Bayesian reasoning for physics-informed neural networks

We introduce an evidence-driven Bayesian formulation of physics-informed neural networks that enables automatic optimization of loss weights between residuals of partial differential equations, boundary conditions, and observational data. Unlike existing Bayesian Physics-Informed Neural Network 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.