A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives

TL;DR

This paper introduces a novel descent algorithm for solving optimal control problems involving PDEs with unknown components approximated by ReLU neural networks, avoiding issues caused by smoothing such networks.

Contribution

The paper develops a direct nonsmooth optimization algorithm for control problems with ReLU-approximated PDEs, improving solution accuracy and stability.

Findings

The algorithm effectively solves control problems with nonsmooth neural network components.

Numerical examples demonstrate the efficiency and robustness of the proposed method.

Avoids issues related to smoothing ReLU networks in PDE control problems.

Abstract

We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations. The latter is a class of PDEs with constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that a direct smoothing of the ReLU network with the aim to make use of classical numerical solvers can have certain disadvantages, namely potentially introducing multiple solutions for the corresponding state equation. This motivates us to devise a numerical algorithm that treats directly the nonsmooth optimal control problem, by employing a descent algorithm inspired by a bundle-free method. Several numerical examples are provided and the efficiency of the algorithm is shown.