Bayesian Inference for Partial Differential Equations via Neural Network Surrogates

Abstract

Partial differential equations (PDEs) provide the fundamental framework for describing physical systems; yet, in many practical applications, these equations contain unknown parameters that must be inferred from experimental observations. Solving such inverse problems using traditional mesh-based numerical methods is often computationally intensive; furthermore, because these solvers cannot be easily differentiated with respect to model parameters, they create significant bottlenecks for gradient-based inference. To address these challenges, we train parameterized Physics-Informed Neural Networks (PINNs) for two distinct systems: the Allen–Cahn and Cahn–Hilliard (AC–CH) phase field equations and diffusion models for cyclic voltammetry (CV). These surrogates demonstrate strong generalizability across continuous parameter spaces and serve as differentiable components for gradient-based Bayesian parameter estimation via the No-U-Turn Sampler (NUTS). This work verifies the feasibility of a unified PINN-surrogate-Bayesian workflow for parameter estimation, offering a promising complement to existing methods for solving inverse problems with uncertainty quantification.