# Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers

arXiv: https://arxiv.org/abs/2212.12474 · HTML: https://mlsanity.com/paper/2212.12474/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 78 references that are in the map.
- First posted on arXiv 23 Dec 2022.

## Builds on (most cited first, 17 of 17)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions](https://mlsanity.com/paper/1901.01002/) (2019)
- [Equivariant Learning of Stochastic Fields: Gaussian Processes and Steerable Conditional Neural Processes](https://mlsanity.com/paper/2011.12916/) (2020)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems](https://mlsanity.com/paper/1701.04006/) (2017)
- [A Modern Retrospective on Probabilistic Numerics](https://mlsanity.com/paper/1901.04457/) (2019)
- [Gaussian processes with linear operator inequality constraints](https://mlsanity.com/paper/1901.03134/) (2019)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2110.11847/) (2021)
- [Probabilistic Linear Solvers for Machine Learning](https://mlsanity.com/paper/2010.09691/) (2020)
- [ProbNum: Probabilistic Numerics in Python](https://mlsanity.com/paper/2112.02100/) (2021)
- [Small Sample Spaces for Gaussian Processes](https://mlsanity.com/paper/2103.03169/) (2021)

## Built on by (most cited first, 30 of 41)

- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [A spectrum of physics-informed Gaussian processes for regression in engineering](https://mlsanity.com/paper/2309.10656/) (2023)
- [Solving High Frequency and Multi-Scale PDEs with Gaussian Processes](https://mlsanity.com/paper/2311.04465/) (2023)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagation](https://mlsanity.com/paper/2304.10242/) (2023)
- [Sample Path Regularity of Gaussian Processes from the Covariance Kernel](https://mlsanity.com/paper/2312.14886/) (2023)
- [FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning](https://mlsanity.com/paper/2407.13711/) (2024)
- [Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inference](https://mlsanity.com/paper/2411.01036/) (2024)
- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [Gaussian process learning of nonlinear dynamics](https://mlsanity.com/paper/2312.12193/) (2023)
- [Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations](https://mlsanity.com/paper/2404.15618/) (2024)
- [Position: Biology is the Challenge Physics-Informed ML Needs to Evolve](https://mlsanity.com/paper/2510.25368/) (2025)
- [Conditioning Gaussian Processes on Almost Anything](https://mlsanity.com/paper/2605.21041/) (2026)
- [Equivariant geometric convolutions for emulation of dynamical systems](https://mlsanity.com/paper/2305.12585/) (2023)
- [Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics](https://mlsanity.com/paper/2409.10972/) (2024)
- [Uncertainty-Aware Diagnostics for Physics-Informed Machine Learning](https://mlsanity.com/paper/2510.26121/) (2025)
- [Scaling up Probabilistic PDE Simulators with Structured Volumetric Information](https://mlsanity.com/paper/2406.05020/) (2024)
- [Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations](https://mlsanity.com/paper/2411.16663/) (2024)
- [Turbine location-aware multi-decadal wind power predictions for Germany using CMIP6](https://mlsanity.com/paper/2408.14889/) (2024)
- [Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields](https://mlsanity.com/paper/2503.08343/) (2025)
- [Online Conformal Probabilistic Numerics via Adaptive Edge-Cloud Offloading](https://mlsanity.com/paper/2503.14453/) (2025)
- [Towards scientific machine learning for granular material simulations -- challenges and opportunities](https://mlsanity.com/paper/2504.08766/) (2025)
- [Robust and Computation-Aware Gaussian Processes](https://mlsanity.com/paper/2505.21133/) (2025)
- [Exploring Efficient Quantification of Modeling Uncertainties with Differentiable Physics-Informed Machine Learning Architectures](https://mlsanity.com/paper/2506.18247/) (2025)
- [Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometry](https://mlsanity.com/paper/2509.02617/) (2025)
- [Physics-based deep kernel learning for parameter estimation in high dimensional PDEs](https://mlsanity.com/paper/2509.14054/) (2025)
- [Physics-augmented Multi-task Gaussian Process for Modeling Spatiotemporal Dynamics](https://mlsanity.com/paper/2510.13601/) (2025)
- [Propagating Model Uncertainty through Filtering-based Probabilistic Numerical ODE Solvers](https://mlsanity.com/paper/2503.04684/) (2025)
- [Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems](https://mlsanity.com/paper/2508.09623/) (2025)
- [Comparing EPGP Surrogates and Finite Elements Under Degree-of-Freedom Parity](https://mlsanity.com/paper/2511.04518/) (2025)

## Abstract (arXiv)

Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical methods based on discretization are used to solve PDEs. They generally use an estimate of the unknown model parameters and, if available, physical measurements for initialization. Such solvers are often embedded into larger scientific models with a downstream application and thus error quantification plays a key role. However, by ignoring parameter and measurement uncertainty, classical PDE solvers may fail to produce consistent estimates of their inherent approximation error. In this work, we approach this problem in a principled fashion by interpreting solving linear PDEs as physics-informed Gaussian process (GP) regression. Our framework is based on a key generalization of the Gaussian process inference theorem to observations made via an arbitrary bounded linear operator. Crucially, this probabilistic viewpoint allows to (1) quantify the inherent discretization error; (2) propagate uncertainty about the model parameters to the solution; and (3) condition on noisy measurements. Demonstrating the strength of this formulation, we prove that it strictly generalizes methods of weighted residuals, a central class of PDE solvers including collocation, finite volume, pseudospectral, and (generalized) Galerkin methods such as finite element and spectral methods. This class can thus be directly equipped with a structured error estimate. In summary, our results enable the seamless integration of mechanistic models as modular building blocks into probabilistic models by blurring the boundaries between numerical analysis and Bayesian inference.