Energy stable neural network for gradient flow equations

Partial differential equations are important tools in solving a wide range of problems in science and engineering fields. Over the past twenty years, deep neural networks (DNNs) [12, 19] have demonstrated their power in science and engineering applications, and efforts have been made to employ DNNs to solve complex partial differential equations as an alternative to the traditional numerical schemes, especially for problems in high dimensions. Early works [5, 17] use feedforward neural network to learn the initial/boundary value problem by constraining neural networks using differential equation. Methods using continuous dynamical systems to model high-dimensional nonlinear functions used in machine learning were proposed in [6]. A deep learning-based approach to solve high dimensional parabolic partial differential equations (PDEs) based on the formulation of stochastic differential equations was developed in [14].