NEURAL NETWORK SURROGATES WITH UNCERTAINTY QUANTIFICATION FOR INVERSE PROBLEMS IN PARTIAL DIFFERENTIAL EQUATIONS

Inverse problems for partial differential equations (PDEs) require inferring unknown model parameters from noisy, incomplete observations. While Bayesian frameworks are ideal for this task because they quantify uncertainty, they typically require millions of forward model evaluations, making them computationally prohibitive for complex systems. This paper addresses the need for efficient, uncertainty-aware surrogate models that can replace expensive numerical solvers in Bayesian inference.

The authors introduce DeepGaLA (Deep Galerkin via Laplace Approximation), a framework that combines the Deep Galerkin Method (DGM) with a Bayesian neural network approach. By reinterpreting the neural network's loss function as a maximum a posteriori (MAP) estimate, the authors apply a Laplace approximation to the network's weights. This allows the model to produce not just a point estimate of the solution, but a distribution that captures uncertainty. To validate the fidelity of these surrogate-induced posteriors, the authors propose using a short run of delayed-acceptance Markov chain Monte Carlo (MCMC) as a diagnostic tool.

DeepGaLA demonstrates performance comparable to established Gaussian process surrogates while offering better scalability as the dimensionality of the parameter space increases. The inclusion of uncertainty quantification is shown to be critical; it prevents overconfident, inaccurate inferences when training data are scarce. Furthermore, the framework is flexible enough to incorporate differential equation constraints, allowing it to handle both linear and nonlinear PDEs effectively. The authors provide theoretical justification for using delayed-acceptance MCMC to verify the surrogate's accuracy, offering a practical way to assess reliability without needing the true posterior.