非线性系统
领域(数学分析)
数学
偏微分方程
分解
区域分解方法
虚域法
计算机科学
应用数学
数学优化
数学分析
有限元法
工程类
物理
结构工程
生物
生态学
量子力学
作者
Suchuan Dong,Zongwei Li
标识
DOI:10.1016/j.cma.2021.114129
摘要
We present a neural network-based method for solving linear and nonlinear partial differential equations, by combining the ideas of extreme learning machines (ELM), domain decomposition and local neural networks. The field solution on each sub-domain is represented by a local feed-forward neural network, and $C^k$ continuity is imposed on the sub-domain boundaries. Each local neural network consists of a small number of hidden layers, while its last hidden layer can be wide. The weight/bias coefficients in all hidden layers of the local neural networks are pre-set to random values and are fixed, and only the weight coefficients in the output layers are training parameters. The overall neural network is trained by a linear or nonlinear least squares computation, not by the back-propagation type algorithms. We introduce a block time-marching scheme together with the presented method for long-time dynamic simulations. The current method exhibits a clear sense of convergence with respect to the degrees of freedom in the neural network. Its numerical errors typically decrease exponentially or nearly exponentially as the number of degrees of freedom increases. Extensive numerical experiments have been performed to demonstrate the computational performance of the presented method. We compare the current method with the deep Galerkin method (DGM) and the physics-informed neural network (PINN) in terms of the accuracy and computational cost. The current method exhibits a clear superiority, with its numerical errors and network training time considerably smaller (typically by orders of magnitude) than those of DGM and PINN. We also compare the current method with the classical finite element method (FEM). The computational performance of the current method is on par with, and oftentimes exceeds, the FEM performance.
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