Bayesian Inverse Generative Neural Operator

Abstract

Inverse problems governed by partial differential equations (PDEs) are ill-posed, and responsible use of their solutions requires quantifying the uncertainty in recovered parameters. Neural operator methods for inverse problems offer fast surrogates for classical solvers, but placing posteriors over network weights is intractable at scale. This thesis extends the Inverse Generative Neural Operator (IGNO) to full Bayesian posterior sampling by adding a normalising flow prior term to the inversion objective and replacing gradient-based optimisation with the No-U-Turn Sampler (NUTS). The extension requires no retraining of any network component. We evaluate the method on four inverse problems spanning Darcy flow, electrical impedance tomography (EIT), and the viscous Burgers equation. On in-domain test instances, the posterior achieves 93% to 100% empirical coverage at the 95% nominal level across all four benchmarks and responds appropriately to changes in observation noise and sensor count. The posterior mean matches or improves on the maximum a posteriori (MAP) point estimate in every case. A Laplace approximation baseline, which fits a Gaussian posterior at the MAP estimate, fails on two of the four problems and does not consistently outperform NUTS on the two where it converges. Because the posterior formulation separates data, physics, and prior into additive terms, physical constraints can be incorporated during sampling alongside the data likelihood. Including PDE residuals as a virtual likelihood is most beneficial when observations alone leave the posterior under-determined, as demonstrated by EIT, where boundary-only measurements provide no direct information about the interior conductivity. The uncertainty estimates are unreliable for out-of-distribution coefficient fields. The learned prior pulls the posterior toward the training distribution, producing credible intervals that can be both narrow and wrong.