Deep Latent Regularity Network for Modeling Stochastic Partial Differential Equations

Many problems in science and engineering can be modeled mathematically by a group of Partial Differential Equations (PDEs). Mechanism-based Scientific Computing 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, apart from a few Differential Equations where analytical solutions are directly available, more equations must rely on numerical approaches to be solved approximately, e.g., the Finite Difference Method and Finite Element Method. These numerical methods usually divide a continuous problem domain into discrete grids and then concentrate on solving the system at each of those points or elements. Although these traditional numerical methods show effectiveness in solving PDEs, the vast number of iterative operations accompanying each step forward greatly 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 of this interesting subdivision of AI for Science is still lacking. This survey aims to categorize and review the current progress on Deep Neural Networks 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/medical scenarios. The origin, developing history, character, and sort, as well as the future trends in each potential direction of this area are also introduced.