Solving elliptic optimal control problems via neural networks and optimality system

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.