Partial Differential Equation Benchmarks

Neural operators evaluate parametric partial differential equations cheaply but degrade sharply outside their training distribution. Physics-informed neural networks avoid dependence on labeled data, yet their optimization can be basin-fragile: when the governing residual admits multiple solutions, a PINN trained from scratch may converge to a physically incorrect state despite achieving a small residual. We show that these failure modes can be addressed jointly: an imperfect NO provides the structural prior needed to place a PINN in the correct solution basin, while the PDE residual refines the solution beyond the operator's accuracy. We introduce a three-stage framework that freezes the spatial basis of a physics-informed NO, extrapolates its solution branch to an out-of-distribution parameter using a polynomial continuation prior, and distills the resulting field into a fresh PINN. The NO need not be accurate at the target; it transfers solution-branch information, while PDE residual minimization in the PINN governs convergence. We evaluate the framework on three nonlinear PDEs: 1D viscous Burgers, 2D steady Allen-Cahn near a pitchfork bifurcation, and 2D steady lid-driven cavity flow. For Allen-Cahn, where the trivial solution satisfies the PDE residual exactly, a standard PINN collapses to the trivial zero branch, whereas distillation from the crude extrapolated operator recovers the non-trivial branch that matches the finite-difference reference. For the lid-driven cavity, extrapolating to a Reynolds number of Re = 3200 accelerates convergence to the correct physical state, achieving competitive accuracy using fewer parameters and optimization steps than recent literature baselines. These results establish a simple principle: an NO need not accurately predict the solution to be useful; it only needs to identify the correct basin from which PINN optimization can recover it.

Physics-informed neural networks and finite element methods provide two different paradigms for the numerical approximation of partial differential equations: the former are commonly trained by minimizing pointwise strong residuals, whereas the latter are naturally built from weak variational formulations and the finite-dimensional systems obtained after discretization. In this work, we introduce a common framework based on the discretization of functional Gauss--Newton problems by finite families of linear measurements. We show that, through an appropriate duality pairing, the linear measurements can be represented by test functions. The resulting Gauss--Newton system is then precisely a Petrov--Galerkin discretization of the linearized functional problem. This perspective recovers pointwise collocation and natural-gradient constructions as particular cases, while making the choice of test functions an explicit algorithmic design choice. We specialize this framework to elliptic problems, where it naturally leads to weak residual formulations and to a hybrid finite element--neural construction acting on complementary approximation spaces. Numerical experiments support the proposed framework and demonstrate the effectiveness of weak Gauss--Newton formulations and hybrid finite element--neural approximations.

Latent neural operators improve the efficiency of operator learning for partial differential equations (PDEs) by performing the main computation on compact latent representations. However, directly compressing the input representation to obtain such compact representations may discard solution-relevant spatial information, especially for PDE solutions with multiscale structures. To address this problem, we propose HiLNO, a hierarchical latent neural operator that constructs a fine-to-coarse-to-fine latent space and further introduces multi-scale supervision (MSS) and anisotropic Gaussian attention. The hierarchy mitigates potential information loss during compression, while MSS aligns intermediate predictions with downsampled target fields, encouraging solution-relevant structures to be captured across multiple spatial scales. Anisotropic Gaussian attention enables feature transfer across the hierarchy, making HiLNO applicable to general geometries. Experiments on representative PDE benchmarks and a large-scale automotive aerodynamics task show that HiLNO achieves competitive predictive accuracy, while reducing the parameter count by an average of 84.4% and FLOPs by an average of 69.2% compared with LinearNO. Additional experiments demonstrate effective generalization to unseen spatial resolutions. Code is available at https://github.com/JcLimath/HiLNO.

Neural operators such as the Fourier Neural Operator (FNO) achieve remarkable accuracy in approximating solutions to partial differential equations (PDEs). However, providing rigorous uncertainty estimates remains an open challenge. We propose Physics-Informed Conformal Prediction (PI-CP), a framework that embeds PDE residuals into the nonconformity score of split conformal prediction, producing prediction intervals that are (i) distribution-free with provable coverage guarantees, and (ii) spatially adaptive when the PDE residual correlates with prediction error -- tighter where physics is well-satisfied, wider where it is violated. Additionally, we prove that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, and show that coordinate channels resolve this with up to 63x error reduction. We validate PI-CP across six physics scenarios -- heat conduction (2D/3D), structural mechanics (2D/3D), Darcy flow, and Navier-Stokes -- demonstrating consistent 89-91% coverage for all four Conformal methods, while MC Dropout and Deep Ensembles are unstable (82-100%). FNO outperforms CNN and DeepONet by 10-12x.

We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.

In this paper, we derive an explicit solution of the stationary prediction with expert advice PDE for five experts. The formula is given in three regions. In the first two regions, it is the four-expert solution plus a single integral with an elementary positive density. In the third region, it is a finite sum of hyperbolic products whose coefficients are determined by one scalar quadrature. Our formula establishes that the direction $(1,0,1,0,0)$ is optimal throughout the ordered sector, and that the COMB strategy $(1,0,1,0,1)$ is optimal only on a lower dimensional subset of the sector (where $x_1=x_2$ and $x_3=x_4$). This disproves the COMB optimality conjecture of Gravin, Peres and Sivan (2016). The verification of the Hamiltonian inequalities is a tedious task, part of which is completed with a computer assisted proof. The verification reduces to 21 scalar inequalities, which we prove using 147 exact rational Bernstein polynomial certificates. The exact certificates and their independent arithmetic checks are included in a supplement to this paper.

Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing partial differential equations (PDEs). Recently, deep learning, as represented by neural operators, has provided a data-driven paradigm for addressing such tasks. However, most existing global neural operators for PDEs require large training datasets and many learnable parameters, with limited interpretability and generalization. We propose the local gradient neural operator (LGNO) as a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems. The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels that resemble discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, rendering the learned operators more transparent. For problems with symmetries, network folding shares equivalent components and reduces parameter counts. We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases. Results show that LGNO maintains accuracy, parameter efficiency, and rollout stability across these tasks, and further exhibits wide applicability to mechanical problems including diffusion, flow, and quantum phenomena.

Existing approaches for synthesizing Partial Differential Equation (PDE) solvers predominantly rely on autoregressive models, yet their global left-to-right decoding incurs substantial redundancy when addressing inherently localized bugs. In this work, we challenge this inefficient paradigm and propose DiffPDE, a framework leveraging discrete diffusion language models for targeted code repair. By introducing a localized re-masking and infilling strategy, DiffPDE regenerates only erroneous regions while preserving correct context, naturally aligning generation with the sparse nature of PDE errors. Furthermore, to handle coupled bugs requiring sequential interventions, we present Iterative Debugging GRPO (ID-GRPO), a reinforcement learning scheme that enables multi-round debugging within single trajectories via intermediate rewards. Experiments on PDEBench show that DiffPDE achieves competitive accuracy, outperforms same-scale AR models, and significantly accelerates repair.

Classical numerical solvers for partial differential equations (PDEs) are computationally expensive to solve repeatedly across varying initial conditions, motivating the need for learned surrogates. In this paper, we propose a trainable Neural Cellular Automata (NCA) based surrogate model for learning long time PDE dynamics. Rather than mapping an entire initial field to a full trajectory in one shot, our proposed model learns a small, local, homogeneous update rule that is applied identically and repeatedly at every grid cell, mirroring the locality of differential operators. We benchmark this framework against three baselines: PDE - Net, a modified physics-informed neural network (PINN), and a Fourier Neural Operator (FNO), on five canonical PDEs (heat, advection, Burgers, Allen - Cahn, and Fisher - KPP), evaluated at temporal domain two times beyond the training temporal domain. The proposed model achieves the lowest long-horizon relative errors on the majority of the experiments.

Learning solution operators for partial differential equations (PDEs) on irregular and geometry-dependent domains remains a central challenge in scientific machine learning. While spectral methods provide strong inductive biases for modeling global interactions, they are typically limited to regular domains, and existing neural approaches often require domain warping, interpolation, or costly geometric embeddings. We introduce the \textbf{Graph Spectral Neural Operator (GSNO)}, a neural operator that combines spatial graph spectral decompositions with temporal Fourier transforms through a unified space--time spectral kernel. This formulation enables globally coherent operator learning on non-Cartesian discretizations without domain warping or autoregressive rollouts. By replacing learned geometric embeddings with a graph Laplacian spectral basis, GSNO provides geometry-aware spectral learning with low parameter complexity. Across steady and unsteady PDE benchmarks on irregular and geometry-dependent domains, GSNO achieves strong accuracy with reduced runtime and parameter counts, while demonstrating robust zero-shot generalization across mesh resolutions and geometry families.

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.

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.

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.

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.

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.

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.

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.

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.

Parameterized and coupled partial differential equations (PDEs) are central to modeling phenomena in science and engineering, yet neural operator methods that address both aspects remain limited. We extend Fourier neural operators (FNOs) with minimal architectural modifications along two directions. For parameterized dynamics, we propose a hypernetwork-based modulation that conditions the operator on physical parameters. For coupled systems, we conduct a systematic exploration of architectural choices, examining how operator components can be adapted to balance shared structure with cross-variable interactions while retaining the efficiency of standard FNOs. Evaluations on benchmark PDEs, including the one-dimensional capacitively coupled plasma equations and the Gray-Scott system, show that our methods achieve up to 55-72% lower errors than strong baselines, demonstrating the effectiveness of principled modulation and systematic design exploration.

In this study, we introduce latent PDE mapping, a broadly applicable physics-informed learning technique designed to enable efficient geometric generalization with sparse training data. Latent PDE mapping pulls back geometry-specific PDE residuals and boundary conditions to a predefined latent geometry via the deformation gradient, thereby enabling the automated calculation of geometry-consistent shape gradients that are missing in conventional physics-informed machine learning formulations. We demonstrate the utility of latent PDE mapping in solving the anisotropic Aliev-Panfilov PDE of cardiac electrophysiology using both physics-informed neural networks and physics-informed deep operator networks. The Aliev-Panfilov PDE serves as a challenging exemplar: a nonlinear, time-dependent PDE benchmark with sharp gradients that are expensive to capture using traditional numerical solvers. To represent the limited data regime, we train the networks using just fifteen geometric samples drawn from parameterized distributions in two and three spatial dimensions. While modest improvements appear for geometries parameterized by affine and shear deformations, latent PDE mapping demonstrates significant benefits on select geometric families, achieving a factor ~4-6 reduction in mean relative L2 error. Furthermore, our results show that the computational cost of applying latent PDE mapping was modest during network training, and negligible at inference. Taken together, our study highlights how latent PDE mapping facilitates the creation of generalizable physics-informed machine learning models from limited sets of training geometries.

Scientific machine learning methods such as neural operators and physics-informed neural networks have advanced engineering applications and inverse problems, but their training typically requires large volumes of simulated data. This makes data preparation and model training expensive. We propose Graph Wavelet Compressed Sensing (GWCS), a learning-based framework for offline compression of graph signals by representing them as sparse, interpretable wavelet-domain representations using the spectral graph wavelet transform. The framework combines a nonparametric multilevel importance sampler, which retains high-energy wavelet coefficients within each scale for a given compression ratio, with a scale-aware graph neural network that reconstructs the signal from the sparse coefficients. We evaluate the proposed framework on synthetic approximately band-limited graph signals over random graphs and four PDE simulation datasets over meshes, which include Turbulent Radiative Layer, Viscoelastic Instability, Kolmogorov Flow, and Dynamic Stall. We compare against graph signal sampling methods and graph autoencoder baselines. Results demonstrate that the framework achieves high reconstruction fidelity and substantial data compression compared to existing benchmarks.

Accurately solving partial differential equations (PDEs) on arbitrary geometries and a variety of meshes is an important task in science and engineering applications. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the kernel integral formulation of SSMs. This is achieved by constructing Takenaka-Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and can approximate the solution manifold of PDEs on a wide range of geometries and meshes. In several challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and irregular domains, AMO consistently outperforms state-of-the-art solvers in terms of relative $L^2$ error. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.

Physics-Informed Neural Networks (PINNs) solve PDEs by incorporating physical constraints into neural-network training, but large-scale problems are limited by automatic-differentiation memory overhead and inefficient execution of grid-based PDE operators. We present FlashPDE, a drop-in fused operator library for grid-based scientific machine learning. FlashPDE replaces fragmented PyTorch finite-difference execution with differentiable Triton kernels. Each operator integrates fused stencil evaluation, an analytic discrete-adjoint backward pass, and boundary-gradient correction within a unified PyTorch autograd Function interface. The library provides 14 differentiable PDE operators covering 17 configurations across 1D--3D elliptic, parabolic, and Navier--Stokes systems, while remaining independent of neural architectures and training strategies. Experiments on an NVIDIA A100 GPU show that FlashPDE reduces peak memory usage by up to 37.0x compared with coordinate-based automatic differentiation and reduces CUDA kernel launches by up to 3.5x compared with eager PyTorch finite-difference implementations. Across six representative PDE benchmarks, FlashPDE achieves up to 2.30x end-to-end time-to-solution speedup and up to 19.2x kernel-level acceleration while maintaining numerical agreement with PyTorch finite-difference references. FlashPDE provides a hardware-efficient execution layer that bridges differentiable PDE solvers and GPU-optimized numerical computation within the PyTorch ecosystem.

Physics-Informed Neural Networks (PINNs) provide a meshless approach for solving partial differential equations (PDEs), but suffer severe degradation in stiff and shock-dominated problems, where small PDE residuals can correspond to globally inaccurate solutions. We show these failures are multi-causal, arising from the concurrent interplay of (i) spectral bias against sharp features, (ii) imbalanced multi-term optimization and loss-weight collapse, (iii) violation of temporal causality, and (iv) under-resolved collocation. We present SPARC-Net, a unified architecture and training framework that jointly addresses all four pathologies. SPARC-Net leverages an adaptive multi-scale spectral encoder with a learnable spectral gate, a gated residual backbone, adaptive activations, and a hard-constraint output ansatz that exactly enforces initial and boundary conditions, structurally eliminating loss-weight collapse. Training employs stabilized gradient-norm loss balancing, floored causality-respecting residual weighting, and residual-based adaptive collocation (RAD). Validated against exact analytic and high-order spectral reference solutions across four canonical benchmarks -- viscous Burgers', Allen-Cahn, convection (beta=30), and reaction -- SPARC-Net yields substantial improvements over vanilla PINNs: relative L2 error drops from 1.47e-1 to 1.14e-1 on Burgers' (22% reduction), 9.93e-1 to 5.78e-2 on Allen-Cahn (94% reduction), and 9.82e-1 to 3.54e-3 on reaction (100% reduction). A characteristic-coordinate encoder for hyperbolic transport further reduces convection error from 5.14e-1 to 9.88e-5 (100% reduction). We report five-seed mean +/- standard deviation errors, Wilcoxon significance tests, full ablation studies, hyperparameter sensitivities, an extension to the 2D heat equation, and comparisons against parameter-matched baselines.

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.

We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions. A head-orthogonalization penalty removes degeneracies in the latent representation and stabilizes the principal-component spectrum across training realizations. Because the initial condition is built into the network output by construction, these principal components measure the additional variability the network learns on top of the initial profile, not the full solution itself. We apply the method to the one-dimensional viscous Burgers equation, with the heat and wave equations as robustness checks. For a latent dimension $n_b=20$, the learned manifolds exhibit pronounced effective dimensional reduction: for Burgers dynamics, only $2$-$4$ principal components capture about $95\%$ of the latent-space variance, while $4$-$7$ capture about $99\%$, depending on the initial-condition family; the same qualitative compression holds for the heat and wave equations. We also split the wavenumber axis into bands (``Fourier shells'') and measure how much each band contributes to every principal component. The resulting frequency profile is invariant under the change-of-basis freedom that the orthogonalization penalty leaves in the latent space, and is therefore reproducible across independent training runs. More broadly, this establishes the learned spectral profiles and principal components as robust observables of solution-manifold geometry.

Differentiable partial differential equation (PDE) solvers underpin solver-in-the-loop ML training, gradient-based optimal control, and inverse problems, yet the practical cost of obtaining correct, usable gradients from a given solver on a given problem is largely undocumented. Integration effort, computational cost, gradient accuracy, and numerical conditioning vary widely across solvers and are discoverable only by trial and error. We introduce Mosaic, an extensible benchmarking framework for differentiable PDE solvers that standardizes access to solver gradients. Each solver is packaged as a containerized component (Tesseract) exposing a uniform gradient API regardless of language or automatic differentiation (AD) strategy, enabling researchers to evaluate, compare, and build on non-trivial physical solvers. Our evaluation of 14 solvers across fluid dynamics, structural mechanics, and heat transfer demonstrates that the benchmark surfaces practically relevant differences: order-of-magnitude variation in computational cost and Jacobian conditioning, alongside structural incompatibilities that eliminate solvers from realistic tasks entirely. Despite this variation, all solvers that produce gradients converge to similar optima, indicating that the practical barriers are memory limits, numerical stability, and setup compatibility rather than gradient accuracy alone. Mosaic is open-source and available at https://github.com/pasteurlabs/mosaic.

Physics-informed neural networks (PINNs) embed governing equations in their loss function, enabling mesh-free solutions to partial differential equations. Low training loss is treated as evidence that the learned solution is physically correct. This paper shows that assumption breaks down when encoded physics are incorrect. By perturbing PDE parameters before training, a setting we describe as physics parameter poisoning or parameter misspecification, we produce models that train to low loss but give incorrect answers; we treat the perturbation schedule as sensitivity analysis rather than only as a security threat, and none of our claims requires an adversary. Achieving low residual loss does not discriminate accurate from inaccurate solutions: poisoned models reach losses at or below the clean baseline yet differ by large margins, so driving the residual down is not evidence of physical accuracy. Across three PDE systems (Burgers equation, Navier-Stokes cavity, and convection-diffusion), poisoned models match or beat the clean-model training loss while their solutions differ by up to 71% in the fixed sweep and up to 128% under adversarial search; at Cavity Re=400 the poisoned loss falls below the clean baseline. We define a detection difficulty ratio R (solution error divided by training loss) to summarize how invisible the corruption is, though cross-PDE comparison is complicated by differences in loss scale. We test six candidate defenses, none of which reliably detects corruption across all regimes. We propose a post-hoc defense: sweeping the PDE residual loss across parameter values without retraining. The loss minimum recovers the true training parameter without external data, and generalizes across all three PDE systems. The effect holds across five network architectures (8.7K to 133K parameters), is bidirectional, and is confirmed across multiple random seeds.

Physics-informed neural networks (PINNs) provide an effective way to solve partial differential equations (PDEs) by embedding physical principles into the learning process. However, the conventional PINN formulation, in which all constraints are imposed as soft penalty terms within a composite loss, often exhibits slow convergence, sensitivity to loss weight scaling, and inaccurate boundary enforcement due to poor conditioning of the optimization landscape. To address these limitations, this study proposes a unified hard--soft physics--informed neural network (HSPINN) with adaptive loss weighting. In this framework, Dirichlet and periodic boundary conditions are enforced exactly by construction through analytical or polynomial lifting, masking functions, and periodic feature mappings, while the governing PDE residuals, Neumann fluxes, and initial conditions are treated as soft constraints. An inverse-share softmax strategy dynamically balances the relative importance of individual loss components during training, eliminating manual penalty tuning and improving gradient stability. This formulation ensures boundary admissibility throughout optimization and enhances convergence efficiency and numerical robustness. Applications to representative elliptic (Poisson), parabolic (Burgers), and hyperbolic (convection with periodic boundaries) problems demonstrate that HSPINN consistently achieves faster convergence, higher accuracy, and greater stability than conventional PINNs, establishing a general and scalable foundation for physics-constrained deep learning across science and technology.

Non-stationary partial differential equations (PDEs) arise throughout scientific computing, where the dominant frequency content and energy distribution can drift over time. While efficient in PDE solving, many spectral neural operators apply a shared spectral response across rollout stages, leading to mismatch with time-varying spectra in non-stationary systems. To address this issue, we propose Time-Frequency Gated Spectral Neural Operator (TF-SNO), a state-adaptive framework with learnable time-frequency gating inside spectral blocks. TF-SNO extracts compact frequency-domain and physical-space statistics from the current state to generate modulation coefficients, enabling the spectral response to evolve with the dynamics. TF-SNO learns temporal variation implicitly from the evolving state without introducing an explicit time dimension or time embedding, keeping the modeling complexity low. We further embed the adaptive operator blocks to accurately capture the multi-scale features, thereby improving long-horizon stability. Experiments on six non-stationary PDE benchmarks in 1D and 2D demonstrate that TF-SNO significantly reduces prediction errors and improves robustness compared to strong baselines, with particularly clear gains in long rollout, suggesting the effectiveness of state-dependent spectral adaptation in modeling non-stationary physical systems.

Mathematicians understand a PDE solution through mathematical structures rather than tables of computed values. Historically, this has been the product of mathematical analysis, carried out by hand for each problem individually. Neither numerical simulation nor neural networks produce those structures directly. We propose Agentic Symbolic Search (ASYS), a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. This makes the search an automated form of inductive-bias injection rather than blind symbolic regression. For problems with known analytical forms, ASYS recovers these forms naturally; for other problems, ASYS constructs analytical approximations which can guide mathematicians toward further analysis. In our experiments, across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, in settings where no closed-form description was previously available. ASYS shows the possibility of a new paradigm for characterizing PDE solutions, beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.

Physics-informed neural networks (PINNs) have become a powerful framework for solving PDEs by embedding physical laws into differentiable objectives. Despite their advances, training PINNs remains fragile: recent conflict-averse optimization schemes alleviate gradient interference between residual and boundary losses, but we show that their effectiveness deteriorates as model capacity increases. In this paper, we identify a capacity-induced failure mode, where overparameterized networks undergo functional modularity, self-partitioning into task-exclusive modules that suppress cross-objective interaction and hinder convergence toward Pareto-stationary points. To address this issue, we propose a novel framework, Modular-Sparsity Synchronization (ModSync), which integrates structural optimization into conflict-averse training by penalizing task-exclusive connections while preserving interaction-promoting pathways. Extensive experiments across diverse PDE benchmarks demonstrate that ModSync consistently prevents capacity-driven failures, sustains robust cross-objective coupling, and achieves state-of-the-art accuracy. Codes are available at \url{https://github.com/heejokong/ModSync}.

Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems. While traditional numerical solvers are robust, they often incur prohibitive computational costs due to mesh dependencies, whereas recent Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but frequently suffer from slow convergence and optimization instability. To bridge this gap, this article proposes the Physics-Informed Broad Learning System (PIBLS), a novel backpropagation-free framework that reformulates PDE solving as a direct least-squares optimization. We improved an algorithm within this framework to handle nonlinear PDEs efficiently and provide a rigorous mathematical proof establishing the universal approximation property of PIBLS for these equations. Experiments on linear and nonlinear PDEs demonstrate that PIBLS is one to three orders of magnitude faster than conventional PINNs while achieving significantly higher solution accuracy. This framework provides a computationally efficient paradigm for scientific machine learning, offering a practical, high-speed alternative for real-time simulation and design optimization tasks.

Neural operators provide fast surrogates for PDEs but their deterministic predictions limit their use in tasks requiring uncertainty quantification (UQ), especially under geometric variability. Existing approaches primarily model uncertainty in network parameters, largely overlooking the geometry-aware representations learned by the operator itself. We propose REEF-GP (Residual on Embedded Features Gaussian Process), a post-hoc UQ framework that fits a GP to the residuals of a frozen neural operator whose internal embeddings define the kernel feature space. Rather than learning a separate feature map, REEF-GP adapts the operator's intrinsic coordinate-feature representations to construct geometry-aware uncertainties. To ensure stability and scalability on unstructured domains, REEF-GP incorporates spectral-normalized projections, heteroscedastic geometry-aware noise, and efficient subset-based training that avoids restrictive low-rank approximations. Across five PDE benchmarks with varying geometries, REEF-GP preserves predictive accuracy while achieving calibrated uncertainty estimates competitive with deep ensembles but at a fraction of their cost. Our approach remains robust under geometric distribution shift, with uncertainty concentrating in physically meaningful regions (e.g., shock fronts). Our results demonstrate that accurate and scalable post-hoc UQ for neural operators can be achieved directly in their learned feature space, offering a practical alternative to parameter-centric approaches.

Transformer-based neural operators have shown remarkable performance for approximating solution operators of partial differential equations on complex geometries. However, existing approaches implicitly assume a fixed domain size, which limits their ability to generalize at inference. In this work, we investigate domain extension, namely zero-shot inference on spatial domains that are significantly larger than those encountered during training. We argue that this setting fundamentally requires spatial locality and translation equivariance. We propose to implement this locality via a decomposable bias in the attention logits computation, enabling finely controllable locality while remaining fully decomposable into query-key inner products and directly compatible with optimized attention kernels. Combined with rotary positional embeddings, it enables expressive embeddings with controllable spatial support without altering the transformer architecture. We empirically show that our approach substantially improves zero-shot generalization to larger domains across two PDE benchmarks and a 3D industrial atmospheric flow application. Our code and datasets are available at https://github.com/cerea-daml/domain-extension.

Numerical solvers for partial differential equations (PDEs) are core computational tools in science and engineering. Building reliable PDE solvers requires not only executable code, but a numerical solver strategy, a set of decisions about discretization, stabilization, solver configuration, and resolution control, that matches the PDE structure. Recent LLM-based coding agents have begun to reduce the programming burden by generating and debugging solver implementations. However, they typically move directly from a PDE problem to solver code, leaving the solver strategy implicit in implementation details. Feedback from a failed solve is therefore routed back to code edits rather than to the underlying strategy, so numerical decisions remain hard to check before code is generated and hard to revise using numerical evidence when it fails. To address this limitation, we propose AutoPDE, a code agent that maintains the solver strategy as an explicitly represented object throughout the solving process: an independent, inspectable object that is built before any code is written and can be revised, using numerical evidence, whenever a solve fails. AutoPDE builds and maintains this object in three stages, all drawing from a library of reusable PDE-solving skills: PDE analysis identifies the equation type and algebraic structure; numerical method selection chooses a numerical method that matches the analysis result and commits to a discretization, stabilization, and linear solver accordingly; and adaptive tuning runs low-cost pilot solves to calibrate resolution and tolerances under the prescribed accuracy and runtime budget. We evaluate AutoPDE on the PDE Agent Bench, where experimental results show that AutoPDE achieves a pass rate of $54.5$, improving over the strongest baseline by $14.2$ percentage points.

We investigate the approximation of solution operators for partial differential equations (PDEs) using sparse high-dimensional techniques. Building on a dimension-incremental framework, we combine product basis expansions with sparse recovery methods, specifically orthogonal matching pursuit (OMP), to substantially reduce the required sample size compared with a previously considered cubature-based approach. We evaluate the resulting method numerically on several examples, comparing it against both cubature-based sparse approximation and Fourier neural operators in terms of accuracy, runtime, and sample size. The experiments show that our approach considerably reduces the number of required PDE solves relative to its predecessor while maintaining competitive accuracy, particularly when the solution admits a sparse representation in the chosen basis. Furthermore, the recovered sparse index sets yield interpretable insights into the relevant variables and parameter interactions.

Recently, partial differential equations (PDEs) have been used to directly model the measurement process in signal processing, although their evaluation is costly. In this paper, we propose a novel alternating direction method of multipliers (ADMM)-based algorithm called physics-aware linearized ADMM (PA-LADMM) for inverse problems from PDE-based measurement processes. The key idea is the linearization of the subproblem with PDEs, leading to a cost-efficient update rule that calls only a PDE solver and its gradient evaluation per iteration. The algorithm has a theoretical convergence guarantee under certain conditions. In addition, we combine it with deep unfolding to unroll the PA-LADMM and train its internal parameters using supervised data. Two distinct experiments, compressed sensing with optical fiber communication and image restoration from noisy anisotropic diffusion, demonstrated the effectiveness of the proposed algorithms.

Neural operators provide fast surrogate models for PDE simulations, but standard architectures often treat geometry and discretization as secondary to field data. Physical states are usually represented as grid-channel stacks, even when different quantities naturally belong on vertices, edges, faces, cells, boundaries, or interfaces and must satisfy compatibility constraints. We propose Cellular Sheaf Neural Operators, a discretization-aware framework for structure-preserving neural PDE surrogates. The method represents PDE states on oriented cell complexes, couples local feature spaces through learned restriction maps, and uses incidence/Hodge-informed message passing to follow computational geometry. Learned update heads pass through coboundary or flux maps, allowing selected constraints to arise from cell-complex structure rather than only from loss penalties. For magnetohydrodynamics, this yields face-based magnetic-flux updates driven by edge electromotive fields and finite-volume-style fluid updates driven by learned face fluxes and cell sources. On turbulent MHD and fusion-equilibrium surrogate tasks, the method improves structure-sensitive diagnostics, including rollout behavior, divergence control, spectral error, and equilibrium-regression accuracy. These results indicate that cellular-sheaf structure is a useful inductive bias for neural PDE surrogates in constrained multiphysics systems.

We introduce the Laplace-Fourier Neural Operator (LFNO), a unified framework for modeling dynamical systems across transient and steady-state regimes by integrating the spectral advantages of Laplace and Fourier Neural Operators. LFNO employs a dual-branch architecture that explicitly decomposes system dynamics into transient and steady-state components. We evaluate LFNO on nine benchmarks, including three ODE systems (Duffing, Lorenz, and Pendulum) and six PDE systems (Euler-Bernoulli beam, Heat, Reaction-diffusion, Brusselator, Burgers, and Navier-Stokes). LFNO significantly outperforms existing operators on ODE systems, where transient dynamics dominate, and consistently surpasses LNO while achieving performance competitive with FNO on PDE benchmarks. Furthermore, LFNO offers improved stability and physical interpretability through its component-wise decomposition. These results demonstrate that LFNO provides a robust and unified approach for learning complex dynamical systems across multiple temporal scales.

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear combination of basis functions integrated into the network, with coefficients determined by a direct least-squares solve, thereby bypassing traditional gradient-based training. The key methodological contribution include: (1) an efficient tensor-product scheme that generates multi-dimensional basis functions from combinations of two sets of subnetwork outputs, significantly reducing model complexity and parameter count while maintaining expressivity; (2) a block time-marching strategy to improve computational efficiency in long-time simulations; and (3) a linear reformulation strategy for handling nonlinear PDEs by treating known nonlinear terms as sources. TPNet achieves superior accuracy and shorter training times than conventional neural network solvers. This performance gain stems from its structured design and deterministic least-squares fitting, which contrast with the iterative, often computationally intensive optimization required by mainstream methods like Physics-Informed Neural Networks (PINNs).

Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.

While Physics-Informed Neural Networks (PINNs) are powerful for solving Partial Differential Equations (PDEs), their training is often paralyzed by gradient pathology. The gradients from the PDE residuals and boundary constraints oppose each other, trapping the model in local minima. Current solutions, such as adaptive weighting or hard constraints, either fail to fundamentally resolve this ill-conditioning or are limited to simple geometries. In this study, we systematically analyze the possible causes of this gradient pathology from the perspectives of loss landscapes and optimization dynamics. Based on the obtained conclusion, we propose Constraint-Aligned loss with Manifold Lifting (CAML). By reformulating all zeroth-order terms into aligned constraints, our method effectively mitigates gradient conflicts. In addition, we introduce a delay factor to help the optimizer skip the high-curvature area. Experiments demonstrate that our CAML significantly enhances numerical stability and efficiency in highly complex PINN problems. Our code is open-sourced on https://github.com/YichenLuo-0/CAML.

This work introduces the Wavelet-Laplace Neural Operator (WLNO), a novel neural operator that fuses Haar wavelet multi-scale spatial decomposition with the Laplace-domain pole-residue formulation of the Laplace Neural Operator (LNO). While LNO captures transient and steady-state dynamics through learnable system poles and residues, it lacks an explicit mechanism for extracting spatially localized multi-scale features inherent in complex PDE solutions. WLNO addresses this by augmenting the LNO core with a parallel single-level Haar discrete wavelet transform (DWT) branch that decomposes the lifted feature map into four frequency subbands: approximation (LL), horizontal detail (LH), vertical detail (HL), and diagonal detail (HH) and applies independent learned $1\times1$ convolutions to each subband before reconstruction via the inverse DWT. The two branches are fused through a learnable sigmoid-gated weight $α_\mathrm{wav}$, initialized to give a small initial contribution to the wavelet branch, allowing the model to adaptively balance Laplace-domain dynamics against spatial multi-scale features throughout training. WLNO is evaluated against LNO on five benchmark PDE problems using identical hyperparameters, training data, and evaluation protocols: the diffusion equation, the Burgers equation, the reaction-diffusion system, Darcy flow, and the two-dimensional Navier-Stokes equation. WLNO consistently outperforms LNO on all five problems, with the most pronounced improvement on problems with strong spatial multi-scale structure, such as the Burgers equation with sharp shock fronts and the Navier-Stokes equation with coherent vortical structures, while remaining consistent across smoother and elliptic problems. These results demonstrate that wavelet-based multi-scale spatial decomposition is a principled and effective complement to Laplace-domain operator learning.

We present an improved neural field architecture for solving partial differential equations (PDEs). Current physics-informed neural networks (PINNs) provide a flexible framework for solving PDEs, but they struggle to achieve highly accurate solutions and require computation that scales poorly with parameter count. Our model, which we call beignet (Bandlimited Embedding with Interpolated Grid Network), replaces the random Fourier feature embedding used by existing PINN models with a trainable multi-resolution Fourier feature pyramid. To query beignet at a continuous coordinate, we use Fourier interpolation at each level of the pyramid to return features at the input coordinate, and then decode this vector with a fully-connected neural network trunk. Our model provides multiple benefits: 1) Spatial derivatives can be computed efficiently by using the chain rule to compose derivatives of the neural network computed with automatic differentiation with derivatives of the feature grid computed spectrally by the Fast Fourier transform (FFT). 2) beignet can achieve higher accuracy in a compute-efficient manner by scaling the parameter count of this Fourier feature pyramid, instead of the less-efficient strategy of scaling the neural network architecture. 3) beignet can directly control the representation bandlimit, resulting in more stable optimization for difficult PDEs. We demonstrate that beignet finds significantly more accurate solutions on PDE benchmarks using fewer parameters than state-of-the-art PINN methods. We further evaluate beignet on the self-similar inviscid Burgers blowup problem and show that it can minimize residuals to near machine precision using Adam, an accuracy regime previously attained only by using computationally expensive higher-order optimizers.

Physics-informed diffusion models typically enforce PDE constraints only on final outputs, leaving intermediate representations unconstrained and prone to shortcut learning under shifted boundary conditions. We introduce REPA-P, a teacher-free, architecture-agnostic framework that aligns intermediate features with physical states using first-principles residuals. REPA-P attaches lightweight $1{\times}1$ projection heads to selected layers, decodes hidden activations into physical quantities, and applies PDE residual losses during training. These heads are discarded at inference, introducing zero overhead. Across four PDE tasks, including Darcy flow, topology optimization, electrostatic potential, and turbulent channel flow, REPA-P accelerates convergence by up to $2{\times}$, reduces physics residuals by up to $66.4\%$, and improves out-of-distribution robustness by up to $49.3\%$, with consistent gains on both U-Net and Diffusion Transformer backbones. Ablations show that supervising a small set of intermediate layers captures most benefits and complements output-level physics losses. Code is available at https://github.com/Hxxxz0/REPA-P.

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing approaches typically approximate such features within continuous function spaces, often requiring increased model capacity and high-resolution data. In this work, we propose Cut-DeepONet, a two-stage training framework that explicitly models discontinuities while reducing learning complexity. Our approach reformulates the problem via a lifting strategy, partitioning the domain into smooth subregions while representing discontinuities as boundaries in a higher-dimensional space. This separation aligns the operator learning task with the inductive bias of neural networks and avoids directly approximating discontinuities. An additional network predicts input-dependent discontinuity locations for unseen inputs, which are then used to guide the neural operator in generating smooth components within each region. Experiments on benchmark PDEs show that Cut-DeepONet outperforms state-of-the-art methods, even when trained on low-resolution datasets. The method excels on problems with discontinuities and sharp transitions, while using fewer trainable parameters. Our results highlight the benefits of changing the representation of operator learning rather than increasing model complexity.

This work presents an end-to-end strategy for solving inverse problems constrained by Partial Differential Equations within a fully differentiable Machine Learning framework. The proposed formulation provides a unified and user-friendly methodology applicable to a wide range of problems, from data assimilation to closure modeling. Our approach combines a baseline differentiable PDE solver, which predicts the state w from the nonlinear system $R(w) = 0$, with a generic additive, parametrized, and differentiable correction $f_φ(w)$, with trainable parameters $φ$. We show how to optimize phi within a fully differentiable Python workflow by reformulating the PDE as an implicit layer, enabling its integration into arbitrary objective functions, while leveraging PyTorch's automatic differentiation graph. The method is demonstrated on the Reynolds-Averaged Navier-Stokes equations for compressible flows, where the closure term, or a portion of it, is modeled using trainable parameters or a Neural Network. The first application considers the 2D NASA Wall-Mounted Hump test case, where a production-term parameter is optimized against time-averaged LES data. A second application is carried out on the VKI LS-59 turbine blade, where the Spalart-Allmaras eddy viscosity field is reconstructed through the optimization of a trainable spatial field. A dataset is generated starting from the VKI LS-59 turbine blade geometry using the differentiable BROADCAST solver with the Spalart-Allmaras turbulence model. The results highlight the flexibility of the framework, showing its applicability beyond turbulence modeling to a broader class of physics-informed PDE-constrained problems with data-driven components.

Physics-informed neural networks (PINNs) offer a mesh-free framework for solving partial differential equations (PDEs), yet training often suffers from gradient pathologies, spectral bias, and poor convergence, especially for problems with strong nonlinearity, sharp gradients, or multiscale features. We propose the Curriculum-Guided Gaussian Mixture Physics-Informed Neural Network (CGMPINN), which integrates Gaussian mixture modeling with dynamic curriculum learning. Specifically, a GMM is periodically fitted to the PDE residual distribution to quantify spatially varying learning difficulty. A smooth curriculum schedule progressively shifts training focus from easy to harder regions, while precision-based variance modulation suppresses unreliable clusters during early optimization. This dual curriculum is governed by a shared curriculum parameter and can be combined with self-adaptive loss balancing. We further establish theoretical guarantees, including sublinear convergence of the gradient norm for the induced time-varying loss, uniform equivalence between the curriculum-weighted and standard PDE losses, and a generalization bound with an explicit weighting-induced bias characterization. Experiments on six benchmark PDEs spanning elliptic, parabolic, hyperbolic, advection-dominated, and nonlinear reaction-diffusion types show that CGMPINN consistently achieves the lowest relative $L_2$ and maximum absolute errors among all compared methods, reducing relative $L_2$ error by up to 97.8% over the standard PINN at comparable cost. Our code is publicly available at https://github.com/Mathematics-Yang/CGMPINN.

The source distribution in conditional flow matching is a design parameter that can be calibrated to data, not a default isotropic prior. We exploit this in Multi-Fidelity Flow Matching (MFFM), a cascade refinement framework for parametric PDE solutions: the source is calibrated to the empirical low-to-high-fidelity residual scale with local Gaussian-blur correlation, and the velocity network is conditioned on the low-fidelity solution. Conditioning makes the residual refinement problem substantially easier than unconditional field generation, while residual-calibrated source noise improves the flow-matching training geometry. A multi-resolution cascade applies the same construction independently between adjacent fidelities. After level-wise flow-matching pretraining, we fine-tune the composed cascade end-to-end with a deterministic one-step rollout, which makes one velocity evaluation per cascade level the optimized operating point at inference. The result is a learned analog of multigrid refinement that reaches the finest grid in $L$ deterministic network evaluations per query. We validate MFFM on eight benchmarks: two super-resolution problems and six spatiotemporal forecasting tasks from PDEBench, The Well, and the FNO Navier--Stokes dataset.

Partial differential equations (PDEs) are fundamental for modeling complex natural and physical phenomena. In many real-world applications, however, observational data are extremely sparse, which severely limits the applicability of both classical numerical solvers and existing neural approaches. While neural methods have shown promising results under moderately sparse observations, their inference efficiency at high resolutions is limited, and their accuracy degrades substantially in the extremely sparse regime. In this work, we propose the Di-BiLPS, a unified neural framework that effectively handle both forward and inverse PDE problems under extremely sparse observations. Di-BiLPS combines a variational autoencoder to compress high-dimensional inputs into a compact latent space, a latent diffusion module to model uncertainty, and contrastive learning to align representations. Operating entirely in this latent space, the framework achieves efficient inference while retaining flexible input-output mapping. In addition, we introduce a PDE-informed denoising algorithm based on a variance-preserving diffusion process, which further improves inference efficiency. Extensive experiments on multiple PDE benchmarks demonstrate that Di-BiLPS consistently achieves SOTA performance under extremely sparse inputs (as low as 3%), while substantially reducing computational cost. Moreover, Di-BiLPS enables zero-shot super-resolution, as it allows predictions over continuous spatial-temporal domains.

Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structured distribution shifts remains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, the Fourier Neural Operator (FNO) and the Deep Operator Network (DeepONet). To examine their generalization under distribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations in frequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization under distribution shifts in operator learning, underscoring the role of architectural representation bias in developing more reliable neural operators for physics-based PDE simulations beyond the training distribution.

Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free framework that removes this bottleneck completely. We decouple basis discovery from the solving process. We meta-train a dual-branch neural network on diverse Gaussian Random Fields. This offline process creates a universal dictionary of neural basis functions. At test time, we freeze the network. We solve the PDE by assembling a collocation matrix. We find the solution through a single linear least squares step. For non-linear PDEs, we apply the Newton-Raphson method to achieve fast quadratic convergence. Our experiments across six 2D and 3D PDEs show massive improvements. MetaColloc reaches state-of-the-art accuracy on smooth and non-linear problems. It also reduces test-time computation by several orders of magnitude. Finally, we provide a detailed frequency sweep analysis. This analysis reveals a critical mismatch between function approximation and operator stability at extremely high frequencies. This profound finding opens a clear path toward future operator-aware meta-learning.

Most learned PDE solvers follow a global-surrogate paradigm: a neural operator is trained to map full problem descriptions to full solution fields for a prescribed distribution of geometries, boundary conditions, and coefficients. This has enabled fast inference within fixed problem families, but limits reuse across new domains and makes large-scale deployment dependent on expensive problem-specific data generation. We introduce $\textbf{NEST}$ ($\textbf{Ne}$ural-$\textbf{S}$chwarz $\textbf{T}$iling), a local-to-global framework that shifts learning from full-domain solution operators to reusable local physical solvers. The central premise is that, although global PDE solutions depend on geometry, scale, and boundary conditions, the physical response on small neighborhoods can be learned locally and composed into global solutions through classical domain decomposition. NEST learns a neural operator on minimal voxel patches ($3 \times 3 \times 3$) with diverse local geometries and boundary/interface data. At inference time, an unseen voxelized domain is tiled into overlapping patches, the learned local solver is applied patchwise, and global consistency is enforced through iterative Schwarz coupling with partition-of-unity assembly. In this way, generalization is shifted from a monolithic neural model to the combination of local physics learning and algorithmic global assembly. We instantiate NEST on nonlinear static equilibrium in compressible neo-Hookean solids and evaluate it on large, geometrically complex 3D domains far outside the scale of the training patches. Our results show that local neural building blocks, coupled through Schwarz iteration, offer a reusable local-training path toward scalable learned PDE solvers that generalize across domain size, shape, and boundary-condition configurations.

Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically requires large input-output paired datasets generated by costly high-fidelity PDE solvers. Unsupervised operator learning frameworks alleviate data dependency but remain hindered by computational bottlenecks. To address this, we propose Neural Variational Quantum Linear Solver (NVQLS), the first hybrid quantum-classical operator learning framework leveraging the Legendre--Galerkin weak formulation. We critically resolve the sign ambiguity in VQLS energy minimization, preventing erroneous solution representations. Additionally, we introduce a neural embedding, a novel encoding scheme to map varying forcings and PDE coefficients into parameterized quantum circuit representations. These structural innovations provide theoretical computational complexity advantages under efficient state preparation schemes, while achieving superior accuracy compared to a representative classical baseline. Validations on 1D and 2D parametric PDEs under diverse boundary conditions demonstrate NVQLS's capability to simultaneously process varying inputs, offering a scalable unsupervised approach to quantum-enhanced operator learning.

Physics-informed neural networks (PINNs) are a promising approach for solving partial differential equations (PDEs). Their training, however, is often difficult because multiple loss terms induced by PDE residuals and boundary or initial conditions must be optimized simultaneously. To address this difficulty, existing approaches often construct update directions by explicitly enforcing particular desirable properties, such as scale robustness and simultaneous descent. While effective in many cases, such property-by-property designs can make it unclear which conditions are essential, what geometric principle determines the selected update direction, and how different methods are structurally related. In this work, we formulate update-direction selection for PINN training as a Chebyshev-center problem in the dual cone. The proposed formulation selects a normalized direction that maximizes the minimum distance to the cone facets. The resulting formulation admits an efficient dual problem in a much lower-dimensional space and yields a convergence guarantee in the nonconvex setting. It also recovers the key desirable properties targeted by existing approaches without imposing them separately; rather, they follow from the single geometric criterion underlying the formulation. This makes the selected direction interpretable through a single geometric rule and provides a unified basis for systematically comparing related direction-selection methods. Experiments on several PINN benchmarks further demonstrate strong empirical performance of the proposed method.

PDE-to-solver code generation aims to automatically synthesize executable numerical solvers from partial differential equation (PDE) specifications. This task requires not only understanding the mathematical structure of PDEs, but also selecting appropriate discretization schemes and solver configurations, and correctly implementing the resulting formulations in finite-element method (FEM) libraries. Existing code generation benchmarks mainly evaluate syntactic correctness, or success on predefined test cases. To our knowledge, there is currently no publicly available benchmark specifically for PDE-to-solver code generation, and general-purpose code benchmarks do not fully capture the unique challenges of numerical PDE solution, such as ensuring solver accuracy, efficiency, and compatibility with professional FEM libraries. We introduce PDEAgent-Bench, to the best of our knowledge, the first multi-metric, multi-library benchmark for PDE-to-solver code generation. PDEAgent-Bench contains 645 instances across 6 mathematical categories and 11 PDE families, with common FEM libraries for DOLFINx, Firedrake, and deal.II. Each instance provides an agent-facing problem specification, a reference solution on a prescribed evaluation grid, and case-specific accuracy and runtime targets. PDEAgent-Bench adopts a staged evaluation framework in which generated solvers must sequentially pass executability, numerical accuracy, and computational efficiency checks. Experiments with representative LLMs and code agents show that models can often produce runnable code, but their pass rate drops substantially once accuracy and efficiency requirements are enforced. These results indicate that current agents remain limited in producing numerically reliable and efficient PDE solvers, and that PDEAgent-Bench provides a reproducible testbed grounded in the practical requirements of numerical PDE solving.

Partial differential equation (PDE) solvers underpin scientific computing, but real-world deployment is bounded by compute. Classical Monte Carlo solvers such as Walk-on-Spheres (WoS) are unbiased and geometry-agnostic but are slow. Learned solvers are fast but biased and brittle under distribution shift. We present \textbf{MC$^2$}, a hybrid WoS-Neural Network (WoS-NN) PDE solver that treats a low-budget Monte Carlo solution as a structured estimator of the true field and learns a single-pass neural correction to recover a high-fidelity solution. MC$^2$ matches the accuracy of solutions using over $1000\times$ more Monte Carlo compute, outperforming all evaluated classical, denoising, and neural-operator baselines. To enable reproducible study of finite-compute PDE solving, we additionally release \textbf{PDEZoo}, the largest standardized elliptic PDE benchmark to date: 2M PDEs spanning five elliptic families and unlimited geometric compositions, with analytic ground truth and multi-budget Monte Carlo trajectories. Together \textbf{MC$^2$} and \textbf{PDEZoo} (1) empirically establish that finite-sample Monte Carlo error is structured, learnable, and correctable in a single forward pass, (2) show that we can solve PDEs $\sim$\textbf{1000x} faster than with just WoS, and (3) provide the evaluation infrastructure the field has so far lacked.