Neural operators with residual-based correction: Applications to optimization and Bayesian inference

Abstract

A book-chapter treatment connecting the mathematics of residual-based correction to the two places a neural-operator surrogate is actually used: PDE-constrained optimization, and Bayesian inference over model parameters. The correction is built from the residual of the governing equations and its linearization, so it needs no new labeled data. Applied inside an optimization loop it moves a DeepONet minimizer from 56.86% error to 0.05% in the reported case; applied to inference it addresses queries that fall outside the training distribution, including hyperelasticity. The tests are two-dimensional on a single mesh with synthetic inference data, and no end-to-end wall-time comparison against a full solve is reported.