DeLISA: Deep learning based iteration scheme approximation for solving PDEs

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, Schrodinger, 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. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper proposes a deep learning framework based iteration scheme approximation, called DeLISA, for solving high dimensional partial differential equations. By integrating the physical information into time iteration schemes and introducing time-dependent input, our method achieves continuous time prediction and outperforms existing techniques on accuracy and flexibility.