Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations
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Abstract (arXiv)
Working with systems of partial differential equations (PDEs) is a fundamental task in computational science. Well-posed systems are addressed by numerical solvers or neural operators, whereas systems described by data are often addressed by PINNs or Gaussian processes. In this work, we propose Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs), a novel probabilistic framework for constructing GP priors that satisfy both general systems of linear PDEs with constant coefficients and linear boundary conditions and can be conditioned on a finite data set. We explicitly construct GP priors for representative PDE systems with practical boundary conditions. Formal proofs of correctness are provided and empirical results demonstrating significant accuracy and computational resource improvements over state-of-the-art approaches.
Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 28 of its 89 references that are in the map. First posted on arXiv 25 Nov 2024.
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28 of 89 references.
References · 28
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
- 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
- 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
- 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
- 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 214
- 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 137
- 2015Probabilistic Numerics and Uncertainty in Computationscited by 60
- 2023Learning Physical Models that Can Respect Conservation Lawscited by 59
- 2017Linearly constrained Gaussian processescited by 42
- 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 41
- 2015Modeling and interpolation of the ambient magnetic field by Gaussian processescited by 28
16 more
- 2014Probabilistic ODE Solvers with Runge-Kutta Meanscited by 22
- 2018Algorithmic Linearly Constrained Gaussian Processescited by 20
- 2020Linearly Constrained Neural Networkscited by 18
- 2022Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficientscited by 17
- 2020Gaussian Process Regression constrained by Boundary Value Problemscited by 15
- 2020Linearly Constrained Gaussian Processes with Boundary Conditionscited by 14
- 2022Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equationscited by 14
- 2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
- 2020Learning Constrained Dynamics with Gauss Principle adhering Gaussian Processescited by 13
- 2024Linearization Turns Neural Operators into Function-Valued Gaussian Processescited by 12
- 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
- 2023Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagationcited by 11
- 2020APIK: Active Physics-Informed Kriging Model with Partial Differential Equationscited by 9
- 2022On boundary conditions parametrized by analytic functionscited by 4
- 2016Differential Algebra for Model Comparisoncited by 2
- 2023Exact and soft boundary conditions in Physics-Informed Neural Networks for the Variable Coefficient Poisson equationcited by 2
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Citing papers per year: 2025: 1; 2026: 1.
2 papers.