Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System

TL;DR

This paper introduces a neural network-based method for solving elliptic optimal control problems, providing error bounds and demonstrating effectiveness through numerical examples, advancing the application of deep learning in control theory.

Contribution

It develops a neural network approach for elliptic optimal control problems, including error analysis and comparison with existing methods, which is a novel integration of deep learning and control theory.

Findings

The method achieves accurate solutions with quantifiable error bounds.

Neural networks effectively approximate solutions to elliptic control problems.

Numerical experiments validate the approach and compare favorably with existing methods.

Abstract

In this work, we investigate a neural network based solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. It utilizes a coupled system derived from the first-order optimality system of the optimal control problem, and employs deep neural networks to represent the solutions to the reduced system. We present an error analysis of the scheme, and provide $L^2(\Omega)$ error bounds on the state, control and adjoint in terms of neural network parameters (e.g., depth, width, and parameter bounds) and the numbers of sampling points. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the method and compare it with two existing ones.