Local Randomized Neural Networks with Discontinuous Galerkin Methods for Partial Differential Equations

06/11/2022
by   Jingbo Sun, et al.
0

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.

READ FULL TEXT

page 24

page 28

research
08/06/2023

Randomized Neural Networks with Petrov-Galerkin Methods for Solving Linear Elasticity Problems

We develop the Randomized Neural Networks with Petrov-Galerkin Methods (...
research
01/31/2022

Deep Petrov-Galerkin Method for Solving Partial Differential Equations

Deep neural networks are powerful tools for approximating functions, and...
research
12/04/2020

Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations

We present a neural network-based method for solving linear and nonlinea...
research
03/20/2021

Understanding Loss Landscapes of Neural Network Models in Solving Partial Differential Equations

Solving partial differential equations (PDEs) by parametrizing its solut...
research
01/10/2022

Neural Networks to solve Partial Differential Equations: a Comparison with Finite Elements

We compare the Finite Element Method (FEM) simulation of a standard Part...
research
04/05/2021

A decision-making machine learning approach in Hermite spectral approximations of partial differential equations

The accuracy and effectiveness of Hermite spectral methods for the numer...
research
03/21/2020

Parameter robust preconditioning by congruence for multiple-network poroelasticity

The mechanical behaviour of a poroelastic medium permeated by multiple i...

Please sign up or login with your details

Forgot password? Click here to reset