DeLISA: Deep learning based iteration scheme approximation for solving PDEs
DeLISA: Deep learning based iteration scheme approximation for solving PDEs
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DeLISA：基于深度学习的迭代方案近似求解偏微分方程
DOI：
10.1016/j.jcp.2021.110884
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发表时间：
2021
影响因子：
4.1
通讯作者：
Shihui Ying
中科院分区：
文献类型：
--
作者：
Ying Li;Zuojia Zhou;Shihui Ying
用经典数值方法求解高维偏微分方程是一项具有挑战性的任务。由于具有处理高维数据的能力，深度学习自然被认为是解决偏微分方程的方法。本文提出了一种基于迭代方案近似的深度学习框架，称为DeLISA。首先，采用隐式多步法和龙格-库塔法求解时间迭代方案。然后用神经网络对该迭代方案进行逼近。该方法将控制方程的物理信息整合到时间迭代方案中，并引入时变输入，实现了不需要大量内点的连续时间预测。这里，具有自适应变量的激活函数在迭代过程中自我调整。最后，我们给出了一些基准偏微分方程的数值实验结果，包括Burgers， Allen-Cahn, Schrödinger，渗碳和Black-Scholes方程，并验证了所提出的方法在准确性和灵活性方面优于最先进的技术。此外，本文还通过不同迭代下预测值的变化来说明频率原理。
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.
影响因子：
4.1
作者：
Raissi, M.;Perdikaris, P.;Karniadakis, G. E.
通讯作者：
Karniadakis, G. E.