints, objectives, sampling, and optimization) while abstracting away low-level implementation details. A deterministic compiler maps each valid search state to an executable solver. In effect, ADSL-PDE reshapes the search space: it removes large regions of invalid programs, increases the density of meaningful candidates, and preserves the compositional freedom needed to discover previously unseen designs. Solver evolution can thus operate over design decisions rather than code artifacts. Built on this representation, our evolutionary agent iteratively proposes, evaluates, and refines solver search states using empirical feedback. Across multiple PDE benchmarks, ADSL-PDE improves both search efficiency and optimization stability, achieving an improvement of more than 52% within the first ten evolution iterations. These results suggest a broader principle for LLM-driven auto-design: effective agents do not merely require stronger reasoning, but rather a search representation that concentrates exploration on valid and consequential decisions.

Deep learning models for scientific spatio-temporal downscaling often minimize reconstruction error while failing to preserve physically meaningful multi-scale structure. For sea surface temperature prediction, this can yield outputs that are numerically plausible yet overly smooth, missing mesoscale variability critical to regional ocean dynamics. Existing methods often focus on pixel-wise objectives or single-context conditioning, which limits their ability to preserve spectral fidelity and generalize across regions. To address this, we propose EddyFlow, a representation learning framework for kilometer-scale sea surface temperature downscaling that balances predictive accuracy, scale-dependent structure, and regional generalization. EddyFlow is trained on the Gulf of St.~Lawrence and evaluated in zero-shot and few-shot settings on the Bay of Fundy and the Gulf of Mexico. EddyFlow demonstrates that physics-informed representation learning reduces zero-shot RMSE by 21%, achieves up to 85.6% skill relative to persistence on unseen domains, and maintains near-ideal spectral fidelity with a PSD ratio of $\approx 1.00$.

Physics-informed neural networks (PINNs) often require high-accuracy quasi-Newton refinement to obtain reliable partial differential equation solutions, but their residual objectives can exhibit indefinite, nearly singular, and poorly scaled local curvature. Regularized quasi-Newton methods provide established mechanisms for stabilizing secant models, while self-concordant methods provide local-metric rules for curvature-dependent step selection. Building on these two lines of work, we propose SCORE, a self-concordance-inspired quasi-Newton method with decrement-coupled shifted secant geometry for PINN training. Its distinguishing mechanism is that a single quasi-Newton decrement computed from the learned inverse metric jointly determines a strong-Wolfe-tested candidate step and an adaptive shift used to define the next secant geometry. The shifted displacement represents the action of an averaged shifted metric along the accepted step, while requiring neither Hessian construction nor Hessian-vector products. Under a local spectral-equivalence condition, we show that the quasi-Newton decrement and candidate step remain comparable to their counterparts in a positive shifted metric, and recover the normalized self-concordant rule in the matched-metric case. Strong Wolfe acceptance, fallback line search, and standard curvature safeguards provide globalization without modifying the underlying PINN objective. Experiments on the viscous Burgers, Kuramoto--Sivashinsky, Korteweg--de Vries, and complex Ginzburg--Landau equations show that SCORE attains lower final errors than the tested BFGS and self-scaled Broyden baselines. The Burgers ablation further indicates that shifted curvature stabilization and decrement-based step selection make complementary contributions to high-accuracy refinement.

Engineered Skeletal Muscle Tissues (ESMs) have become a key structure for biomedical disease modeling and pharmacological screening, yet their functional characterization often relies on simplistic metrics like peak force, discarding critical kinetic information. This is partially due to the high level of mathematical complexity which mechanistic models introduce to capture these dynamics. Hence, exactly the complexity prevents scalable application and widespread adaptation in the field. Here we present a Physics-Flavored Neural Network (PFNN) that automates the kinetic phenotyping of ESMs. Our architecture integrates a stretched-exponential physical model into a CNN-Transformer, enabling the extraction of physically meaningful parameters directly from force-time profiles. To address the scarcity of labeled biological data, we employ a hybrid training paradigm: the model develops a "physical intuition" on synthetic data before undergoing unsupervised self-alignment on unlabeled real-world measurements. Our results demonstrate that this physics-flavored approach achieves high-fidelity parameterization across diverse contractile phenotypes and cell lines, including Duchenne Muscular Dystrophy models. Our scalable, self-improving pipeline bridges the gap between idealized biophysics and noisy \emph{in vitro} data, providing a robust tool for high-throughput biophysical research.

Partial differential equations (PDEs) become actionable in science and engineering not as isolated formulae, but as executable workflows that connect modelling assumptions, governing equations, numerical solvers, diagnostics, and decisions. Large language models (LLMs) are beginning to support such workflows by linking natural language, symbolic mathematics, code, solver outputs, and feedback. Here we examine recent advances in LLM-assisted PDE research across three stages: the discovery and formulation of governing models, the generation and revision of executable numerical solvers, and the use of simulation feedback to support control, design, and optimization. Across these stages, current systems act primarily as workflow-level interfaces. Despite this progress, the field remains limited by the scarcity of high-quality datasets and benchmarks, especially for knowledge discovery and real-world applications, where expert annotation, executable problem construction, and task-level feedback require substantial domain effort. A further challenge is the persistent gap between simulation-based results and real-world scientific and engineering systems, which limits the direct transfer of numerical simulations, control policies, and optimized designs to practical settings. These challenges make LLM-assisted PDE workflows a critical testbed for developing scientific AI systems that can connect language, computation, physical constraints, and real-world decision-making.

Non-intrusive reduced-order models (NIROMs) have become a standard tool for approximating parametric partial differential equations from computer design of experiments while significantly reducing computational costs. However, assessing the reliability of their predictions remains a major challenge, particularly in extrapolation regimes or under limited training data. In this work, we introduce a framework for quantifying model-form uncertainty in NIROMs by combining a perturbative stochastic representation of reduced bases with distribution-free conformal-type methods. Starting from a deterministic reduced basis constructed from snapshot matrices, we model uncertainty through random perturbations defined on the Stiefel manifold, directed along the discarded modes, yielding stochastic reduced-order approximations whose induced variance reflects the basis-truncation error. A transport approximation gives a closed-form posterior variance that separates basis-induced from regression-induced uncertainty, without re-training the underlying Gaussian processes. We include this posterior variance within a conformal risk control calibration framework, that provides prediction sets with coordinate miscoverage guarantees. The calibration factor produced by this framework is itself an interpretable, scalar diagnostic of the quality of the uncertainty estimate. The methodology is evaluated on parametric PDE benchmarks and an industrial tire-manufacturing calendering process. Numerical experiments demonstrate reliable, locally informative uncertainty quantification that goes beyond the Gaussian predictive variance.

Magnetic components in high-frequency, high-power-density converters are increasingly driven by non-sinusoidal flux-density waveforms with fast transitions, minor-loop operation, dc bias, and temperature variation. Under these conditions, steady-state core-loss formulas and single-valued material curves cannot fully capture transient magnetization responses. This work proposes the Physics-Informed Hybrid Neural Operator (PI-HNO), a compact material-specific neural model with B-H energy-consistency regularization for core-loss-oriented transient magnetization prediction. Given the measured B(t)-H(t) history, the input B(t) series over the prediction interval and operating-condition information, PI-HNO predicts the H(t) series and the corresponding reconstructed B-H trajectory. The model integrates a local recurrent branch for boundary-state representation and rate-dependent response evolution with a Preisach-inspired global branch that extracts waveform-level hysteresis context. Evaluation on the MagNetX transient database using material-specific models for 14 ferrite materials demonstrates that PI-HNO achieves a compact trade-off between sequence accuracy and B(t)-H(t) energy consistency, with the mean and 95th percentile B(t)-H(t) energy consistency errors of 1.92% and 7.60%, respectively, using only 4777 trainable parameters per model. Ablation studies further demonstrate that the local, global, and energy-aware regularized components provide distinct contributions to transient magnetization prediction.

Classical neural networks frequently produce overconfident predictions on ambiguous or out-of-distribution (OOD) data, a liability that grows with each AI system deployed in safety-critical real-world scenarios. Bayesian neural networks (BNNs) provide a principled framework for uncertainty-aware prediction by replacing deterministic parameters with probability distributions, but repeated sampling increases latency, memory traffic, and energy consumption. Photonic probabilistic computing offers a promising alternative by exploiting intrinsic optical stochasticity for fast and parallel sampling. However, photonic BNNs are not ideal samplers: analog constraints on quantization, programming error, dynamic range, and representable mean and variance restrict the variational families that can be implemented in hardware. In this work, we study which hardware-imposed constraints limit scalable photonic BNN inference, how these constraints can be represented, and which ranges can be tolerated by photonic BNNs beyond small proof-of-concept networks. We formulate photonic BNN inference as constrained stochastic variational inference and perform a systematic ablation study over stochasticity location, stochasticity modality, quantization, programming error, and mean/variance bounds. From these results, we derive concrete co-design guidelines that distinguish hardware constraints that can be compensated by training from those requiring hardware or architecture intervention. We validate these guidelines under coupled, hardware-realistic constraints on Dirty-MNIST, CIFAR-10, and CINIC-10, using Fashion-MNIST and SVHN as OOD benchmarks, showing that hardware-aware training recovers predictive performance and uncertainty quality whenever the required variational family remains representable, whereas violations of representational limits require targeted hardware modifications.

Physics-informed neural networks (PINNs) combine sparse observations with physical equations, providing an important approach for modeling complex plasma processes and inferring unknown physical quantities. The steep-gradient pedestal of high-confinement-mode tokamaks is closely linked to plasma confinement and edge transport. Analyzing ion-temperature-gradient (ITG) drift waves in this region requires jointly identifying complex eigenfrequencies and reconstructing two-dimensional complex-valued mode fields. Localized high-frequency oscillations, strong real-imaginary coupling, and nonlinear coupling between the mode field and eigenfrequency challenge PINN representation and joint optimization. To address these challenges, we propose a physics-informed neural framework combining Fourier feature encoding, complex-valued feature propagation, and three-stage training. Under sparse observations and physical constraints, it jointly solves for the complex eigenfrequency and mode field of a representative ground-state ITG branch. Experiments show that the framework accurately recovers the target complex eigenfrequency and two-dimensional complex-valued mode field and outperforms representative PINN baselines. It also provides a basis for analyzing higher-order and multiple-branch drift-wave modes.

Robust to intensity inhomogeneity, the local Chan--Vese (LCV) model extends the classical Chan--Vese (CV) image segmentation method by incorporating local statistical information around each pixel. Originally, the LCV model was solved using a finite difference scheme, following the approach used for the CV model. As an alternative to the finite difference scheme, a more efficient algorithm based on the Merriman-Bence-Osher (MBO) scheme was later developed for the CV model. In this paper, we derive a similar MBO-based algorithm to solve the LCV model and propose an efficient implementation. The algorithm is developed for both two-phase and multiphase segmentation, and an extension to color images is also discussed. To demonstrate the effectiveness of the proposed approach, we apply it to a variety of grayscale and color images, including medical and microscopy images.

Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.

Hard-constrained recurrent physics-informed networks (HRPINNs) embed known dynamics inside a recurrent numerical integrator and restrict a neural branch to learning only the residual dynamics that the first-principles model does not capture. Kolmogorov--Arnold Networks (KANs) have been proposed as parameter-efficient replacements for multilayer perceptrons (MLPs) in such residual branches, but their learnable B-spline activations follow a markedly different execution profile. Building on prior work that characterized when a vanilla B-spline KAN matches or underperforms an MLP as an HRPINN residual branch in discovery accuracy, this paper asks whether that parameter efficiency survives deployment. Using identical trained weights, we measured execution latency, energy per integration step, and dependability under post-training quantization in the closed recurrent loop on a RISC-V RV64GC platform without vector extensions (StarFive VisionFive~2, SiFive U74). For the two accuracy-comparable pairs, the KAN residual branch executed $13.5\times$ and $8.0\times$ slower and consumed $11.3\times$ and $5.6\times$ more energy per integration step (3.7,$μ$J against 0.33,$μ$J for the smallest pair); across all four parameter-matched size tiers the ranges are $4.7\times$--$14.5\times$ and $4.7\times$--$18.7\times$. Under INT8 quantization, KAN trajectories diverged up to $43\times$ earlier than matched MLPs; the damage traces to weight quantization, not to input-side knot-interval misassignment. These results indicate that the parameter efficiency reported for KANs does not transfer to deployment cost on scalar embedded cores, and that an MLP residual branch is the more dependable default for embedded HRPINN deployment unless specific quantization co-design is used.

Nonlocal partial differential equations arise in many applications but are often difficult to model and learn because of the presence of nonlocal operators. We present a flow-map learning (FML) framework for modeling unknown nonlocal PDEs directly from solution data. Rather than learning or approximating the underlying nonlocal operators, the proposed approach learns the finite-time evolution operator in either modal or nodal space. Two complementary formulations are developed for spectral and grid-based solution representations. Numerical experiments on one- and two-dimensional fractional diffusion and wave equations demonstrate accurate and stable long-time prediction using only short observation windows. The proposed approach provides an effective data-driven framework for learning unknown nonlocal dynamics without explicit evaluation of nonlocal operators.

This paper proposes a localization framework that uses spatiotemporal fields governed by partial differential equations (PDEs) as localization signatures. Two PDE classes are considered: the shallow water equations, which describe free-surface flows in coastal and riverine environments, and the advection-diffusion equation, which models the transport and mixing of scalar quantities such as temperature, salinity, and dissolved oxygen. A numerical PDE solver provides predicted fields over the domain, and multiple field channels are fused as multimodal measurements to improve localization accuracy. We formulate the problem within a Rao-Blackwellized particle filter (RBPF) that partitions the vehicle state into a nonlinear component sampled by particles and a linear sensor bias component tracked analytically via per-particle Kalman filters. This factorization reduces the required number of particles compared to a standard particle filter while accounting for realistic sensor drift. Simulation studies on both PDE scenarios show that the RBPF consistently outperforms a standard particle filter in terms of final position error and Root Mean Square Error (RMSE) across varying particle counts. Field experiments with an autonomous surface vehicle measuring salinity, temperature, and dissolved oxygen validate that PDE-governed environmental fields provide sufficient spatial variability for practical localization. Related experimental videos are available at https://localization-environmental-pdes.github.io/.

Spatial atomic layer deposition (SALD) is a leading atmospheric-pressure, high-throughput route to industrial ALD, but design and control are limited by the cost of predicting surface coverage: high-fidelity CFD is far too slow for operating-window scans, while analytic models miss transport modulation such as the gas curtain. We present a physics-chemistry-informed neural network (PCINN), a hybrid surrogate with CFD-level accuracy at real-time speed: a query returns coverage in about 7 ms, roughly 5x10^4 times faster than a CFD solve, reaching a test R^2_log = 0.998 (leave-one-out R^2_raw = 0.974) from only 30 training cases spanning four orders of magnitude in coverage. The architecture is not a black box: a small network learns only the operating-condition to near-wall concentration closure, while the known surface kinetics is a hard-coded, trainable chemistry layer integrated along the substrate trajectory. This single-scalar bottleneck keeps it accurate under sparse data, interpretable and invertible. We add a full identifiability analysis (Fisher information, profile likelihood). The adsorption energy E_ads and desorption rate k_des are robustly identifiable; k_ads is not separately identifiable at a single temperature (only k_ads*c_wall is). Across four temperatures the prefactor nu and E_ads bind along a weakly identifiable degeneracy valley of slope 0.065 eV/decade, derived analytically as k_B T_eff ln(10) and turned into a reliability diagnostic: a seven-chemistry mismatch matrix shows it is invariant under any single-Arrhenius mismatch and shifts only when a second thermally activated process appears, so a slope departure flags unmodelled site heterogeneity. Data come from simulation with known ground truth inverted by the same kinetic form, so the study verifies pipeline self-consistency and the identifiability boundary, not real parameters.

PDE discovery from sparse observations requires reconstructing a continuous field and selecting the correct differential terms. Our analysis of optimization paths in coupled neural PDE discovery reveals three behaviors: the exact support can persist to the end of training, appear only transiently, or fail to emerge. To decouple equation selection from neural optimization, we develop a freeze-then-select method combining a structured field adapter with Stability-Validated Weak Selection (SVWS). Trained from observations without a PDE residual, the adapter factorizes the field into learned spatial features and temporal coefficients represented by cubic splines. After freezing the field, SVWS identifies recurrent terms across independent weak-form systems, refits candidate supports, and selects the final equation on held-out weak-form systems. Beyond fixed libraries, we apply the same principle to expressions generated by genetic programming and recover the power-law form of an unknown nonlinear diffusion function from sparse, noisy observations. Across all six sparse MDBench regimes, our method attains the highest exact support recovery rate, with its clearest gains over classical and neural baselines on challenging Kuramoto-Sivashinsky dynamics.

Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive prediction remains challenging: local errors accumulate as spectral inconsistency, phase misalignment, or mean drift. Existing methods mainly improve state representations and operator backbones, while leaving the repeatedly applied latent transition increment weakly structured, allowing spectral errors and unstable channel couplings to accumulate during rollout. To address these issues, we propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction. GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands derived from the increment spectral energy distribution. To reduce physical-space reconstruction errors, GeoIncNO further introduces a mean--fluctuation decoupled reconstruction mechanism, where stable mean structures and dynamic fluctuations are fused separately, and phase correction is applied only to the zero-mean fluctuation component. Extensive experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO achieves consistently strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.

Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this problem, we evaluate Koopman-based upsampling techniques implemented with dynamic mode decomposition (DMD), extended DMD (EDMD), and optimized DMD. These methods learn finite-dimensional approximations of Koopman evolution on selected observables and are used to interpolate and denoise snapshots inside the observed time window before derivative estimation and sparse regression. The empirical benchmark comprises two ODE systems, Lorenz-63 and Van der Pol, and three periodic PDE systems, Burgers, Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP), and linear advection-diffusion, over sparse and noisy sampling regimes. Polynomial EDMD gives the strongest ODE results, especially in coefficient accuracy. The PDE results are system-dependent: low-rank DMD-assisted reconstructions improve Burgers and advection-diffusion discovery, while the raw baseline (without upsampling) remains competitive for the Fisher-KPP data. A comparison against linear and smoothing-spline interpolation techniques shows that the selected Koopman-based preprocessors provide overall performance gains over these non-dynamical alternatives. We also demonstrate that DMD-assisted upsampling can stabilize Pareto-based non-oracle support-size selection. Overall, Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step that can reduce derivative-estimation error when its observable representation and low-rank structure are appropriate for the data.

Despite the many variants of DeepONet that have been proposed, query-based operator networks still struggle with shock-dominated and low-viscosity PDEs, whose sharp moving discontinuities and slowly decaying solution spectra challenge finite-dimensional separable representations. In this work, we propose \emph{Feature Interaction Modeling Operator} (FM-Operator), a point-wise query neural operator that explicitly models feature construction and interactions between sensor observations and query coordinates. Our design is motivated by a reinterpretation of the canonical DeepONet aggregation through the lens of multiplicative interactions. Specifically, the branch--trunk inner product admits the equivalent form $\boldsymbol{b}(u)^\top \boldsymbolτ(y)=\boldsymbol{1}^\top \operatorname{diag}(\boldsymbol{b}(u))\,\boldsymbolτ(y)$, revealing that the two representations interact only along corresponding latent dimensions and therefore constitute a diagonally constrained multiplicative interaction. This observation suggests that, beyond improving the individual branch and trunk networks, the structure through which function and query representations interact is itself an important inductive bias in point-wise operator learning. FM-Operator accordingly redesigns both feature construction and feature interaction, enabling structured information exchange beyond the conventional branch--trunk coupling while retaining point-wise query evaluation. Experiments across multiple PDE benchmarks demonstrate that FM-Operator consistently outperforms vanilla DeepONet and achieves clear improvements over the strong Shift-DeepONet baseline. These results suggest that explicitly designing representation construction and interaction provides a promising direction for improving the effectiveness of DeepONet-style query-based neural operators.

This proceedings contribution elaborates on the findings of arXiv:2605.26234v2: a joint work with Marco Usula, where we introduced a machine learning framework based on physics-informed neural networks (PINNs), aimed at constructing near-minimal discs in hyperbolic space asymptotic to a prescribed knot at infinity. We used this method to provide numerical evidence for a conjecture of Joel Fine relating minimal surfaces in $H^{4}$ to the coefficients of the HOMFLY polynomial. This is a methodological companion to that paper, based on a presentation given at the 2026 edition of the workshop "DANGER: Data, Numbers, and Geometry". Rather than reviewing the results, which are presented extensively in the preprint above, we discuss the two aspects of the framework which, in our experience, determined whether the method worked at all. First, the geometry of the problem must be encoded in the architecture of the model, so that the boundary condition and asymptotics at infinity hold exactly for every value of the learnable parameters - leaving us with a single-component loss function; second, the evaluation of the PDE residual must be engineered with care to ensure that complete trainings can be performed in a reasonable time. On the latter point, we describe two implementation techniques which are not spelled out in detail in the original paper: replacing nested reverse-mode automatic differentiation with the forward propagation of second-order jets, and compiling the computational graph of the residual once instead of rebuilding it at every optimisation step. Together, on identical hardware, these two changes reduce the cost of a training step by a factor of roughly forty to fifty. We hope these methodological discussions can be useful for researchers in differential geometry and geometric analysis who wish to deploy PINNs on problems of their own.

It is well known that Newton's method converges faster when the initial guess is closer to a root of a system of nonlinear equations. In this paper, a two-stage Newton initial guess strategy is proposed by learning features from a parameter-space sampling and a database of precomputed solutions. The method uses discrete Newton trajectories to construct two complementary reduced spaces: a solution feature space, built from converged states, and a corrective search direction feature space, built from intermediate Newton increments. For an unseen parameter, a regression model is used to predict a surrogate solution approximation. Then, in a second step, a residual-minimizing correction is computed using a dedicated GMRES-based approach. The resulting state is then used as an initial guess for the high-fidelity Newton method, which completes convergence. The corrective step is computationally inexpensive since it only requires residual evaluations and the solution of a small least-squares problem. The methodology is weakly intrusive once the high-fidelity residual fields and a script-based programming interface are available. This strategy reduces the number of Newton iterations and decreases the overall CPU time. Numerical experiments on representative PDE problems show quantifiable speedups compared with standalone surrogate initialization. Significant speedups are observed. This generic approach can be applied to a broad class of large-scale nonlinear problems.

We establish a dimension-efficient neural network approximation theory for solutions to fractional parabolic equations with lower-order drift and potential terms. By introducing anisotropic spectral Barron spaces, which measure temporal and spatial regularity separately in frequency space, we first develop a dimension-independent maximal regularity theory for these equations, using dimension-independent multiplication estimates and the method of continuity to incorporate the lower-order terms. A key technical novelty is the application of the Vandermonde matrix to the global-in-time extension of the finite-time fractional heat semigroup with sufficient regularity at the initial time, thereby enabling analysis of the forward-in-time evolution via the global space-time Fourier structure of anisotropic Barron norms. We also show that a corresponding uniform-in-time estimate of the spectral Barron regularity generally fails. Finally, we derive $n^{-1/2}$ two-layer approximation bounds in mixed Sobolev norms for non-constant periodic activations and, under additional anisotropic Barron regularity, for non-periodic activations satisfying a polynomial-decay condition.

Transient-stability assessment determines whether a power system can recover after a disturbance and is therefore essential to preventing generator trips and cascading outages. A key metric is the critical clearing time (CCT), which specifies the maximum time available to clear a fault before synchronism is lost. Reliable CCT estimation is challenging because complicated fault-clearing dynamics require repeated simulations over many fault severities and clearing times. We propose an event-structured physics-informed neural network (ES-PINN) that aligns its representation with the pre-fault, fault-on, and post-clearing swing dynamics and enforces exact state chaining across event interfaces. A smooth trajectory-induced stability margin defines a differentiable approximation of the CCT boundary, enabling accurate boundary extraction, local sensitivity analysis, and optional direct CCT prediction through a distilled readout. We further prove a local residual-to-trajectory-to-CCT error estimate, in which exact event chaining eliminates separate state-interface defect terms. Experiments on IEEE 9-, 14-, and 30-bus systems show that ES-PINN consistently improves held-out trajectory and stability-boundary accuracy over matched neural-surrogate baselines across mechanical and electrical contingencies with multiple clearing configurations. Additional full-network DAE validation, multi-fault experiments, and runtime analyses further demonstrate the effectiveness and computational efficiency of the proposed framework.

We develop two parametric data-driven reduced models: a physics-informed neural network (PINN) and a non-intrusive tensorial reduced-order model (TROM), and apply both approaches to the parametrized one-dimensional shallow-water dam-break problem. Both reduced models do not require time integration and learn a direct solution map from space, time, and dam-break parameters to the physical state. We present a detailed comparison for out-of-sample and extrapolated parameter values. In addition, we demonstrate that it is essential to introduce shock-aware collocation to improve the robustness of the PINN model.

Physics-informed machine learning incorporates physical principles --often expressed via differential operators-- into data-driven models. While physics-informed neural networks (PINNs) dominate empirical applications, the complexity of neural network architectures and optimization landscapes hinders the development of a corresponding learning theory. In turn, kernel methods offer an appealing alternative with closed-form solutions and analytical tractability, yet existing guarantees primarily cover the well-specified setting where the target belongs to the native Reproducing Kernel Hilbert Space (RKHS). This imposes unrealistic regularity assumptions that physical targets often fail to satisfy. In this paper, we introduce and analyze Physics-Informed Kernel methodS (PIKS). We establish the universal consistency of PIKS for linear differential constraints, proving that for universal kernels (such as Gaussian or Matérn), the estimator asymptotically learns the target while satisfying physical constraints. We further derive finite-sample bounds under suitable source conditions. Our analysis is based on extending classical operator-theoretic analysis of kernel methods to physics-informed machine learning. Numerical experiments demonstrate that PIKS can be competitive with PINNs and traditional finite element methods.

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs), yet their performance heavily relies on the manual, trial-and-error engineering of neural representations, loss formulations, and optimization dynamics. While Large Language Models (LLMs) offer a promising avenue for automated design, unconstrained code generation often yields mathematically invalid or numerically unstable solutions under strict scientific computing constraints. To bridge this gap, we propose \textbf{EvoPINN}, an agentic framework that reformulates PINN development from labor-intensive manual design into a rigorous, execution-grounded algorithm discovery problem. EvoPINN navigates a modular search space by decoupling neural representations from training programs, utilizing an LLM agent to iteratively propose memory-conditioned programmatic modifications. To ensure scientific validity, all candidates undergo strict structural verification and budget-matched PDE evaluation. Extensive experiments across diverse PDE regimes (oscillatory, elliptic, dissipative, and nonlinear transport) demonstrate that EvoPINN discovers PDE-specialized learning algorithms that significantly reduce relative $L_{2}$ error compared to baselines. Crucially, EvoPINN autonomously invented SLRC-PINN, a novel architecture whose performance gains persist under rigorous parameter-matched comparisons, establishing the viability of execution-grounded agents for discovering genuinely new scientific computing mechanisms.

Brain tumors such as glioma, meningioma, and pituitary adenoma alter the mechanical behavior of soft brain tissue, yet common diagnostic methods rely on static imaging that cannot capture tumor growth, tissue displacement, or changes in stiffness over time. Deep learning models for this task typically require pooling patient data at one site, which conflicts with privacy rules such as GDPR and HIPAA and limits generalization across institutions, a challenge that is pronounced in neuro oncology given patient diversity. This study presents a federated physics informed neural network combining federated learning with a physics informed loss built on the equations of linear elasticity. Three simulated clinical sites each train a local network on patient specific MRI data using a physics informed loss, and only model weights are shared with a central server through the FedAvg protocol over one hundred rounds, keeping raw data at its site of origin. The federated model reached an overall accuracy of 91.4%, against 90.0% for a non federated baseline trained on pooled data, an average AUC of 0.985 across tumor classes, and a rise in pituitary tumor accuracy from 85.6 to 94.5%. Training produced smooth, divergence free displacement fields consistent with expected tissue deformation, showing that federated training can be paired with physics based constraints without a meaningful loss in performance.

Identifying cost-effective indigenous building materials that minimise heat penetration through walls is critical for indoor thermal comfort in low-income rural housing in hot-dry climates, where summer temperatures routinely exceed 45 C. We present a two-stage computational framework for thermal ranking of five low-cost indigenous wall materials: mud brick, clay-straw adobe, lime-stabilised bamboo panel, fired clay brick, and lime-mud composite. First, a validated Crank-Nicolson finite difference method (FDM) solves the one-dimensional transient heat equation with Robin boundary conditions under diurnal solar and outdoor air-temperature forcing, generating 1500 periodic-day solutions across a nine-dimensional parameter space by Latin Hypercube sampling. Second, a Physics-Informed Neural Operator (PINO) with a Fourier Neural Operator (FNO) backbone learns the parameter-to-solution operator mu -> T(x,t), enforcing both data fidelity and PDE consistency. The trained PINO attains a relative L2 field error of 5.14e-4 and a 0.201 K mean absolute error on the peak inner surface temperature, preserving the FDM material ranking exactly; PINO trained on 150 FDM samples matches a data-only FNO trained on twice as many, so the physics loss is most valuable when data are scarce. The periodic-day formulation also yields the ISO 13786 time lag and decrement factor, reproduced to within 0.99 h and 0.010. At nominal hot-dry summer conditions, clay-straw adobe achieves the best cost-performance index among widely available materials. A climate sweep, confirmed by FDM spot checks, reveals a regime boundary: under sub-ambient outdoor conditions the ranking inverts to conductive fired clay brick, delineating heat-exclusion and heat-rejection regimes. The framework supports evidence-based material selection for post-flood reconstruction in hot-dry regions.

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding governing physical laws into deep neural networks. However, their reliance on computationally expensive gradient-based optimization and deep architectures often results in slow training, high computational cost, and limited scalability. In this work, we propose a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs. The proposed formulation embeds the governing differential operator and the associated initial and boundary constraints directly into a linear output-layer optimization problem, thereby replacing nonlinear gradient-based training with a deterministic least-squares solution obtained via the pseudoinverse. Consequently, the entire learning process is reduced to a single linear optimization stage while preserving the underlying physical constraints. As a result, PI-BLS offers an efficient learning paradigm for a physics-informed learning framework for solving PDEs that eliminates iterative backpropagation while preserving the underlying physical constraints. Experimental results on representative forward PDE benchmarks demonstrate that PI-BLS achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.

Elasto-plastic boundary value problems in geotechnical engineering are conventionally solved by the Finite Element Method (FEM), which incurs high computational cost from incremental-iterative procedures. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but suffer from spectral bias, failing to resolve the sharp gradients arising at elastic-plastic boundaries and within localized plastic zones. This limitation is particularly consequential for the non-associative Mohr-Coulomb model, whose pressure-dependent yield surface and dilatant flow rule generate narrower plastic zones and steeper stress gradients than pressure-independent criteria. This study proposes a Fourier Feature Physics-Informed Neural Network (FF-PINN) for two-dimensional elasto-plastic problems governed by this model. Random Fourier feature mapping is embedded into the input layer to mitigate spectral bias, supported by a multi-objective loss function enforcing equilibrium, constitutive relations, and Karush-Kuhn-Tucker conditions against high-fidelity FEM data, together with a strain-adaptive sampling strategy. Benchmarked across three test cases, FF-PINN achieves superior accuracy across most predicted fields, with error reductions up to approximately 66 percent in displacement and 27 percent in stress components, and reproduces the plastic failure zone geometry with markedly closer fidelity to FEM. Sensitivity analysis confirms robustness across training data size, collocation density, loss weighting, and noise levels up to 2.0 percent. FF-PINN converges in half the training epochs required by the conventional PINN, halving wall-clock training time while achieving greater predictive accuracy. The framework therefore offers a computationally efficient and physics-consistent alternative to FEM for elasto-plastic geotechnical analysis.

Deep learning surrogate models trained with mean-squared-error loss produce statistically accurate but physically unconstrained flood predictions: water may flow uphill, appear spontaneously, or smooth over street-level corridors. We develop a physics-informed training framework for CNN-LSTM models that predict urban flood depths at 15 min intervals over a 128x128 spatial grid. Three differentiable penalty terms are embedded into the loss: (i) a gravity loss penalizing depth increases against the water-surface-elevation gradient, (ii) a continuity loss enforcing local mass conservation with rainfall-adaptive thresholds, and (iii) a topography-aware false-alarm penalty modulated by the topographic wetness index (TWI). We evaluate on the Norfolk, Virginia flood dataset spanning two storm events (August 2017 and September 2022, 300 samples), with all variants trained on identical splits and robustness assessed over repeated random splits and leave-one-storm-out tests. A road-proximal evaluation restricted to a TWI-derived street mask quantifies street-level skill. The physics-constrained model achieves near-zero gravity violations (order 1e-6) and the highest street-channel recall (0.77 +/- 0.09 vs 0.44 +/- 0.10 for the unconstrained baseline), the capability most relevant to traffic routing, and its advantage more than doubles on a held-out storm; a uniform false-alarm variant attains 16% lower mean absolute error but suppresses street recall to 0.25. The TWI-modulated penalty reconciles this trade-off: it improves on the uniform variant on every metric, recovering 60% higher street recall at the lowest MAE among constrained variants and the best street-level F1. These results expose a fundamental tension between aggregate pixel-level error and application-specific physical plausibility, and show that terrain-aware loss modulation offers a principled resolution.

The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning. In these methods, a neural network is trained to approximate the PDE solution by using (stochastic) gradient descent to minimize the PDE residual of the neural network. Due to the non-convexity of the PDE residual objective function, the trained neural network may, in principle, only converge to a local minimizer of the objective function (which would not be a solution of the PDE). Therefore, there is a longstanding question regarding the mathematical foundations of these algorithms, and it is highly valuable to establish that the trained neural network will converge to the PDE solution. For a class of semi-linear PDEs (nonlinear in the solution and its first derivative), we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution.

Transformer architectures have attracted increasing attention for solving partial differential equations (PDEs), owing to their flexibility in handling irregular discretizations and their ability to capture long-range physical dependencies. However, unlike discrete language tokens or fixed-resolution image patches, observed physical fields are finite samples of underlying infinite-dimensional functions. Consequently, effectively applying Transformers to PDEs requires a tokenizer that respects the functional nature of physical fields and constructs physically expressive tokens from arbitrary discretizations.To this end, we propose \methodname{Physics Transformer}, a function-projection-based Transformer architecture for physical field prediction. Physics Transformer treats a physical field as a continuous function and partitions its discretization into locality-preserving spatial patches. Within each patch, it dynamically learns a set of adaptive local basis functions and projects the sampled field onto these bases to obtain compact physics tokens. The resulting tokens capture diverse latent physical states while preserving fine-scale spatial structures, enabling efficient global interaction through factorized attention across space and physical states. The projected representation further supports efficient decoding at arbitrary query locations. Extensive experiments on diverse benchmarks, ranging from two-dimensional PDE dynamics to industrial-scale three-dimensional CFD simulations, demonstrate that Physics Transformer accurately captures fine-grained physical structures and achieves state-of-the-art predictive performance. These results establish function projection as a practical and effective foundation for designing Transformer architectures for PDE solving.

Neural spectral PDE solvers often learn an entire unresolved vector field even when an inexpensive approximate model can already capture most of the trajectory. Here we introduce Perturbative-NeuSA, a residual formulation that decomposes the target solution into a low-fidelity background and a high-resolution perturbation, so that only the unresolved dynamics is learned. Starting from the exact perturbation equation, the method combines a fixed spectral operator, a background-dependent correction, the background defect in the target PDE, and an optional neural closure. This construction makes the roles of physical structure and neural closure separately measurable. Across 2D Burgers, Klein-Gordon, and heterogeneous 2D wave equations, the deterministic structured solver outperforms the trained NeuSA baseline while requiring no neural-network training. The largest gains occur on Burgers, where the deterministic correction reduces training and extrapolation errors by factors of 24 and 44, respectively. In addition, a Klein-Gordon sweep over seven background resolutions shows that the effect of the closure is conditional: it improves a poor background by 3.6 times, becomes neutral at intermediate resolutions, and degrades a well-resolved background. For the wave equation, however, the closure provides an additional 18% reduction when the remaining residual is interface-localized. Multi-initial-condition diagnostics further show that the useful closure regime depends on the initial-condition spectrum and can disappear in extrapolation when structured correction already captures the dominant Burgers dynamics. Perturbative-NeuSA therefore reframes neural closure as a conditional, diagnosable correction governed by background fidelity, residual organization, and compatibility with the closure model.

Physics-Informed Neural Networks (PINNs) solve differential equations by minimizing the residual of a nonlinear operator over a neural parameterization of the solution. However, monolithic PINNs often suffer from ill-conditioning, spectral bias, and optimization instability. We introduce a variational boosting framework in which solutions are constructed additively in function space. Each stage trains a weak learner whose converged correction satisfies a local orthogonality condition, equivalent to a projected functional gradient descent step onto the tangent space of the network's function manifold. Because each correction network is deliberately small, the restricted minimization admits full Newton or conjugate gradient updates, which are typically infeasible in large PINNs. The resulting method separates global nonlinear refinement into a sequence of well-conditioned subproblems while preserving the full variational structure of the operator. This framework provides a geometric interpretation of multi-stage PINNs as projected functional gradient descent and enables stable second-order optimization for nonlinear differential equations.

Partial differential equation (PDE) discovery aims to identify from data the governing law of a physical system. Constituting a cornerstone of scientific advancement, it has become during the past decade a major line of research in the rapidly evolving field of Physics-informed Machine Learning (PiML). Among the remaining open problems to address in this domain, the post-hoc evaluation of discovered PDEs raises the particular difficulty of being multifaceted. Indeed, it requires jointly considering predictive accuracy, physical consistency, interpretability, and out-of-distribution generalization capacity. Given that some of these properties are conflicting, it is worth noting that the wide range of existing evaluation metrics only partially address the overall problem, potentially leading to overly interpreted conclusions about the validity of a presumed new physical theory. From an abundant literature spanning machine learning, numerical analysis, information theory or symbolic regression, we propose, to our knowledge, the first taxonomy of PDE evaluation metrics, and discuss their advantages and limitations in depth. Based on the observation that evaluation is often achieved on a case-by-case basis and that a universally accepted methodology remains elusive, we further provide recommendations with the aim of promoting standardized and reliable practices, before sketching promising future lines of research in this field. We argue that this paper is intended both for ML experts who design new PDE discovery algorithms and for users of these methods aiming, in real applications, to discover and validate well-founded scientific laws.