Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields

• URL: http://arxiv.org/abs/2503.08343v1
• Date: Tue, 11 Mar 2025 11:53:21 GMT
• Title: Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields
• Authors: Tim Weiland, Marvin Pförtner, Philipp Hennig,
• Abstract summary: We show how to leverage GP priors to make probabilistic PDE solvers practical, even for large-scale nonlinear PDEs.<n>Our approach also allows for flexible and physically meaningful priors beyond what can be modeled with covariance functions.
• Score: 23.654711580674885
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Mechanistic knowledge about the physical world is virtually always expressed via partial differential equations (PDEs). Recently, there has been a surge of interest in probabilistic PDE solvers -- Bayesian statistical models mostly based on Gaussian process (GP) priors which seamlessly combine empirical measurements and mechanistic knowledge. As such, they quantify uncertainties arising from e.g. noisy or missing data, unknown PDE parameters or discretization error by design. Prior work has established connections to classical PDE solvers and provided solid theoretical guarantees. However, scaling such methods to large-scale problems remains a fundamental challenge primarily due to dense covariance matrices. Our approach addresses the scalability issues by leveraging the Markov property of many commonly used GP priors. It has been shown that such priors are solutions to stochastic PDEs (SPDEs) which when discretized allow for highly efficient GP regression through sparse linear algebra. In this work, we show how to leverage this prior class to make probabilistic PDE solvers practical, even for large-scale nonlinear PDEs, through greatly accelerated inference mechanisms. Additionally, our approach also allows for flexible and physically meaningful priors beyond what can be modeled with covariance functions. Experiments confirm substantial speedups and accelerated convergence of our physics-informed priors in nonlinear settings.

Related papers

• Mechanistic PDE Networks for Discovery of Governing Equations [52.492158106791365]
  We present Mechanistic PDE Networks, a model for discovery of partial differential equations from data.<n>The represented PDEs are then solved and decoded for specific tasks.<n>We develop a native, GPU-capable, parallel, sparse, and differentiable multigrid solver specialized for linear partial differential equations.
  arXiv  Detail & Related papers  (2025-02-25T17:21:44Z)
• Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous media [1.8416014644193066]
  We propose a novel, data-driven framework for learning surrogates for parametrized Partial Differential Equations (PDEs) It consists of a probabilistic, learning objective in which weighted residuals are used to probe the PDE and provide a source of em virtual data i.e. the actual PDE never needs to be solved. This is combined with a physics-aware implicit solver that consists of a much coarser, discretized version of the original PDE.
  arXiv  Detail & Related papers  (2024-05-29T12:01:49Z)
• Unisolver: PDE-Conditional Transformers Are Universal PDE Solvers [55.0876373185983]
  We present the Universal PDE solver (Unisolver) capable of solving a wide scope of PDEs. Our key finding is that a PDE solution is fundamentally under the control of a series of PDE components. Unisolver achieves consistent state-of-the-art results on three challenging large-scale benchmarks.
  arXiv  Detail & Related papers  (2024-05-27T15:34:35Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients [4.327763441385371]
  Partial differential equations (PDEs) are important tools to model physical systems. We propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell's equations.
  arXiv  Detail & Related papers  (2022-12-29T14:28:32Z)
• Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers [32.57938108395521]
  A class of mechanistic models, Linear partial differential equations, are used to describe physical processes such as heat transfer, electromagnetism, and wave propagation. specialized numerical methods based on discretization are used to solve PDEs. By ignoring parameter and measurement uncertainty, classical PDE solvers may fail to produce consistent estimates of their inherent approximation error.
  arXiv  Detail & Related papers  (2022-12-23T17:02:59Z)
• 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)
• Learning to Solve PDE-constrained Inverse Problems with Graph Networks [51.89325993156204]
  In many application domains across science and engineering, we are interested in solving inverse problems with constraints defined by a partial differential equation (PDE) Here we explore GNNs to solve such PDE-constrained inverse problems. We demonstrate computational speedups of up to 90x using GNNs compared to principled solvers.
  arXiv  Detail & Related papers  (2022-06-01T18:48:01Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Semi-Implicit Neural Solver for Time-dependent Partial Differential Equations [4.246966726709308]
  We propose a neural solver to learn an optimal iterative scheme in a data-driven fashion for any class of PDEs. We provide theoretical guarantees for the correctness and convergence of neural solvers analogous to conventional iterative solvers.
  arXiv  Detail & Related papers  (2021-09-03T12:03:10Z)
• Stochastic Normalizing Flows [52.92110730286403]
  We introduce normalizing flows for maximum likelihood estimation and variational inference (VI) using differential equations (SDEs) Using the theory of rough paths, the underlying Brownian motion is treated as a latent variable and approximated, enabling efficient training of neural SDEs. These SDEs can be used for constructing efficient chains to sample from the underlying distribution of a given dataset.
  arXiv  Detail & Related papers  (2020-02-21T20:47:55Z)

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.