Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations

Abstract

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 Ck 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.