Learning Generalizable Neural Operators for Inverse Problems

TL;DR

This paper introduces B2B^{-1}, a neural operator framework that decouples function representation from inverse mapping, enabling scalable, robust, and generalizable solutions for ill-posed inverse PDE problems.

Contribution

The paper proposes B2B^{-1}, a novel neural operator architecture that separates basis function learning from inverse modeling, improving generalization and robustness in inverse problems.

Findings

Consistent re-simulation across ill-posedness levels

Effective probabilistic modeling capturing uncertainty

Robustness to measurement noise through implicit denoising

Abstract

Inverse problems challenge existing neural operator architectures because ill-posed inverse maps violate continuity, uniqueness, and stability assumptions. We introduce B2B${}^{-1}$, an inverse basis-to-basis neural operator framework that addresses this limitation. Our key innovation is to decouple function representation from the inverse map. We learn neural basis functions for the input and output spaces, then train inverse models that operate on the resulting coefficient space. This structure allows us to learn deterministic, invertible, and probabilistic models within a single framework, and to choose models based on the degree of ill-posedness. We evaluate our approach on six inverse PDE benchmarks, including two novel datasets, and compare against existing invertible neural operator baselines. We learn probabilistic models that capture uncertainty and input variability, and remain robust to measurement noise due to implicit denoising in the coefficient calculation. Our results show consistent re-simulation performance across varying levels of ill-posedness. By separating representation from inversion, our framework enables scalable surrogate models for inverse problems that generalize across instances, domains, and degrees of ill-posedness.