Solving Elliptic Optimal Control Problems using Physics Informed Neural
  Networks
Bangti Jin Ramesh Sau Luowei Yin Zhi Zhou
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.