Numerical Analysis of Unsupervised Learning Approaches for Parameter Identification in PDEs

TL;DR

This paper surveys recent unsupervised neural network methods for PDE parameter identification, focusing on diffusion coefficients, and provides a numerical analysis framework with error bounds and stability considerations.

Contribution

It offers a comprehensive review of neural network approaches for PDE parameter identification and introduces a framework for rigorous error analysis using classical numerical methods.

Findings

Neural networks show impressive empirical performance in PDE parameter identification.

Error bounds can be derived using Galerkin finite element and hybrid methods.

Conditional stability estimates are crucial for error analysis.

Abstract

Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve PDE parameter identifications. These approaches employ neural networks as ansatz functions to approximate the parameters and / or the states, and have demonstrated impressive empirical performance. In this paper, we provide a comprehensive survey on these unsupervised learning techniques on one model problem, diffusion coefficient identification, from the classical numerical analysis perspective, and outline a general framework for deriving rigorous error bounds on the discrete approximations obtained using the Galerkin finite element method, hybrid method and deep neural networks. Throughout we highlight the crucial role of conditional stability estimates in the error analysis.