用神经网络求解偏微分方程最优控制，间接法比直接法更准更稳。
PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methods
- 将最优控制问题转化为物理信息神经网络的直接与间接两种形式
- 间接PINN在保持方程约束和最优性结构上表现更优，结果更准确
- 神经网络本身具有隐式正则化作用，生成更平滑的控制解
本文研究物理信息神经网络（PINNs）作为半线性偏微分方程最优控制问题的数值工具。首先回顾了经典直接法与间接法的最优控制框架，随后提出两种PINN建模方式：一种基于状态约束下的目标最小化（直接法），另一种基于一阶最优性系统（间接法）。针对一类半线性抛物型方程，推导出状态方程、伴随方程及平稳性条件，其形式符合连续时间庞特里亚金型最优性条件。进一步将框架应用于Allen-Cahn控制问题，比较三种方法：(i) 先离散后优化的伴随法，(ii) 直接PINN，(iii) 间接PINN。数值结果显示，PINN参数化具有隐式正则化效应，倾向于产生更平滑的控制解；且间接PINN更忠实保留PDE约束与最优性结构，对神经网络近似精度更高。
原文摘要 · Abstract (English)
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.
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