# Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations

arXiv: https://arxiv.org/abs/2411.16663 · HTML: https://mlsanity.com/paper/2411.16663/

## 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.

## Builds on (most cited first, 28 of 28)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)
- [Probabilistic ODE Solvers with Runge-Kutta Means](https://mlsanity.com/paper/1406.2582/) (2014)
- [Algorithmic Linearly Constrained Gaussian Processes](https://mlsanity.com/paper/1801.09197/) (2018)
- [Linearly Constrained Neural Networks](https://mlsanity.com/paper/2002.01600/) (2020)
- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Gaussian Process Regression constrained by Boundary Value Problems](https://mlsanity.com/paper/2012.11857/) (2020)
- [Linearly Constrained Gaussian Processes with Boundary Conditions](https://mlsanity.com/paper/2002.00818/) (2020)
- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)
- [Solving High Frequency and Multi-Scale PDEs with Gaussian Processes](https://mlsanity.com/paper/2311.04465/) (2023)
- [Learning Constrained Dynamics with Gauss Principle adhering Gaussian Processes](https://mlsanity.com/paper/2004.11238/) (2020)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagation](https://mlsanity.com/paper/2304.10242/) (2023)
- [APIK: Active Physics-Informed Kriging Model with Partial Differential Equations](https://mlsanity.com/paper/2012.11798/) (2020)
- [On boundary conditions parametrized by analytic functions](https://mlsanity.com/paper/2205.03185/) (2022)
- [Differential Algebra for Model Comparison](https://mlsanity.com/paper/1603.09730/) (2016)
- [Exact and soft boundary conditions in Physics-Informed Neural Networks for the Variable Coefficient Poisson equation](https://mlsanity.com/paper/2310.02548/) (2023)

## Built on by (most cited first, 2 of 2)

- [Comparing EPGP Surrogates and Finite Elements Under Degree-of-Freedom Parity](https://mlsanity.com/paper/2511.04518/) (2025)
- [Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data](https://mlsanity.com/paper/2605.20514/) (2026)

## 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.