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.

翻译：本文研究了一种基于神经网络的最优控制问题求解器（考虑/不考虑箱式约束），适用于线性和半线性二阶椭圆问题。该方法利用最优控制问题一阶最优性系统导出的耦合系统，并采用深度神经网络表示降阶系统的解。我们给出了该方案的误差分析，并提供了关于神经网络参数（如深度、宽度和参数边界）及采样点数量的状态、控制和伴随变量的$L^2(\Omega)$误差界。分析中的主要工具包括偏移Rademacher复杂度以及神经网络函数的有界性和Lipschitz连续性。我们通过多个数值算例验证了该方法，并与两种现有方法进行了比较。