# An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equation

arXiv: https://arxiv.org/abs/2503.00213 · HTML: https://mlsanity.com/paper/2503.00213/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 71 references that are in the map.
- First posted on arXiv 28 Feb 2025.

## Builds on (most cited first, 14 of 14)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [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)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothness](https://mlsanity.com/paper/2001.10818/) (2020)
- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Physics-informed Information Field Theory for Modeling Physical Systems with Uncertainty Quantification](https://mlsanity.com/paper/2301.07609/) (2023)
- [Gaussian processes for Bayesian inverse problems associated with linear partial differential equations](https://mlsanity.com/paper/2307.08343/) (2023)
- [A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2401.03492/) (2024)
- [A path-integral approach to Bayesian inference for inverse problems using the semiclassical approximation](https://mlsanity.com/paper/1312.2974/) (2013)
- [An information field theory approach to Bayesian state and parameter estimation in dynamical systems](https://mlsanity.com/paper/2306.02150/) (2023)

## Abstract (arXiv)

Many inverse problems require reconstructing physical fields from limited and noisy data while incorporating known governing equations. A growing body of work within probabilistic numerics formalizes such tasks via Bayesian inference in function spaces by assigning a physically meaningful prior to the latent field. In this work, we demonstrate that Brownian bridge Gaussian processes can be viewed as a softly-enforced physics-constrained prior for the Poisson equation. We first show equivalence between the variational problem associated with the Poisson equation and a kernel ridge regression objective. Then, through the connection between Gaussian process regression and kernel methods, we identify a Gaussian process for which the posterior mean function and the minimizer to the variational problem agree, thereby placing this PDE-based regularization within a fully Bayesian framework. This connection allows us to probe different theoretical questions, such as convergence and behavior of inverse problems. We then develop a finite-dimensional representation in function space and prove convergence of the projected prior and resulting posterior in Wasserstein distance. Finally, we connect the method to the important problem of identifying model-form error in applications, providing a diagnostic for model misspecification.