Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations
Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations
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DOI：
10.1016/j.cma.2021.114129
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发表时间：
2021-09-11
影响因子：
7.2
通讯作者：
Li, Zongwei
中科院分区：
文献类型：
--
作者：
Dong, Suchuan;Li, Zongwei
结合极端学习机（ELM）、区域分解和局部神经网络的思想，提出了一种求解线性和非线性偏微分方程的神经网络方法.每个子域上的场解由一个局部前馈神经网络表示，子域边界上施加C-k连续性条件。每个局部神经网络由少量的隐藏层组成，而其最后一个隐藏层可以很宽。在局部神经网络的所有隐藏层中的权重/偏置系数被预先设置为随机值并且在整个计算过程中是固定的，并且仅局部神经网络的输出层中的权重系数是训练参数。整个神经网络通过线性或非线性最小二乘计算来训练，而不是通过反向传播型算法来训练。我们介绍了一个块时间推进计划与所提出的方法长时间模拟的时间依赖的线性/非线性偏微分方程。目前的方法表现出一个清晰的意义上的收敛相对于神经网络中的自由度。其数值误差通常随着训练参数的数量、训练数据点的数量或子域的数量的增加而呈指数或接近指数地减小。大量的数值实验已经进行了证明所提出的方法的计算性能。我们还证明了它的能力，长时间的动态模拟与一些测试问题。我们比较了所提出的方法与深层伽辽金方法（DGM）和物理信息神经网络（PINN）方法的精度和计算成本。目前的方法表现出明显的优越性，其数值误差和网络训练时间大大小于（通常由数量级）比DGM和PINN。我们还比较了目前的方法与经典的有限元法（FEM）。目前的方法的计算性能与精度和计算成本方面的FEM性能相当，并且经常超过FEM性能。(C)2021爱思唯尔有限公司版权所有。
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 conditions are 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 the hidden layers of the local neural networks are pre-set to random values and fixed throughout the computation, and only the weight coefficients in the output layers of the local neural networks 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 simulations of time-dependent linear/nonlinear partial differential equations. 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 training parameters, or the number of training data points, or the number of sub-domains increases. Extensive numerical experiments have been performed to demonstrate the computational performance of the presented method. We also demonstrate its capability for long-time dynamic simulations with some test problems. We compare the presented method with the deep Galerkin method (DGM) and the physics-informed neural network (PINN) method 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 often exceeds, the FEM performance in terms of the accuracy and computational cost. (C) 2021 Elsevier B.V. All rights reserved.