Discontinuous hybrid neural networks for the one-dimensional partial differential equations

TL;DR

This paper introduces a novel discontinuous hybrid neural network approach for solving one-dimensional partial differential equations, combining variational principles, interface conditions, and boundary constraints to achieve high accuracy and convergence.

Contribution

It presents a new hybrid neural network method with a specialized loss functional and combined optimization techniques for PDEs, enhancing solution accuracy and convergence guarantees.

Findings

Achieves high-accuracy solutions for PDEs.

Guarantees convergence of the loss functional.

Reduces computational complexity with discontinuous networks.

Abstract

A feedforward neural network, including hidden layers, motivated by nonlinear functions (such as Tanh, ReLU, and Sigmoid functions), exhibits uniform approximation properties in Sobolev space, and discontinuous neural networks can reduce computational complexity. In this work, we present a discontinuous hybrid neural network method for solving the partial differential equations, construct a new hybrid loss functional that incorporates the variational of the approximation equation, interface jump stencil and boundary constraints. The RMSprop algorithm and discontinuous Galerkin method are employed to update the nonlinear parameters and linear parameters in neural networks, respectively. This approach guarantees the convergence of the loss functional and provides an approximate solution with high accuracy.