Neural network augmented inverse problems for PDEs

TL;DR

This paper introduces a method combining neural networks with classical inverse problem techniques for PDEs, leveraging neural networks as priors to improve robustness and smoothness in coefficient recovery from noisy data.

Contribution

It presents a novel approach that augments traditional inverse problem methods with neural networks, eliminating the need for explicit regularization and enhancing robustness.

Findings

Neural network augmentation improves robustness to noise and incomplete data.

The method successfully recovers smooth coefficients in Poisson equations across multiple dimensions.

Neural networks serve as effective priors for inverse problems in PDEs.

Abstract

In this paper we show how to augment classical methods for inverse problems with artificial neural networks. The neural network acts as a prior for the coefficient to be estimated from noisy data. Neural networks are global, smooth function approximators and as such they do not require explicit regularization of the error functional to recover smooth solutions and coefficients. We give detailed examples using the Poisson equation in 1, 2, and 3 space dimensions and show that the neural network augmentation is robust with respect to noisy and incomplete data, mesh, and geometry.