Solving Elliptic Optimal Control Problems using Physics Informed Neural Networks

In this work, we present and analyze a numerical solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. The approach is based on a coupled system derived from the first-order optimality system of the optimal control problem, and applies physics informed neural networks (PINNs) to solve the coupled system. We present an error analysis of the numerical scheme, and provide $L^2(\Omega)$ error bounds on the state, control and adjoint state in terms of deep neural network parameters (e.g., depth, width, and parameter bounds) and the number of sampling points in the domain and on the boundary. 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 approach and compare it with three existing approaches.