The modeling and control of complex physical dynamics are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.
翻译:复杂物理动态的建模和控制在现实世界问题中至关重要。我们提出了一个新的框架,它一般适用于解决受PDE制约的最佳控制问题,办法是为具有特殊规范的PDE解决方案操作者采用替代模式。拟议框架的程序分为两个阶段:为PDE制约因素进行解决方案操作者学习(第1阶段)和寻求最佳控制(第2阶段),一旦第1阶段的替代模式经过培训,可以在第2阶段进行最佳控制,而不进行密集计算。我们的框架可以适用于数据驱动和无数据案例。我们展示了我们如何成功地运用我们的方法,对从Poisson方程式到Burgers方程式等式等不同的PDE变量进行各种最佳控制。