Bayesian Reasoning for Physics Informed Neural Networks

TL;DR

This paper presents a Bayesian approach to physics-informed neural networks using Laplace approximation for efficient hyperparameter tuning and uncertainty quantification.

Contribution

It introduces a novel evidence-driven Bayesian formulation that avoids sampling, enabling automatic loss weight optimization and model comparison in PINNs.

Findings

Accurate solutions for heat, wave, and Burgers' equations.

Efficient hyperparameter tuning without posterior sampling.

Natural integration of noisy data and governing equations with uncertainty estimates.

Abstract

We introduce an evidence-driven Bayesian formulation of physics-informed neural networks that enables automatic optimization of loss weights between PDE residuals, boundary conditions, and observational data. Unlike existing Bayesian PINN approaches based on sampling or variational inference, the proposed method uses a Laplace approximation to compute model evidence analytically, enabling efficient hyperparameter tuning and model comparison without posterior sampling. We demonstrate the method on the heat, wave, and Burgers' equations, obtaining solutions in agreement with exact or reference results. In the Burgers' equation example, we further show that the framework naturally integrates information from governing equations and noisy measurements, providing predictive uncertainties within a unified Bayesian setting.