DeLISA: Deep learning based iteration scheme approximation for solving PDEs

Abstract

Solving the high dimensional partial differential equations (PDEs) with the classical numerical methods is a challenge task. As possessing the power of progressing high dimensional data, deep learning is naturally considered to solve PDEs. This paper proposes a deep learning framework based iteration scheme approximation, called DeLISA. First, we adopt the implicit multistep method and Runge-Kutta method for time iteration scheme. Then, such iteration scheme is approximated by a neural network. Due to integrating the physical information of governing equation into time iteration schemes and introducing time-dependent input, our method achieves the continuous time prediction without a mass of interior points. Here, the activation function with adaptive variable adjusts itself during the iteration process. Finally, we present numerical experiments results for some benchmark PDEs, including Burgers, Allen-Cahn, Schrödinger, carburizing and Black-Scholes equations, and verify that the proposed approach is superior to the state-of-the-art techniques on accuracy and flexibility. Moreover, the Frequency Principle is also illustrated by the changes of prediction at different iterations in this paper.