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.

翻译：本文提出并分析了一种数值求解器，用于处理带有/不带箱体约束的线性和半线性二阶椭圆问题的最优控制问题。该方法基于最优控制问题的一阶最优性系统推导出的耦合系统，并应用物理信息神经网络（PINNs）求解该耦合系统。我们给出了该数值方案的误差分析，以深度神经网络参数（如深度、宽度和参数界）以及域内和边界上的采样点数量为变量，提供了状态变量、控制变量和伴随状态变量的$L^2(\Omega)$误差界。分析中的主要工具包括偏移拉德马赫复杂度以及神经网络函数的有界性和利普希茨连续性。我们通过多个数值算例展示了该方法，并与三种现有方法进行了比较。