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

2025-02-28

AI Summary

This work addresses the challenge of unifying physical constraints—such as partial differential equations (PDEs)—and experimental data within a Bayesian framework for physics-informed machine learning (PIML). Methodologically, it establishes, for the first time, a rigorous Bayesian correspondence between Brownian bridge Gaussian processes (GPs) and the physics-informed loss function associated with the Poisson equation, explicitly interpreting PDE constraints as soft priors. It proves that this GP constitutes a natural physical prior for the Poisson equation and demonstrates the equivalence among the PIML loss, kernel ridge regression, and the GP posterior mean. Theoretical contributions include: (i) providing a principled basis for identifying model-form errors; (ii) enabling convergence analysis of inverse problem solutions; and (iii) unifying PIML with probabilistic modeling, thereby establishing an interpretable and generalizable statistical foundation.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Probabilistic Inference
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Physics-informed machine learning is one of the most commonly used methods for fusing physical knowledge in the form of partial differential equations with experimental data. The idea is to construct a loss function where the physical laws take the place of a regularizer and minimize it to reconstruct the underlying physical fields and any missing parameters. However, there is a noticeable lack of a direct connection between physics-informed loss functions and an overarching Bayesian framework. 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 form of the physics-informed loss function for 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 physics-informed loss function minimizer agree. This connection allows us to probe different theoretical questions, such as convergence and behavior of inverse problems. We also connect the method to the important problem of identifying model-form error in applications.

Problem

• Linking physics-informed loss functions to Bayesian frameworks.
• Using Brownian bridge as a prior for Poisson equation.
• Exploring convergence and model-form error identification.

Innovation

• Brownian bridge as physics-informed prior
• Gaussian process for Poisson equation
• Kernel ridge regression objective equivalence