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

Alex Alberts, Ilias Bilionis

TL;DR

The paper develops a Bayesian, function-space framework for inverse problems governed by the Poisson equation by treating the PDE as prior information and employing a Brownian bridge Gaussian process as a physics-informed prior.It establishes a precise link between the Poisson variational energy and a kernel ridge regression objective, showing that the GP posterior mean aligns with the PDE solution under a carefully chosen prior (mean $u^0=Cq$, covariance given by the Brownian-bridge kernel).A finite-dimensional representation is derived with proven convergence of priors and posteriors in Wasserstein distance, and the posterior mean converges to the ground truth with increasing data, even in the presence of model-form error.The hyperparameter $eta$ acts as a soft constraint on the physics, enabling detection and quantification of model-form error through changes in prior variance and posterior uncertainty, and the framework is extended to infinite-dimensional Gaussian measures in 1D and to Sobolev-space settings in higher dimensions.The work lays a theoretical foundation for probabilistic numerics in PDEs and opens avenues for extending to other linear PDEs (and nonlinear settings) and for developing computational approaches that exploit the finite-dimensional priors without repeatedly solving the PDE.

Abstract

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.

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

TL;DR

The paper develops a Bayesian, function-space framework for inverse problems governed by the Poisson equation by treating the PDE as prior information and employing a Brownian bridge Gaussian process as a physics-informed prior.It establishes a precise link between the Poisson variational energy and a kernel ridge regression objective, showing that the GP posterior mean aligns with the PDE solution under a carefully chosen prior (mean $u^0=Cq$, covariance given by the Brownian-bridge kernel).A finite-dimensional representation is derived with proven convergence of priors and posteriors in Wasserstein distance, and the posterior mean converges to the ground truth with increasing data, even in the presence of model-form error.The hyperparameter $eta$ acts as a soft constraint on the physics, enabling detection and quantification of model-form error through changes in prior variance and posterior uncertainty, and the framework is extended to infinite-dimensional Gaussian measures in 1D and to Sobolev-space settings in higher dimensions.The work lays a theoretical foundation for probabilistic numerics in PDEs and opens avenues for extending to other linear PDEs (and nonlinear settings) and for developing computational approaches that exploit the finite-dimensional priors without repeatedly solving the PDE.

Abstract

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.

Paper Structure

This paper contains 18 sections, 35 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Contributions
  4. Outline
  5. Preliminaries
  6. The Brownian bridge as a physics-informed prior
  7. Physics-informed prior as a Gaussian process
  8. Physics-informed prior as a Gaussian measure
  9. Analysis
  10. Regularity
  11. Finite-dimensional representations
  12. Convergence properties
  13. A note on the use of neural networks
  14. On model-form error
  15. Conclusions and outlook
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $k:\Omega \times \Omega \to \mathbb{R}$ be a continuous, positive-definite kernel, and $C_k$ and $(\lambda_n,\psi_n)_{n=1}^{\infty}$ be as given in eq. (eqn:intop) and eq. (eqn:esystem), respectively. Then, for $x,x'\in\Omega$, where the convergence is absolute and uniform.

Figures (1)

  • Figure 1: Physics-informed prior for the Poisson equation with source term $q(x) = 10\exp\{-|x-1/4|^2\}$ for varying values of $\beta$. The solution to this equation is in blue, and sample paths are shown in black.

Theorems & Definitions (65)

  • Definition 2.1: Reproducing kernel Hilbert space kanagawa2018gaussian
  • Theorem 2.1: Mercer's Theorem steinwart2008support
  • Definition 2.2: Gaussian process kanagawa2018gaussian
  • Theorem 2.2: Theorem 3.1 kanagawa2018gaussian
  • Theorem 2.3: Theorem 3.4 kanagawa2018gaussian
  • Proposition 2.1
  • Proposition 2.1
  • Proposition 2.2: Corollary 4.10 kanagawa2018gaussian
  • Proposition 3.1
  • proof
  • ...and 55 more