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

arXiv: https://arxiv.org/abs/2407.19707 · HTML: https://mlsanity.com/paper/2407.19707/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 49 references that are in the map.
- First posted on arXiv 29 Jul 2024.

## Builds on (most cited first, 19 of 19)

- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Recurrent Neural Networks for Time Series Forecasting: Current Status and Future Directions](https://mlsanity.com/paper/1909.00590/) (2019)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks](https://mlsanity.com/paper/2110.14523/) (2021)
- [An artificial neural network approach to bifurcating phenomena in computational fluid dynamics](https://mlsanity.com/paper/2109.10765/) (2021)
- [Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problems](https://mlsanity.com/paper/2209.11134/) (2022)
- [Physics-informed Neural Network for Nonlinear Dynamics in Fiber Optics](https://mlsanity.com/paper/2109.00526/) (2021)
- [A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equations](https://mlsanity.com/paper/2311.11558/) (2023)

## Built on by (most cited first, 5 of 5)

- [Physics-informed neural networks for high-dimensional solutions and snaking bifurcations in nonlinear lattices](https://mlsanity.com/paper/2507.09782/) (2025)
- [Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm](https://mlsanity.com/paper/2602.08515/) (2026)
- [Learning with Mandelbrot and Julia](https://mlsanity.com/paper/2509.00903/) (2025)
- [Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach](https://mlsanity.com/paper/2603.21568/) (2026)
- [Hybrid Joint-Selective Optimization: Reduced-Space Levenberg-Marquardt Refinement of Low-Dimensional Parameters of Interest](https://mlsanity.com/paper/2609.37308/) (2026)

## 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.