  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.
