Residual and agent-based corrections and beyond for reliable neural operators

A companion to the colloquium, aimed at an applied-analysis audience. The starting point is the same: a neural-operator surrogate with small test error can still be an unreliable substitute for the solver inside an optimization or inference loop.

This talk concentrates on the structure of the correction rather than the application. The residual of the governing equations, together with its tangent, defines a correction step that requires no new labeled data; what it buys, and under which conditions it can be expected to help, is the question. The final part looks at agent-based extensions that treat correcting, solving and accepting as choices with costs, along a sequence of queries rather than one at a time.