Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations

TL;DR

This paper develops neural network methods to analyze bifurcation diagrams and linear stability in nonlinear PDEs, demonstrating improved accuracy and efficiency over traditional finite difference approaches.

Contribution

It introduces neural network techniques combined with continuation methods for bifurcation and stability analysis of nonlinear PDEs, a novel application in this field.

Findings

Neural networks outperform finite difference methods in accuracy.

The approach effectively constructs bifurcation diagrams.

Neural networks provide reasonable computational times.

Abstract

This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams from parameterized nonlinear PDEs. Additionally, a neural network approach is also presented for solving eigenvalue problems to analyze solution linear stability, focusing on identifying the largest eigenvalue. The effectiveness of the proposed neural network is examined through experiments on the Bratu equation and the Burgers equation. Results from a finite difference method are also presented as comparison. Varying numbers of grid points are employed in each case to assess the behavior and accuracy of both the neural network and the finite difference method. The experimental results demonstrate that the proposed neural network produces better solutions, generates more accurate bifurcation diagrams, has reasonable computational times, and proves effective for linear stability analysis.