Let data talk: data-regularized operator learning theory for inverse problems

TL;DR

This paper introduces DaROL, a data-regularized operator learning framework for inverse PDE problems, combining neural networks with traditional regularization techniques for improved speed and flexibility.

Contribution

The paper proposes a novel DaROL method that integrates regularization into neural network training for inverse problems, providing theoretical insights and error estimates.

Findings

DaROL effectively separates regularization from neural network training.

Training on regularized data is equivalent to supervised learning of a regularized inverse map.

Theoretical conditions for the smoothness and error bounds of the inverse map are established.

Abstract

Regularization plays a pivotal role in integrating prior information into inverse problems. While many deep learning methods have been proposed to solve inverse problems, determining where to apply regularization remains a crucial consideration. Typical methods regularize neural networks via architecture, wherein neural network functions parametrize the parameter of interest or the regularization term. We introduce a novel approach, denoted as the "data-regularized operator learning" (DaROL) method, designed to address PDE inverse problems. The DaROL method trains a neural network on data, regularized through common techniques such as Tikhonov variational methods and Bayesian inference. The DaROL method offers flexibility across different frameworks, faster inverse problem-solving, and a simpler structure that separates regularization and neural network training. We demonstrate that training a neural network on the regularized data is equivalent to supervised learning for a regularized inverse map. Furthermore, we provide sufficient conditions for the smoothness of such a regularized inverse map and estimate the learning error in terms of neural network size and the number of training samples.