Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometry

• URL: http://arxiv.org/abs/2509.02617v1
• Date: Mon, 01 Sep 2025 02:40:32 GMT
• Title: Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometry
• Authors: Pucheng Tang, Hongqiao Wang, Wenzhou Lin, Qian Chen, Heng Yong,
• Abstract summary: Parametric partial differential equations (PDEs) are fundamental mathematical tools for modeling complex physical systems.<n>We propose a novel physical law-corrected prior Gaussian process (LC-prior GP) surrogate modeling framework.
• Score: 3.3798563347021093
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Parametric partial differential equations (PDEs) are fundamental mathematical tools for modeling complex physical systems, yet their numerical evaluation across parameter spaces remains computationally intensive when using conventional high-fidelity solvers. To address this challenge, we propose a novel physical law-corrected prior Gaussian process (LC-prior GP) surrogate modeling framework that effectively integrates data-driven learning with underlying physical constraints to flexibly handle multi-coupled variables defined on complex geometries. The proposed approach leverages proper orthogonal decomposition (POD) to parameterize high-dimensional PDE solutions via their dominant modes and associated coefficients, thereby enabling efficient Gaussian process (GP) surrogate modeling within a reduced-dimensional coefficient space. A key contribution lies in the incorporation of physical laws together with a limited number of parameter samples to correct the GP posterior mean, thus avoiding reliance on computationally expensive numerical solvers. Furthermore, interpolation functions are constructed to describe the mapping from the full parameter space to the physics-based correction term. This mapping is subsequently backpropagated to constrain the original GP surrogate, yielding a more physically consistent conditional prior. To handle irregular geometries, the radial basis function-finite difference (RBF-FD) method is incorporated during training set computation, with its inherent differentiation matrices providing both computational efficiency and numerical accuracy for physical constraint optimization. The effectiveness of the proposed method is demonstrated through numerical experiments involving a reaction-diffusion model, miscible flooding models, and Navier-Stokes equations with multi-physics coupling defined on irregular domains.

Related papers

• Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks [0.0]
  We develop a parametric-based reduced-order modeling framework for parameter-dependent partial differential equations.<n>We derive rigorous error estimates that explicitly quantify the interplay between spatial discretization and parameter approximation.<n>The proposed framework is applied to inverse problems in quantitative photoacoustic tomography, where we derive potential and parameter reconstruction error estimates.
  arXiv  Detail & Related papers  (2026-02-16T14:01:50Z)
• Multi-Dimensional Visual Data Recovery: Scale-Aware Tensor Modeling and Accelerated Randomized Computation [51.65236537605077]
  We propose a new type of network compression optimization technique, fully randomized tensor network compression (FCTN)<n>FCTN has significant advantages in correlation characterization and transpositional in algebra, and has notable achievements in multi-dimensional data processing and analysis.<n>We derive efficient algorithms with guarantees to solve the formulated models.
  arXiv  Detail & Related papers  (2026-02-13T14:56:37Z)
• LieSolver: A PDE-constrained solver for IBVPs using Lie symmetries [20.541443888815873]
  We introduce a method for solving initial-boundary value problems (IBVPs) that uses Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construction.<n>By leveraging symmetry transformations, the model inherently incorporates the physical laws and learns solutions from initial and boundary data.
  arXiv  Detail & Related papers  (2025-10-29T17:37:27Z)
• Graph Neural Regularizers for PDE Inverse Problems [62.49743146797144]
  We present a framework for solving a broad class of ill-posed inverse problems governed by partial differential equations (PDEs)<n>The forward problem is numerically solved using the finite element method (FEM)<n>We employ physics-inspired graph neural networks as learned regularizers, providing a robust, interpretable, and generalizable alternative to standard approaches.
  arXiv  Detail & Related papers  (2025-10-23T21:43:25Z)
• Tensor Gaussian Processes: Efficient Solvers for Nonlinear PDEs [16.38975732337055]
  TGPS is a machine learning solver for partial differential equations.<n>It reduces the task to learning a collection of one-dimensional GPs.<n>It achieves superior accuracy and efficiency compared to existing approaches.
  arXiv  Detail & Related papers  (2025-10-15T17:23:21Z)
• Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation [55.88862563823878]
  In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics.<n>We demonstrate performance of the algorithm on two PDEs: a linear equation which governs slightly compressible fluid flow in porous media and the wave equation.<n>Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest.
  arXiv  Detail & Related papers  (2025-07-24T11:02:13Z)
• Governing Equation Discovery from Data Based on Differential Invariants [52.2614860099811]
  We propose a pipeline for governing equation discovery based on differential invariants.<n>Specifically, we compute the set of differential invariants corresponding to the infinitesimal generators of the symmetry group.<n>Taking DI-SINDy as an example, we demonstrate that its success rate and accuracy in PDE discovery surpass those of other symmetry-informed governing equation discovery methods.
  arXiv  Detail & Related papers  (2025-05-24T17:19:02Z)
• A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs [46.685771141109306]
  We propose an autoregressive and data-driven method using the analogy with classical numerical solvers for time-dependent, parametric and (typically) nonlinear PDEs.<n>We show that by leveraging DR we can deliver not only more accurate predictions, but also a considerably lighter and faster Deep Learning model.
  arXiv  Detail & Related papers  (2025-02-12T11:16:15Z)
• Shape-informed surrogate models based on signed distance function domain encoding [8.052704959617207]
  We propose a non-intrusive method to build surrogate models that approximate the solution of parameterized partial differential equations (PDEs) Our approach is based on the combination of two neural networks (NNs)
  arXiv  Detail & Related papers  (2024-09-19T01:47:04Z)
• Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems [41.94295877935867]
  numerical simulation and optimization of technical systems described by partial differential equations is expensive.<n>A comparatively new approach in this context is to combine the good approximation properties of neural networks with the classical finite element method.<n>In this paper, we extend this approach to saddle-point and non-linear fluid dynamics problems, respectively.
  arXiv  Detail & Related papers  (2024-09-06T07:17:01Z)
• Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs [0.0]
  We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. We can parametrically solve partial differential equations in a data-free manner and provide accurate sensitivities. Our study focuses on the steady-state heat equation within heterogeneous materials.
  arXiv  Detail & Related papers  (2024-07-04T21:23:12Z)
• A Physics-driven GraphSAGE Method for Physical Process Simulations Described by Partial Differential Equations [2.1217718037013635]
  A physics-driven GraphSAGE approach is presented to solve problems governed by irregular PDEs. A distance-related edge feature and a feature mapping strategy are devised to help training and convergence. The robust PDE surrogate model for heat conduction problems parameterized by the Gaussian singularity random field source is successfully established.
  arXiv  Detail & Related papers  (2024-03-13T14:25:15Z)
• Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equations [34.500484733973536]
  Ordinary differential equations (ODEs) are widely used to describe dynamical systems in science, but identifying parameters that explain experimental measurements is challenging. We propose diffusion tempering, a novel regularization technique for probabilistic numerical methods which improves convergence of gradient-based parameter optimization in ODEs. We demonstrate that our method is effective for dynamical systems of different complexity and show that it obtains reliable parameter estimates for a Hodgkin-Huxley model with a practically relevant number of parameters.
  arXiv  Detail & Related papers  (2024-02-19T15:36:36Z)
• Fully probabilistic deep models for forward and inverse problems in parametric PDEs [1.9599274203282304]
  We introduce a physics-driven deep latent variable model (PDDLVM) to learn simultaneously parameter-to-solution (forward) and solution-to- parameter (inverse) maps of PDEs. The proposed framework can be easily extended to seamlessly integrate observed data to solve inverse problems and to build generative models. We demonstrate the efficiency and robustness of our method on finite element discretized parametric PDE problems.
  arXiv  Detail & Related papers  (2022-08-09T15:40:53Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Fast Gravitational Approach for Rigid Point Set Registration with Ordinary Differential Equations [79.71184760864507]
  This article introduces a new physics-based method for rigid point set alignment called Fast Gravitational Approach (FGA) In FGA, the source and target point sets are interpreted as rigid particle swarms with masses interacting in a globally multiply-linked manner while moving in a simulated gravitational force field. We show that the new method class has characteristics not found in previous alignment methods.
  arXiv  Detail & Related papers  (2020-09-28T15:05:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.