Partial Differential Equations Meet Deep Neural Networks: A Survey

Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics such as computational fluid dynamics, multiphysics simulation, molecular dynamics, or even dynamical systems. It is a vibrant multi-disciplinary field of increasing importance and with extraordinary potential. At the same time, solving PDEs efficiently has been a long-standing challenge. Generally, except for a few differential equations for which analytical solutions are directly available, many more equations must rely on numerical approaches such as the finite difference method, finite element method, finite volume method, and boundary element method to be solved approximately. These numerical methods usually divide a continuous problem domain into discrete points and then concentrate on solving the system at each of those points. Though the effectiveness of these traditional numerical methods, the vast number of iterative operations accompanying each step forward significantly reduces the efficiency. Recently, another equally important paradigm, data-based computation represented by deep learning, has emerged as an effective means of solving PDEs. Surprisingly, a comprehensive review for this interesting subfield is still lacking. This survey aims to categorize and review the current progress on Deep Neural Networks (DNNs) for PDEs. We discuss the literature published in this subfield over the past decades and present them in a common taxonomy, followed by an overview and classification of applications of these related methods in scientific research and engineering scenarios. The origin, developing history, character, sort, as well as the future trends in each potential direction of this subfield are also introduced.