  Randomized neural networks (RNN) are a variation of neural networks in which
the hidden-layer parameters are fixed to randomly assigned values and the
output-layer parameters are obtained by solving a linear system by least
squares. This improves the efficiency without degrading the accuracy of the
neural network. In this paper, we combine the idea of the local RNN (LRNN) and
the discontinuous Galerkin (DG) approach for solving partial differential
equations. RNNs are used to approximate the solution on the subdomains, and the
DG formulation is used to glue them together. Taking the Poisson problem as a
model, we propose three numerical schemes and provide the convergence analyses.
Then we extend the ideas to time-dependent problems. Taking the heat equation
as a model, three space-time LRNN with DG formulations are proposed. Finally,
we present numerical tests to demonstrate the performance of the methods
developed herein. We compare the proposed methods with the finite element
method and the usual DG method. The LRNN-DG methods can achieve better accuracy
under the same degrees of freedom, signifying that this new approach has a
great potential for solving partial differential equations.
