mlsanityData to 7 Oct 2026

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

arXiv 2407.19707 ↗First posted 29 Jul 2024math.NA cs.LG cs.NAcited by 5

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Abstract (arXiv)

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.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 19 of its 49 references that are in the map. First posted on arXiv 29 Jul 2024.

Builds on

19 of 49 references.

References · 19

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7 more
  1. 2021Physics informed neural networks for continuum micromechanicscited by 46
  2. 2020Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approachcited by 35
  3. 2021Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networkscited by 13
  4. 2021An artificial neural network approach to bifurcating phenomena in computational fluid dynamicscited by 11
  5. 2021Physics-informed Neural Network for Nonlinear Dynamics in Fiber Opticscited by 7
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  7. 2023A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equationscited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 2; 2026: 3.

5 papers.

  1. 2025Physics-informed neural networks for high-dimensional solutions and snaking bifurcations in nonlinear latticescited by 3
  2. 2026Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithmcited by 2
  3. 2025Learning with Mandelbrot and Juliacited by 0
  4. 2026Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approachcited by 0
  5. 2026Hybrid Joint-Selective Optimization: Reduced-Space Levenberg-Marquardt Refinement of Low-Dimensional Parameters of Interestcited by 0