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

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  • Shahab, Muhammad Luthfi
  • Susanto, Hadi

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.

Suggested Citation

  • Shahab, Muhammad Luthfi & Susanto, Hadi, 2024. "Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 483(C).
    • Handle: RePEc:eee:apmaco:v:483:y:2024:i:c:s0096300324004466
    DOI: 10.1016/j.amc.2024.128985
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    References listed on IDEAS

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    1. Iqbal, Sehar & Zegeling, Paul Andries, 2020. "An efficient nonlinear multigrid scheme for 2D boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    2. Hewamalage, Hansika & Bergmeir, Christoph & Bandara, Kasun, 2021. "Recurrent Neural Networks for Time Series Forecasting: Current status and future directions," International Journal of Forecasting, Elsevier, vol. 37(1), pages 388-427.
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