Deep Learning Method for Partial Differential Equations and Optimal Problems

Abstract

Scientific computing problems in high dimensions are difficult to solve with traditional methods due to the curse of dimensionality. The recently fast developing machine learning techniques provide us a promising way to resolve this problem, elevating the field of scientific computing to new heights. This thesis collects my works on machine learning to solve traditional scientific computing problems during my Ph.D. studies, which include partial differential equation (PDE) problems and optimal control problems. The numerical algorithms in the works demonstrate significant advantage over traditional methods. Moreover, the theoretical analysis of the algorithms enhances our understanding of machine learning, providing guarantees that enable us to avoid treating it as a black box.