Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps

2026-06-28

AI Summary

This study addresses the learning of the mapping from random obstacle fields to solutions of elliptic variational inequalities—a highly nonlinear problem governed by intricate geometric couplings among the obstacle, contact set, and free boundary. To tackle this challenge, the authors propose a post-training least-squares readout refitting strategy based on the Fourier Neural Operator (FNO): after freezing the nonlinear backbone, the affine readout layer is refitted in closed form to yield an output mapping that minimizes empirical squared error. Experimental results demonstrate that the resulting method, termed FNO-LS, significantly outperforms standard FNO, DeepONet, and its POD variants—particularly under high-amplitude obstacle scenarios—achieving improved accuracy across the entire solution field, better recovery of the contact set, and reduced obstacle violation, all with minimal computational overhead. This approach is especially effective when the backbone network captures relevant features but has not fully converged.

Abstract

We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of the random input, we learn the map directly from sampled obstacle realizations on a fixed grid. This problem is challenging because the solution is governed not only by the obstacle field itself, but also by the induced contact set and free-boundary geometry. We introduce a post-training least-squares readout refit for the Fourier neural operator (FNO). After the FNO is trained end to end, its nonlinear backbone is frozen and the final affine readout is recomputed by solving the induced linear least-squares problem over all training samples and grid points. The refit yields the empirical squared-error optimal readout for the learned frozen features while leaving the nonlinear representation unchanged. We compare vanilla DeepONet, POD-DeepONet, a two-stage DeepONet baseline, FNO, and FNO with least-squares readout refit (FNO-LS) on two obstacle ensembles with different amplitude levels. Numerical results show that FNO-LS achieves the strongest overall performance among the tested models, particularly for higher-amplitude obstacles with more complex contact geometry. The method improves average field accuracy, contact-set recovery, and obstacle-violation metrics at low additional cost, especially when the FNO backbone is informative but not fully converged. These results suggest that least-squares readout refit is a simple and effective post-training enhancement for learning random obstacle-to-solution maps.

Problem

• operator learning
• random obstacle-to-solution maps
• elliptic variational inequalities
• contact set
• free-boundary geometry

Innovation

• Fourier Neural Operator
• least-squares readout refit
• operator learning
• random obstacle-to-solution maps
• free-boundary problems