Discretization of parameter identification in PDEs using Neural Networks

TL;DR

This paper introduces a novel all-at-once neural network approach for solving ill-posed inverse problems in PDEs, enabling simultaneous recovery of unknown nonlinearities and parameters without supervised training data.

Contribution

It develops a discretization scheme for neural network approximation of nonlinearities in PDE inverse problems, with convergence analysis under noise and discretization errors.

Findings

Proves convergence of the method as noise and discretization errors vanish.

Establishes a framework for regularization via Tikhonov and Landweber methods.

Demonstrates the approach's effectiveness in handling noisy data.

Abstract

We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the unknown state. Hence, it is not possible to construct input-output data pairs to perform a supervised training process. Proposing an all-at-once approach, we bypass the need for training data and recover all the unknowns simultaneously. In the general case, the approximation via a neural network can be realized as a discretization scheme, and the training with noisy data can be viewed as an ill-posed inverse problem. Therefore, we study discretization of regularization in terms of Tikhonov and projected Landweber methods for discretization of inverse problems, and prove convergence when the discretization error (network approximation error) and the noise level tend to zero.