PDE Solving

PDE: Partial Differential Equation

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Overview

PDE solving seeks approximate solutions to equations describing processes such as heat flow, fluids, waves, and diffusion, where exact solutions are often unavailable. Classical approaches include finite difference, finite element, finite volume, spectral, and domain-decomposition methods; learning-based approaches include physics-informed neural networks, neural operators, and randomized or basis-based solvers. Current work also combines these methods, using learned models for acceleration, adaptive refinement, or surrogate prediction while retaining numerical constraints such as conservation and boundary conditions. A central challenge is balancing speed and flexibility with accuracy, stability, reliable generalization, and guarantees that residual errors reflect true solution errors.

Latest papers

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255 papers30 papers in the last four weeks, up 20% on the four weeks before. 0.2% of all new papers.

Apr 26, 2026

A Coupled Fourth Order Telegraph Diffusion Framework Using Grayscale Indicators for Image Despeckling

Speckle noise severely limits the quality of images acquired from coherent imaging systems such as Synthetic Aperture Radar (SAR) and medical ultrasound. Traditional second-order PDE-based despeckling approaches, although popular, often introduce staircase artifacts and blur fine details. To overcome these limitations, we present a nonlinear, fourth-order coupled hyperbolic-parabolic PDE model that effectively reduces noise while preserving the structure. The framework consists of two evolution equations: one governing fourth-order diffusion for effective speckle reduction and smooth intensity transitions, and another refining an edge indicator to protect textures and structural features. The diffusion coefficient is adaptively constructed using both the image intensity variable u and a grayscale-based indicator function, ensuring structure-aware denoising while avoiding blocky artifacts and preserving fine details. We also prove the existence of a weak solution to the proposed model by applying Schauder fixed-point theorem. A finite-difference scheme with Gauss Seidel iteration is employed for efficient implementation. We compare the proposed model with the existing coupled second-order PDE model (HPCPDE) and the fourth-order telegraph diffusion model (TDFM). The results show that our model consistently outperforms these approaches. Experiments on standard grayscale images, real SAR and ultrasound data, as well as speckle-corrupted color images, demonstrate that the proposed method achieves superior performance over conventional PDE-based techniques in terms of PSNR, MSSIM, and Speckle Index.

Manish Kumar, Rajendra K. Ray

School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi (H.P), 175075, India

PDE SolvingSynthetic Aperture RadarImage Denoising
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Apr 26, 2026

Single Image Defogging Using a Fourth-Order Telegraph PDE Guided by Physical Haze Modeling

In real-world scenarios, image defogging is an inverse problem due to unknown scene depth, atmospheric scattering, and the common absence of ground truth . To resolve the issue, we propose a hybrid defogging model that integrates a fourth-order nonlinear PDE with a physical haze formation model. We used Dark Channel Prior to estimate atmospheric parameters and to generate a guidance image, while the final restoration is performed via a fourth-order PDE-based evolution. A fourth-order PDE of the type telegraph is then evolved, incorporating an edge-adaptive diffusion coefficient and a fidelity term weighted by the transmission map. Fourth-order diffusion effectively suppresses haze while preserving structural details, and the hyperbolic formulation improves numerical stability and convergence behavior. We use relative error norm criteria for the convergence of our PDE. The proposed method is compared with Dark Channel prior, modified Dark Channel prior, and variational-based single-image defogging techniques. When we have ground truth available, we use MSE and SSIM for quantitative evaluation, whereas no-reference metrics, including FADE, Contrast Restoration Index, Average Gradient, and Entropy, are applied to real-world foggy images. Experimental results demonstrate that the proposed hybrid PDE-based method provides comparable visual quality and maintains structural details.

Manish Kumar, Rajendra K. Ray

School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi (H.P), 175075, India

PDE SolvingPhysics-Guided Image RestorationImage RestorationImage Dehazing
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Apr 26, 2026

When PINNs Go Wrong: Pseudo-Time Stepping Against Spurious Solutions

Physics-informed neural networks (PINNs) provide a promising machine learning framework for solving partial differential equations, but their training often breaks down on challenging problems, sometimes converging to physically incorrect solutions despite achieving small residual losses. This failure, we argue, is not merely an optimization difficulty. Rather, it reflects a fundamental weakness of the empirical PDE residual loss, which can admit trivial or spurious solutions during training. From this perspective, we revisit pseudo-time stepping, a technique that has recently shown strong empirical success in PINNs. We show that its main benefit is not simply to ease optimization; instead, when combined with collocation-point resampling, it helps reveal and avoid spurious solutions. At the same time, we find that the effectiveness of pseudo-time stepping depends critically on the choice of step size, which cannot be tuned reliably from the training loss alone. To overcome this limitation, we propose an adaptive pseudo-time stepping strategy that selects the step size from a finite-difference surrogate of the local residual Jacobian, yielding the largest step permitted by local stability without per-problem tuning. Across a diverse set of PDE benchmarks, the proposed method consistently improves both accuracy and robustness. Together, these findings provide a clearer understanding of why PINNs fail and suggest a practical pathway toward more reliable physics-informed learning. All code and data accompanying this manuscript are available at https://github.com/sifanexisted/jaxpi2.

Sifan Wang, Shawn Koohy, Yiping Lu +1

Institute for Foundations of Data Science, Yale University · Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania · Industrial Engineering & Management Science, Northwestern University

PDE SolvingPhysics-Informed ML
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Apr 23, 2026

Transferable Physics-Informed Representations via Closed-Form Head Adaptation

Physics-informed neural networks (PINNs) have garnered significant interest for their potential in solving partial differential equations (PDEs) that govern a wide range of physical phenomena. By incorporating physical laws into the learning process, PINN models have demonstrated the ability to learn physical outcomes reasonably well. However, current PINN approaches struggle to predict or solve new PDEs effectively when there is a lack of training examples, indicating they do not generalize well to unseen problem instances. In this paper, we present a transferable learning approach for PINNs premised on a fast Pseudoinverse PINN framework (Pi-PINN). Pi-PINN learns a transferable physics-informed representation in a shared embedding space and enables rapid solving of both known and unknown PDE instances via closed-form head adaptation using a least-squares-optimal pseudoinverse under PDE constraints. We further investigate the synergies between data-driven multi-task learning loss and physics-informed loss, providing insights into the design of more performant PINNs. We demonstrate the effectiveness of Pi-PINN on various PDE problems, including Poisson's equation, Helmholtz equation, and Burgers' equation, achieving fast and accurate physics-informed solutions without requiring any data for unseen instances. Pi-PINN can produce predictions 100-1000 times faster than a typical PINN, while producing predictions with 10-100 times lower relative error than a typical data-driven model even with only two training samples. Overall, our findings highlight the potential of transferable representations with closed-form head adaptation to enhance the efficiency and generalization of PINNs across PDE families and scientific and engineering applications.

Jian Cheng Wong, Isaac Yin Chung Lai, Pao-Hsiung Chiu +3

Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore · National University of Singapore (NUS), Singapore · Indian Institute of Technology Goa (IIT Goa), India +1

Representation LearningPDE SolvingPhysics-Informed ML
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Apr 23, 2026

A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization

Standard physics-informed neural networks (PINNs) struggle to simulate highly oscillatory Helmholtz solutions in heterogeneous media because pointwise minimization of second-order PDE residuals is computationally expensive, biased toward smooth solutions, and requires artificial absorbing boundary layers to restrict the solution. To overcome these challenges, we introduce a Green-Integral (GI) neural solver for the acoustic Helmholtz equation. It departs from the PDE-residual-based formulation by enforcing wave physics through an integral representation that imposes a nonlocal constraint. Oscillatory behavior and outgoing radiation are encoded directly through the integral kernel, eliminating second-order spatial derivatives and enforcing physical solutions without additional boundary layers. Theoretically, optimizing this GI loss via a neural network acts as a spectrally tuned preconditioned iteration, enabling convergence in heterogeneous media where the classical Born series diverges. By exploiting FFT-based convolution to accelerate the GI loss evaluation, our approach substantially reduces GPU memory usage and training time. However, this efficiency relies on a fixed regular grid, which can limit local resolution. To improve local accuracy in strong scattering regions, we also propose a hybrid GI+PDE loss, enforcing a lightweight Helmholtz residual at a small number of nonuniformly sampled collocation points. We evaluate our method on seismic benchmark models characterized by structural contrasts and subwavelength heterogeneity at frequencies up to 20Hz. GI-based training consistently outperforms PDE-based PINNs, reducing computational cost by over a factor of ten. In models with localized scattering, the hybrid loss yields the most accurate reconstructions, providing a stable, efficient, and physically grounded alternative.

Mohammad Mahdi Abedi, David Pardo, Tariq Alkhalifah

University of the Basque Country (UPV/EHU), Department of Mathematics, Spain · Basque Center for Applied Mathematics (BCAM), Bilbao, Spain · Ikerbasque, Basque Foundation for Science, Bilbao, Spain +1

Neural PDE SolversPDE SolvingPhysics-Informed ML
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Apr 22, 2026

CO2_2 sequestration hybrid solver using isogeometric alternating-directions and collocation-based robust variational physics informed neural networks (IGA-ADS-CRVPINN)

This paper presents the hybrid solver for a CO2CO_2 sequestration problem. The solver uses the IGA-ADS (IsoGeometric Analysis Alternating Directions solver) to compute the saturation scalar field update using the explicit method, and CRVPINN (Collocation-based Robust Variational Physics Informed Neural Networks solver) to compute the pressure scalar field. The study focuses on simulating the physical behavior of CO2CO_2 in porous structures, excluding chemical reactions. The mathematical model is based on Darcy's Law. The CRVPINN is pretrained on the initial pressure configuration, and the time step pressure updates require only 100 iterations of the Adam method per time step. We compare our hybrid IGA-ADS solver, coupled with the CRVPINN method, with a baseline of the IGA-ADS solver coupled with the MUMPS direct solver. Our hybrid solver is over 3 times faster on a single computational node from the ARES cluster of ACK CYFRONET. Future work includes extensive testing, inverse problem solving, and potential application to H2H_2 storage problems.

Askold Vilkha, Tomasz Służalec, Marcin Łoś +1

AGH University of Krakow, Faculty of Computer Science, Al. Mickiewicza 2026 30, Kraków, 30-059, Poland

Neural PDE SolversPorous Media FlowPDE SolvingPhysics-Informed MLComputational Fluid Dynamics
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Apr 19, 2026

Virtual boundary integral neural network for three-dimensional exterior acoustic problems

This paper presents a virtual boundary integral neural network (VBINN) for exterior acoustic problems in three dimensions. The method introduces a virtual boundary inside the scatterer or vibrating body and represents the associated source density with a neural network. Coupled with the acoustic fundamental solution, this representation satisfies the Sommerfeld radiation condition by construction and enables direct evaluation of the acoustic pressure and its normal derivative at arbitrary field points. Because the integration surface is separated from the physical boundary, the formulation avoids the singular and near singular kernel evaluations associated with coincident source and collocation points in conventional boundary integral learning methods. To reduce sensitivity to boundary placement, the geometric parameters of the virtual boundary are optimized jointly with the source density during training. Numerical examples for acoustic scattering, multiple body interaction, and underwater acoustic propagation show close agreement with analytical solutions and COMSOL results, and the Burton Miller extension further improves stability near characteristic frequencies. These results demonstrate the potential of VBINN for exterior acoustic analysis in three dimensions.

Jiahao Li, Qiang Xi, Ilia Marchevskiy +1

College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, China · Department of Applied Mathematics, Bauman Moscow State Technical University, Moscow 105005, Russia

Neural PDE SolversPDE Solving
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Apr 18, 2026

Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches

This thesis develops numerical and theoretical approaches for understanding and analyzing singularity formation in Partial Differential Equations (PDEs). The singularity formation in the Navier-Stokes Equation (NSE) is famously challenging as one of the seven Clay Prize problems. Unlike simpler equations such as the Nonlinear Heat (NLH) or Keller-Segel (KS) equations, where formal asymptotics near blowup are better understood, the intrinsic complexity of NSE makes quantitative analytical treatment difficult, if not impossible, without numerical guidance. Building on numerical insights, we introduce a robust analytical framework to simplify and systematize pen-and-paper proofs for simpler singular PDEs. We present a novel approach based on enforcing vanishing modulation conditions for perturbations around approximate blowup profiles, complemented by singularly weighted energy estimates. We demonstrate the efficacy of our method on PDEs with complicated asymptotics, such as NLH and the Complex Ginzburg-Landau (CGL) equation, and address the open problem of singularity formation in the 3D KS equation with logistic damping. We develop and refine numerical approaches that facilitate deeper insights into singularity formation. We demonstrate that machine learning methods significantly enhance our capability to identify and characterize potential blowup solutions with high precision. We improve on existing Physics-Informed Neural Network (PINN) and Neural Operator (NO) frameworks. Moreover, we present a novel machine learning paradigm, the Kolmogorov-Arnold Network (KAN) architecture, whose interpretability and excellent scaling properties are achieved through learnable nonlinearities.

Yixuan Wang

CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California · [math.NA]

PDE SolvingKolmogorov-Arnold NetworksPDE Operator LearningPhysics-Informed ML
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Apr 17, 2026

A Randomized PDE Energy driven Iterative Framework for Efficient and Stable PDE Solutions

Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require costly training and often suffer from limited generalization. In this work, we proposes a PDE energy driven framework that solves PDEs through physically constrained diffusion iterations, without relying on classical matrix based finite element assembly or data driven neural network training. The proposed method evolves arbitrary random initial fields through PDE energy driven implicit iterations combined with Gaussian smoothing, while strictly enforcing boundary conditions at each iteration. The proposed formulation is applied to representative one dimensional Poisson, Heat, and viscous Burgers equations, covering both steady state and transient problems. Numerical results demonstrate stable convergence to the unique physical solution from random initializations, with accurate resolution of sharp gradients and controlled Mean Squared Error (MSE) across a wide range of discretization parameters. Detailed comparisons with analytical solutions indicate that the framework achieves competitive accuracy and stability. Overall, the proposed framework provides a fast, flexible, and physically consistent alternative to traditional numerical solvers, offering a potential pathway for scalable PDE solutions in both research and engineering applications.

Yi Bing, Zheng Ran, Fu Jinyang +2

School of Traffic and Transportation Engineering, Central South University, Changsha, China · School of Civil Engineering, Central South University, Changsha, China · National Engineering Research Center of High-speed Railway Construction Technology, Changsha, China +2

PDE Solving
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Apr 17, 2026

A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Laws

Hyperbolic conservation laws govern a wide range of transport-driven dynamics featuring shocks, contact discontinuities, and complex wave interactions, posing distinct challenges for deep-learning-based surrogate modeling. While classical numerical methods provide robust and physically admissible solutions, their computational cost restricts applicability in many-query tasks such as parametric studies and design optimization. Conversely, existing neural surrogates offer rapid inference but often fail to respect intrinsic PDE structures, leading to non-physical artifacts, rollout instability, and poor generalization. We present an interpretable, structure-preserving graph neural solver that bridges classical numerical principles with graph neural networks (GNNs). The network is designed as a learned reconstruction-and-flux operator rather than a black-box state updater, thereby inherently preserving key properties such as local conservation and upwinding. Inspired by Arbitrary high-order DERivatives schemes, we further recast message-passing GNNs as high-order space-time predictors, enabling conservative and stable neural updates with large time steps. Evaluation is performed on challenging supersonic flow benchmarks spanning broad parametric variations in geometry, initial/boundary conditions, and flow regimes. The neural solver achieves superior long-horizon rollout stability and accuracy compared with strong surrogate baselines, outperforms low-order discretizations, and delivers orders-of-magnitude runtime speedups over high-resolution simulations.

Jiamin Jiang, Shanglin Lv, Jingrun Chen

School of Artificial Intelligence and Data Science and Suzhou Institute for Advanced Research, University of Science and Technology of China, and Suzhou Big Data & AI Research and Engineering Center, China · School of Mathematical Sciences, University of Science and Technology of China, China · School of Mathematical Sciences and Suzhou Institute for Advanced Research, University of Science and Technology of China, and Suzhou Big Data & AI Research and Engineering Center, China

Graph Neural NetworksNeural PDE SolversPDE SolvingPDE Surrogate ModelingStructure-Preserving Neural Networks
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Apr 16, 2026

Python library supporting Discrete Variational Formulations and training solutions with Collocation-based Robust Variational Physics Informed Neural Networks (DVF-CRVPINN)

We explore the possibility of solving Partial Differential Equations (PDEs) using discrete weak formulations. We propose a programming environment for defining a discrete computational domain, introducing discrete functions defined over a set of points, constructing discrete inner products, and introducing discrete weak formulations employing Kronecker delta test functions. Building on this setup, we propose a discrete neural network representation, training the solution function defined over a discrete set of points and employing discrete finite difference derivatives in the automatic differentiation procedures. As a challenging computational model example, we focus on Stokes equations in two-dimensions, defined over a discrete set of points. We train the solution using the discrete weak residual and the Adamax algorithm with discrete automatic differentiation of the discrete gradients. Despite introducing the python environment, we also provide a rigorous mathematical formulation based on discrete weak formulations, proving the well-posedness and robustness of the loss function. The solution of the discrete weak formulations is based on neural network training employing a robust loss function that is related to the true error. In this way, we have a robust control of the numerical error during the training of the neural networks. Besides the Stokes formulation, we also explain the functionality of the proposed library using the Laplace problem formulation.

Tomasz Służalec, Marcin Łoś, Askold Vilkha +1

AGH University of Krakow, Poland

PDE SolvingPhysics-Informed ML
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Apr 15, 2026

Physics-Informed Neural Networks for Solving Derivative-Constrained PDEs

Physics-Informed Neural Networks (PINNs) recast PDE solving as an optimisation problem in function space by minimising a residual-based objective, yet many applications require additional derivative-based relations that are just as fundamental as the governing equations. In this paper, we present Derivative-Constrained PINNs (DC-PINNs), a general framework that treats constrained PDE solving as an optimisation guided by a minimum objective function criterion where the physics resides in the minimum principle. DC-PINNs embed general nonlinear constraints on states and derivatives, e.g., bounds, monotonicity, convexity, incompressibility, computed efficiently via automatic differentiation, and they employ self-adaptive loss balancing to tune the influence of each objective, reducing reliance on manual hyperparameters and problem-specific architectures. DC-PINNs consistently reduce constraint violations and improve physical fidelity versus baseline PINN variants, representative hard-constraint formulations on benchmarks, including heat diffusion with bounds, financial volatilities with arbitrage-free, and fluid flow with vortices shed. Explicitly encoding derivative constraints stabilises training and steers optimisation toward physically admissible minima even when the PDE residual alone is small, providing reliable solutions of constrained PDEs grounded in energy minimum principles.

Kentaro Hoshisashi, Carolyn E Phelan, Paolo Barucca

Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK

Neural PDE SolversPDE SolvingPhysics-Informed MLPartial Differential Equations
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Apr 15, 2026

Comparative Study of Bending Analysis using Physics-Informed Neural Networks and Numerical Dynamic Deflection in Perforated nanobeam

In this chapter, we investigate the bending behavior of a perforated nanobeam subjected to sinusoidal loading using an efficient and computationally robust Physics-Informed Functional Link Constrained Framework with Domain Mapping (DFL-TFC) method. Our aim is to determine the relationship between static bending response and dynamic deflection of a perforated nanobeam for various perforation cases. The static bending is obtained using the FL-TFC with Domain mapped method, whereas dynamic deflection is determined using the Galerkin method. The proposed approach employs the theory of functional connections (TFC) to systematically embed governing differential equation constraints into a constrained expression (CE), which exactly satisfies all prescribed initial and boundary conditions (ICs and BCs) and domain of differential equation is mapped to domain of orthogonal polynomials. Within this framework, the free function appearing in the constrained expression is expressed through a functional link neural network (FLNN). The cost is minimized by the mean square residual of DE, allowing training without requiring complex deep network architectures. Relationship between static and dynamic defection of simply-supported (S-S) perforated nanobeams has been investigated here. FL-TFC with Domain mapped method eliminates the need for deep and complex neural network architectures while ensuring accuracy, efficiency, and strict satisfaction of boundary conditions as compared to standard PINN.

Ramanath Garai, Iswari Sahu, S. Chakraverty

Department of Mathematics, National Institute of Technology Rourkela, Rourkela-769008, India.

Structural MechanicsPhysics-Informed MLPDE Solving
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Apr 13, 2026

SCNO: Spiking Compositional Neural Operator -- Towards a Neuromorphic Foundation Model for Nuclear PDE Solving

Neural operators have emerged as powerful surrogates for partial differential equation (PDE) solvers, yet they are typically trained as monolithic models for individual PDEs, require energy-intensive GPU hardware, and must be retrained from scratch when new physics emerge. We introduce the Spiking Compositional Neural Operator (SCNO), a modular architecture combining spiking and conventional components that addresses all three limitations. SCNO maintains a library of small spiking neural operator blocks, each trained on a single elementary differential operator (convection, diffusion, reaction), and composes them through a lightweight input-conditioned aggregator to solve coupled PDEs not seen during block training. A small correction network learns cross-coupling residuals while keeping all blocks and the aggregator frozen, preserving zero-forgetting modular expansion by construction. We evaluate SCNO on eight PDE families including five coupled systems and a nuclear-relevant 1-group neutron diffusion equation. SCNO with correction achieves the lowest relative L2L^2 error on four of five coupled PDEs, outperforming both a monolithic spiking DeepONet (by up to 62%, mean over 3 seeds) and a standard ANN DeepONet (by up to 65%), while requiring only 95K trainable parameters versus 462K for the monolithic baseline. To our knowledge, this is the first compositional spiking neural operator and the first proof-of-concept for modular neuromorphic PDE solving with built-in forgetting-free expansion.

Samrendra Roy, Souvik Chakraborty, Rizwan-uddin +1

University of Illinois Urbana-Champaign Department of Nuclear, Plasma & Radiological Engineering Urbana, IL, USA · Indian Institute of Technology Delhi Department of Applied Mechanics New Delhi, India · National Center for Supercomputing Applications Urbana, IL, USA

PDE SolvingPDE Surrogate ModelingNeural OperatorsModular Neural NetworksNeuromorphic Computing
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Apr 13, 2026

Generative Path-Finding Method for Wasserstein Gradient Flow

Wasserstein gradient flows (WGFs) describe the evolution of probability distributions in Wasserstein space as steepest descent dynamics for a free energy functional. Computing the full path from an arbitrary initial distribution to equilibrium is challenging, especially in high dimensions. Eulerian methods suffer from the curse of dimensionality, while existing Lagrangian approaches based on particles or generative maps do not naturally improve efficiency through time step tuning. We propose GenWGP, a generative path finding framework for Wasserstein gradient paths. GenWGP learns a generative flow that transports mass from an initial density to an unknown equilibrium distribution by minimizing a path loss that encodes the full trajectory and its terminal equilibrium condition. The loss is derived from a geometric action functional motivated by Dawson Gartner large deviation theory for empirical distributions of interacting diffusion systems. We formulate both a finite horizon action under physical time parametrization and a reparameterization invariant geometric action based on Wasserstein arclength. Using normalizing flows, GenWGP computes a geometric curve toward equilibrium while enforcing approximately constant intrinsic speed between adjacent network layers, so that discretized distributions remain nearly equidistant in the Wasserstein metric along the path. This avoids delicate time stepping constraints and enables stable training that is largely independent of temporal or geometric discretization. Experiments on Fokker Planck and aggregation type problems show that GenWGP matches or exceeds high fidelity reference solutions with only about a dozen discretization points while capturing complex dynamics.

Chengyu Liu, Xiang Zhou

Department of Data Science, City University of Hong Kong, Kowloon, Hong Kong SAR · Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR

Fokker–Planck EquationWasserstein Gradient FlowsPDE Solving
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Apr 10, 2026

Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domains

This paper introduces the Stochastic-Dimension Frozen Sampled Neural Network (SD-FSNN), a novel computational framework for solving high-dimensional Gross-Pitaevskii equation (GPE) on unbounded domain. The proposed method circumvents the curse-of-dimensionality that plagues traditional discretizations and the computational bottlenecks of gradient-based neural network solvers through a synergistic combination of techniques. First, a prescribed Gaussian envelope encodes the far-field decay of the wavefunction, enabling a space-time separation where the spatial approximation is handled by a frozen, single-hidden-layer neural network with data-driven sampled features. This yields a gradient-free formalism where spatial derivatives are analytically precomputed and time-dependence is evolved via reduced ODEs. Second, a stochastic-dimension sampler provides a conditionally unbiased estimate of the spatial operator by evaluating only a small subset of spatial dimensions at each time step, essentially reducing computational and memory costs. Discrete conservation laws are also enforced, ensuring long-term stability. Extensive numerical experiments on GPE in up to 1000 dimensions demonstrate that SD-FSNN achieves significantly higher accuracy and efficiency compared to state-of-the-art methods, including PINNs, randomized feature methods, and tensor-network approaches. The results confirm that SD-FSNN effectively mitigates the Kolmogorov nn-width barrier for frozen-basis models on structured solution manifolds.

Zhangyong Liang

National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China.

Neural PDE SolversPDE Solving
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Apr 10, 2026

Meta-Learned Basis Adaptation for Parametric Linear PDEs

We propose a hybrid physics-informed framework for solving families of parametric linear partial differential equations (PDEs) by combining a meta-learned predictor with a least-squares corrector. The predictor, termed KAPI (Kernel-Adaptive Physics-Informed meta-learner), is a shallow task-conditioned model that maps query coordinates and PDE parameters to solution values while internally generating an interpretable, task-adaptive Gaussian basis geometry. A lightweight meta-network maps PDE parameters to basis centers, widths, and activity patterns, thereby learning how the approximation space should adapt across the parametric family. This predictor-generated geometry is transferred to a second-stage corrector, which augments it with a background basis and computes the final solution through a one-shot physics-informed Extreme Learning Machine (PIELM)-style least-squares solve. We evaluate the method on four linear PDE families spanning diffusion, transport, mixed advection--diffusion, and variable-speed transport. Across these cases, the predictor captures meaningful physics through localized and transport-aligned basis placement, while the corrector further improves accuracy, often by one or more orders of magnitude. Comparisons with parametric PINNs, physics-informed DeepONet, and uniform-grid PIELM correctors highlight the value of predictor-guided basis adaptation as an interpretable and efficient strategy for parametric PDE solving.

Vikas Dwivedi, Monica Sigovan, Bruno Sixou

CREATIS Biomedical Imaging Laboratory INSA, CNRS UMR 5220, Inserm, Université Lyon 1 Lyon 69621, France

PDE SolvingPDE Surrogate ModelingPartial Differential EquationsPhysics-Informed MLMeta-Learning
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Apr 8, 2026

Physics-Informed Functional Link Constrained Framework with Domain Mapping for Solving Bending Analysis of an Exponentially Loaded Perforated Beam

This article presents a novel and comprehensive approach for analyzing bending behavior of the tapered perforated beam under an exponential load. The governing differential equation includes important factors like filling ratio (αα), number of rows of holes (NN), tapering parameters (φφ and ψψ), and exponential loading parameter (γγ), providing a realistic and flexible representation of perforated beam configuration. Main goal of this work is to see how well the Domain mapped physics-informed Functional link Theory of Functional Connection (DFL-TFC) method analyses bending response of perforated beam with square holes under exponential loading. For comparison purposes, a corresponding PINN-based formulation is developed. Outcomes clearly show that the proposed DFL-TFC framework gives better results, including faster convergence, reduced computational cost, and improved solution accuracy when compared to the PINN approach. These findings highlight effectiveness and potential of DFL-TFC method for solving complex engineering problems governed by differential equations. Within this framework, hidden layer is replaced by a functional expansion block that enriches input representation via orthogonal polynomial basis functions, and the domain of DE mapped to corresponding domain of orthogonal polynomials. A Constrained Expression (CE), constructed through the Theory of Functional Connections (TFC) using boundary conditions, ensures that constraints are exactly satisfied. In CE, free function is represented using a Functional Link Neural Network (FLNN), which learns to solve resulting unconstrained optimization problem. The obtained results are further validated through the Galerkin and PINN solutions.

Iswari Sahu, Ramanath Garai, S. Chakraverty

Department of Mathematics, National Institute of Technology Rourkela, India

Physics-Informed MLStructural MechanicsPDE Solving
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Apr 8, 2026

Modelling Gas-Phase Reaction Kinetics with Guided Particle Diffusion Sampling

Physics-guided sampling with diffusion priors has recently shown strong performance in solving complex systems of partial differential equations (PDEs) from sparse observations. However, these methods are typically evaluated on benchmark problems that do not fully demonstrate their ability to generate temporally consistent solutions of time-dependent PDEs, often focusing instead on reconstructing a single snapshot. In this work, we apply these methods to gas-phase reaction kinetics problems governed by the advection-reaction-diffusion (ARD) equation, providing a setting that more closely reflects realistic laboratory experiments. We demonstrate that guided sampling can be used to reconstruct full spatiotemporal trajectories, rather than isolated states. Furthermore, we show that these methods generalise to previously unseen parameter regimes, highlighting their potential for real-world applications.

Andrew Millard, Zheng Zhao, Henrik Pedersen

Division of Statistics and Machine Learning (STIMA), Linköping University, Sweden · STIMA, Linköping University, Sweden · Division of Chemistry, Linköping University, Sweden

PDE SolvingDiffusion Posterior Sampling
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Apr 7, 2026

A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributions

Efficiently solving the Fokker-Planck equation (FPE) is central to analyzing complex parameterized stochastic systems. However, current numerical methods lack parallel computation capabilities across varying conditions, severely limiting comprehensive parameter exploration and transient analysis. This paper introduces a deep learning-based pseudo-analytical probability solution (PAPS) that, via a single training process, simultaneously resolves transient FPE solutions for arbitrary multi-modal initial distributions, system parameters, and time points. The core idea is to unify initial, transient, and stationary distributions via Gaussian mixture distributions (GMDs) and develop a constraint-preserving autoencoder that bijectively maps constrained GMD parameters to unconstrained, low-dimensional latent representations. In this representation space, the panoramic transient dynamics across varying initial conditions and system parameters can be modeled by a single evolution network. Extensive experiments on paradigmatic systems demonstrate that the proposed PAPS maintains high accuracy while achieving inference speeds four orders of magnitude faster than GPU-accelerated Monte Carlo simulations. This efficiency leap enables previously intractable real-time parameter sweeps and systematic investigations of stochastic bifurcations. By decoupling representation learning from physics-informed transient dynamics, our work establishes a scalable paradigm for probabilistic modeling of multi-dimensional, parameterized stochastic systems.

Xiaolong Wang, Jing Feng, Qi Liu +3

School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, China · School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China · MOE Key Laboratory for Complexity Science in Aerospace, Northwestern Polytechnical University, Xi’an, 710072, China +2

PDE SolvingNeural PDE SolversLatent Dynamics Modeling
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Apr 7, 2026

Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies

Fluid flows are governed by the nonlinear Navier-Stokes equations, which can manifest multiscale dynamics even from predictable initial conditions. Predicting such phenomena remains a formidable challenge in scientific machine learning, particularly regarding convergence speed, data requirements, and solution accuracy. In complex fluid flows, these challenges are exacerbated by long-range spatial dependencies arising from distant boundary conditions, which typically necessitate extensive supervision data to achieve acceptable results. We propose the Domain-Decomposed and Shifted Physics-Informed Neural Network (DDS-PINN), a framework designed to resolve such multiscale interactions with minimal supervision. By utilizing localized networks with a unified global loss, DDS-PINN captures global dependencies while maintaining local precision. The robustness of the approach is demonstrated across a suite of benchmarks, including a multiscale linear differential equation, the nonlinear Burgers' equation, and data-free Navier-Stokes simulations of flat-plate boundary layers. Finally, DDS-PINN is applied to the computationally challenging backward-facing step (BFS) problem; for laminar regimes (Re = 100), the model yields results comparable to computational fluid dynamics (CFD) without the need for any data, accurately predicting boundary layer thickness, separation, and reattachment lengths. For turbulent BFS flow at Re = 10,000, the framework achieves convergence to O(10^-4) using only 500 random supervision points (< 0.3 % of the total domain), outperforming established methods like Residual-based Attention-PINN in accuracy. This approach demonstrates strong potential for the super-resolution of complex turbulent flows from sparse experimental measurements.

Prashant Kumar, Rajesh Ranjan

Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016

Domain Decomposition MethodsNeural PDE SolversPhysics-Informed MLPDE SolvingScientific MLComputational Fluid Dynamics
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Apr 3, 2026

A Numerical Method for Coupling Parameterized Physics-Informed Neural Networks and FDM for Advanced Thermal-Hydraulic System Simulation

Severe accident analysis using system-level codes such as MELCOR is indispensable for nuclear safety assessment, yet the computational cost of repeated simulations poses a significant bottleneck for parametric studies and uncertainty quantification. Existing surrogate models accelerate these analyses but depend on large volumes of simulation data, while physics-informed neural networks (PINNs) enable data-free training but must be retrained for every change in problem parameters. This study addresses both limitations by developing the Parameterized PINNs coupled with FDM (P2F) method, a node-assigned hybrid framework for MELCOR's Control Volume Hydrodynamics/Flow Path (CVH/FP) module. In the P2F method, a parameterized Node-Assigned PINN (NA-PINN) accepts the water-level difference, initial velocity, and time as inputs, learning a solution manifold so that a single trained network serves as a data-free surrogate for the momentum conservation equation across all flow paths without retraining. This PINN is coupled with a finite difference method (FDM) solver that advances the mass conservation equation at each time step, ensuring exact discrete mass conservation while replacing the iterative nonlinear momentum solve with a single forward pass. Verification on a six-tank gravity-driven draining scenario yields a water level mean absolute error of 7.85×10−57.85 \times 10^{-5} m and a velocity mean absolute error of 3.21×10−33.21 \times 10^{-3} m/s under the nominal condition with Δt=1.0Δt = 1.0 s. The framework maintains consistent accuracy across time steps ranging from 0.2 to 1.0 s and generalizes to five distinct initial conditions, all without retraining or simulation data. This work introduces a numerical coupling methodology for integrating parameterized PINNs with FDM within a nuclear thermal-hydraulic system code framework.

Jeesuk Shin, Donggyun Seo, Sihyeong Yu +1

Division of Advanced Nuclear Engineering, Pohang University of Science and Technology, Pohang-si, Republic of Korea · Mueunjae School of Undergraduate Studies, Pohang University of Science and Technology, Pohang-si, Republic of Korea · Department of Quantum System Engineering, Jeonbuk National University, Jeonju-si, Republic of Korea

Physics-Based SimulationPDE Surrogate ModelingPhysics-Informed MLPDE Solving
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Apr 3, 2026

WGFINNs: Weak formulation-based GENERIC formalism informed neural networks

Data-driven discovery of governing equations from noisy observations remains a fundamental challenge in scientific machine learning. While GENERIC formalism informed neural networks (GFINNs) provide a principled framework that enforces the laws of thermodynamics by construction, their reliance on strong-form loss formulations makes them highly sensitive to measurement noise. To address this limitation, we propose weak formulation-based GENERIC formalism informed neural networks (WGFINNs), which integrate the weak formulation of dynamical systems with the structure-preserving architecture of GFINNs. WGFINNs significantly enhance robustness to noisy data while retaining exact satisfaction of GENERIC degeneracy and symmetry conditions. We further incorporate a state-wise weighted loss and a residual-based attention mechanism to mitigate scale imbalance across state variables. Theoretical analysis contrasts quantitative differences between the strong-form and the weak-form estimators. Mainly, the strong-form estimator diverges as the time step decreases in the presence of noise, while the weak-form estimator can be accurate even with noisy data if test functions satisfy certain conditions. Numerical experiments demonstrate that WGFINNs consistently outperform GFINNs at varying noise levels, achieving more accurate predictions and reliable recovery of physical quantities.

Jun Sur Richard Park, Auroni Huque Hashim, Siu Wun Cheung +2

Division of Big Data Science, Korea University, Sejong, 30019, Republic of Korea · Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA · Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA.

Physics-Informed MLAI-Assisted Scientific ResearchNeural Network RobustnessPDE Solving
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Apr 2, 2026

General Explicit Network (GEN): A novel deep learning architecture for solving partial differential equations

Machine learning, especially physics-informed neural networks (PINNs) and their neural network variants, has been widely used to solve problems involving partial differential equations (PDEs). The successful deployment of such methods beyond academic research remains limited. For example, PINN methods primarily consider discrete point-to-point fitting and fail to account for the potential properties of real solutions. The adoption of continuous activation functions in these approaches leads to local characteristics that align with the equation solutions while resulting in poor extensibility and robustness. A general explicit network (GEN) that implements point-to-function PDE solving is proposed in this paper. The "function" component can be constructed based on our prior knowledge of the original PDEs through corresponding basis functions for fitting. The experimental results demonstrate that this approach enables solutions with high robustness and strong extensibility to be obtained.

Genwei Ma, Ting Luo, Ping Yang +1

National Center for Applied Mathematics Beijing, Capital Normal University, No. 105 West Third Ring Road North, Beijing, 100048, China · The Academy for Multidisciplinary Studies, Capital Normal University, No. 105 West Third Ring Road North, Beijing, 100048, China · Academy of Information Network Security, People’s Public Security University of China, No.1 Muxidi Nanli, Beijing, 100038, China +2

Neural PDE SolversPDE SolvingPhysics-Informed ML
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Apr 2, 2026

Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing

Partial differential equations (PDEs) govern nearly every physical process in science and engineering, but solving them at scale remains prohibitively expensive. Generative AI has transformed language, vision, and protein science, but learned PDE solvers have not undergone a comparable shift. Existing paradigms each capture part of the problem. Physics-informed neural networks embed residual structure, although they are often difficult to optimize in stiff, multiscale, or large-domain regimes. Neural operators amortize across instances, although they commonly inherit a snapshot-prediction view of solving and can degrade over long rollouts. Diffusion-based solvers model uncertainty, although they are often built on a solver template that still centers on state regression. We argue that the core issue is the abstraction used to train learned solvers. Many models are asked to predict states, while many scientific settings require modeling how uncertainty moves through constrained dynamics. The relevant object is transport over physically admissible futures. This motivates flow learners: models that parameterize transport vector fields and generate trajectories through integration, echoing the continuous dynamics that define PDE evolution. This physics-to-physics alignment supports continuous-time prediction, native uncertainty quantification, and new opportunities for physics-aware solver design. We explain why transport-based learning offers a stronger organizing principle for learned PDE solving and outline the research agenda that follows from this shift.

Yilong Dai, Shengyu Chen, Xiaowei Jia +1

The University of Alabama, Tuscaloosa, Alabama, USA · University of Pittsburgh, Pittsburgh, Pennsylvania, USA

Neural PDE SolversPDE SolvingFlow-Based Generative Modeling
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Apr 1, 2026

Macroscopic transport patterns of UAV traffic in 3D anisotropic wind fields: A constraint-preserving hybrid PINN-FVM approach

Macroscopic unmanned aerial vehicle (UAV) traffic organization in three-dimensional airspace faces significant challenges from static wind fields and complex obstacles. A critical difficulty lies in simultaneously capturing the strong anisotropy induced by wind while strictly preserving transport consistency and boundary semantics, which are often compromised in standard physics-informed learning approaches. To resolve this, we propose a constraint-preserving hybrid solver that integrates a physics-informed neural network for the anisotropic Eikonal value problem with a conservative finite-volume method for steady density transport. These components are coupled through an outer Picard iteration with under-relaxation, where the target condition is hard-encoded and strictly conservative no-flux boundaries are enforced during the transport step. We evaluate the framework on reproducible homing and point-to-point scenarios, effectively capturing value slices, induced-motion patterns, and steady density structures such as bands and bottlenecks. Ultimately, our perspective emphasizes the value of a reproducible computational framework supported by transparent empirical diagnostics to enable the traceable assessment of macroscopic traffic phenomena.

Hanbing Liang, Fujun Liu

PDE SolvingPhysics-Informed ML
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Mar 30, 2026

Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes

Nonlinear Partial Differential Equations (PDEs) are ubiquitous in mathematical physics and engineering. Although Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDE problems, they typically struggle to identify multiple distinct solutions, since they are designed to find one solution at a time. To address this limitation, we introduce Deflation-PINNs, a novel framework that integrates a deflation loss with an architecture based on PINNs and Deep Operator Networks (DeepONets). By incorporating a deflation term into the loss function, our method systematically forces the Deflation-PINN to seek and converge upon distinct finitely many solution branches. We provide theoretical results on the approximation capabilities of our model and demonstrate the efficacy of Deflation-PINNs through numerical experiments on the Landau-de Gennes model of liquid crystals, a system renowned for its complex energy landscape and multiple equilibrium states, and on an Allen--Cahn benchmark whose solution set is provably known. Our results show that Deflation-PINNs can successfully identify and characterize multiple distinct crystal structures: a single unsupervised run recovers all six stable states of the benchmark, each branch certified to lie in the basin of attraction of a different equilibrium, and the discovered branches are refined to percent-level accuracy by a purely neural Deflation--Deep-Ritz stage and to the accuracy of a mesh-converged reference by a classical solver that they initialize.

Sean Disarò, Ruma Rani Maity, Aras Bacho

Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India · Department of Computing and Mathematical Sciences, California Institute of Technology, CA, USA

PDE SolvingPhysics-Informed ML
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Mar 25, 2026

Quantum Neural Physics: Solving Partial Differential Equations on Quantum Simulators using Quantum Convolutional Neural Networks

Neural Physics recasts local discretisations of partial differential equations (PDEs) as fixed convolutional operators, providing a physics-preserving alternative to data-driven surrogate modelling in scientific machine learning. However, existing realizations remain largely confined to classical AI hardware and do not directly connect to quantum structured operator design. To bridge this gap, we introduce a Quantum Neural Physics framework and develop a Hybrid Quantum-Classical CNN Multigrid Solver (HQC-CNNMG). The proposed method maps analytically prescribed stencil operators to local quantum convolutional primitives and embeds them within a classical multilevel W-cycle architecture, combining the operator-centric view of scientific ML with the numerical rigor of multigrid solvers. Using amplitude encoding together with the Linear Combination of Unitaries (LCU) and the Quantum Fourier Transform (QFT), the resulting local quantum operators admit logarithmic-depth implementation, with circuit depth scaling as O(log⁡K)\mathcal{O}(\log K) for an encoded block of size KK under the idealized parallel circuit model considered here. Numerical experiments on Poisson, transient diffusion, convection--diffusion, and incompressible Navier--Stokes problems demonstrate numerical consistency, stable multilevel behaviour, and workflow-level feasibility on noiseless simulators. Comparisons with representative quantum linear solver paradigms further show that the main strength of HQC-CNNMG lies in its balanced trade-off among local circuit depth, numerical robustness, and compatibility with PDE structure, rather than in fully quantum global inversion.

Jucai Zhai, Muhammad Abdullah, Boyang Chen +6

Resource Geophysics Academy, Imperial College London, London, SW7 2AZ UK · Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London, SW7 2AZ UK · Quantum AI Leadership (QUAIL) Ltd, London, WC2H 9JQ UK +2

Neural PDE SolversPDE SolvingQuantum Convolutional Neural NetworksHybrid Quantum-Classical ML
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Mar 24, 2026

Residual Attention Physics-Informed Neural Networks for Robust Multiphysics Simulation of Steady-State Electrothermal Energy Systems

Efficient thermal management and precise field prediction are critical for the design of advanced energy systems, including electrohydrodynamic transport, microfluidic energy harvesters, and electrically driven thermal regulators. However, the steady-state simulation of these electrothermal coupled multiphysics systems remains challenging for physics-informed neural computation due to strong nonlinear field coupling, temperature-dependent coefficient variability, and complex interface dynamics. This study proposes a Residual Attention Physics-Informed Neural Network (RA-PINN) framework for the unified solution of coupled velocity, pressure, electric-potential, and temperature fields. By integrating a unified five-field operator formulation with residual-connected feature propagation and attention-guided channel modulation, the proposed architecture effectively captures localized coupling structures and steep gradients. We evaluate RA-PINN across four representative energy-relevant benchmarks: constant-coefficient coupling, indirect pressure-gauge constraints, temperature-dependent transport, and oblique-interface consistency. Comparative analysis against Pure-MLP, LSTM-PINN, and pLSTM-PINN demonstrates that RA-PINN achieves superior accuracy, yielding the lowest MSE, RMSE, and relative L2L_2 errors across all scenarios. Notably, RA-PINN maintains high structural fidelity in interface-dominated and variable-coefficient settings where conventional PINN backbones often fail. These results establish RA-PINN as a robust and accurate computational framework for the high-fidelity modeling and optimization of complex electrothermal multiphysics in sustainable energy applications.

Yuqing Zhou, Ze Tao, Fujun Liu

Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology, Changchun, 130022, P.R. China

Physics-Informed MLPhysics-Based SimulationPDE Solving
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Mar 23, 2026

A Novel Method for Enforcing Exactly Dirichlet, Neumann and Robin Conditions on Curved Domain Boundaries for Physics Informed Machine Learning

We present a systematic method for exactly enforcing Dirichlet, Neumann, and Robin type conditions on general quadrilateral domains with arbitrary curved boundaries. Our method is built upon exact mappings between general quadrilateral domains and the standard domain, and employs a combination of TFC (theory of functional connections) constrained expressions and transfinite interpolations. When Neumann or Robin boundaries are present, especially when two Neumann (or Robin) boundaries meet at a vertex, it is critical to enforce exactly the induced compatibility constraints at the intersection, in order to enforce exactly the imposed conditions on the joining boundaries. We analyze in detail and present constructions for handling the imposed boundary conditions and the induced compatibility constraints for two types of situations: (i) when Neumann (or Robin) boundary only intersects with Dirichlet boundaries, and (ii) when two Neumann (or Robin) boundaries intersect with each other. We describe a four-step procedure to systematically formulate the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on general quadrilateral domains. The method developed herein has been implemented together with the extreme learning machine (ELM) technique we have developed recently for scientific machine learning. Ample numerical experiments are presented with several linear/nonlinear stationary/dynamic problems on a variety of two-dimensional domains with complex boundary geometries. Simulation results demonstrate that the proposed method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries exactly, with the numerical boundary-condition errors at the machine accuracy.

Suchuan Dong, Yuchuan Zhang

Center for Computational and Applied Mathematics Department of Mathematics Purdue University, USA

PDE SolvingPhysics-Informed ML
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