Relaxation approach for learning neural network regularizers for a class of identification problems

TL;DR

This paper introduces a relaxation method for learning neural network-based regularizers tailored to inverse problems in optimal control, demonstrating improved accuracy and noise compensation through numerical experiments.

Contribution

It proposes a novel relaxation approach to efficiently learn neural network regularizers for inverse optimal control problems, including noise compensation capabilities.

Findings

Feasibility of the relaxation approach demonstrated.

Successful rediscovery of standard $L^2$-regularizers.

Effective design of regularizers for noise compensation.

Abstract

The present paper deals with the data-driven design of regularizers in the form of artificial neural networks, for solving certain inverse problems formulated as optimal control problems. These regularizers aim at improving accuracy, wellposedness or compensating uncertainties for a given class of optimal control problems (inner-problems). Parameterized as neural networks, their weights are chosen in order to reduce a misfit between data and observations of the state solution of the inner-optimal control problems. Learning these weights constitutes the outer-problem. Based on necessary first-order optimality conditions for the inner-problems, a relaxation approach is proposed in order to implement efficient solving of these inner-problems, namely the forward operator of the outer-problem. Optimality conditions are derived for the latter, and are implemented in numerical illustrations dealing with the inverse conductivity problem. The numerical tests show the feasibility of the relaxation approach, first for rediscovering standard $L^2$-regularizers, and next for designing regularizers that compensate unknown noise on the observed state of the inner-problem.