Deep Learning Methods For Solving PDE-constrained Optimal Control Problems

For solving PDE-constrained optimal control problems,traditional methods mainly rely on numerical discrete methods such as finite element method and finite difference method.Although these traditional methods have a reliable accuracy performance,their generally adopt complex geometric grids and thus are only suitable for treating simple low-dimensional cases.On the contrary,deep learning methods are based on grid-free form and have strong nonlinear fitting ability for approximating high-dimensional functions.Recently,scientists began to turn to deep neural networks instead of traditional numerical discrete methods to solve partial differential equations.However,for more complicated PDE-constrained optimal control problems,the corresponding deep learning methods have not been fully studied.Different from the single PDE problem,the solution and parameters of PDE-constrained optimal control problems also need to meet specific optimization objectives,and thus are far more difficult to solve than ordinary PDE problems.This thesis intends to develop some deep learning methods to solve such problems.The research emphasis of the thesis is divided into two parts.The first part mainly considers the distributed control problems.The second part mainly considers the boundary control problems.We first transform the original problem into an optimization objective and construct the corresponding loss function.Then the specific deep neural networks are introduced to solve the optimization objective.The proposed deep learning approaches are general and can solve the problems whose control variables are unconstrained or constrained.Finally,experimental results illustrate the effectiveness of the developed method.