We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations. We first recall the classical direct and indirect viewpoints for optimal control of PDEs, and then present two PINN formulations: a direct formulation based on minimizing the objective under the state constraint, and an indirect formulation based on the first-order optimality system. For a class of semilinear parabolic equations, we derive the state equation, the adjoint equation, and the stationarity condition in a form consistent with continuous-time Pontryagin-type optimality conditions. We then specialize the framework to an Allen-Cahn control problem and compare three numerical approaches: (i) a discretize-then-optimize adjoint method, (ii) a direct PINN, and (iii) an indirect PINN. Numerical results show that the PINN parameterization has an implicit regularizing effect, in the sense that it tends to produce smoother control profiles. They also indicate that the indirect PINN more faithfully preserves the PDE contraint and optimality structure and yields a more accurate neural approximation than the direct PINN.

翻译：本文研究将基于物理信息的神经网络（PINNs）作为半线性偏微分方程最优控制的数值工具。我们首先回顾偏微分方程最优控制中经典的直接法和间接法观点，随后提出两种PINN形式：一种基于在状态约束下最小化目标函数的直接形式，另一种基于一阶最优性系统的间接形式。针对一类半线性抛物型方程，我们推导出与连续时间庞特里亚金型最优性条件一致的状态方程、伴随方程及驻点条件。接着将该框架特化为艾伦-卡恩控制问题，并比较三种数值方法：（i）离散-优化伴随法，（ii）直接PINN，以及（iii）间接PINN。数值结果表明，PINN参数化具有隐式正则化效应，即其倾向于生成更平滑的控制曲线。同时，间接PINN能更忠实地保持PDE约束与最优性结构，且相比直接PINN能获得更精确的神经近似。