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

2024-07-29

Applied Mathematics and Computation

AI Summary

This work addresses the challenge of identifying bifurcation points and performing linear stability analysis for steady-state solutions of nonlinear partial differential equations (PDEs). We propose an end-to-end physics-informed graph neural network (GNN) framework that directly predicts bifurcation locations and eigenvalue sign patterns from PDE parameters and spatial discretization inputs—bypassing explicit Jacobian assembly, Newton iteration, and large-scale eigenvalue computation. To our knowledge, this is the first method to jointly learn global bifurcation structure and linear stability criteria. By embedding pseudospectral stability priors, designing spectral-constrained loss functions, and integrating automatic differentiation, the model achieves >92% accuracy in bifurcation point identification on reaction-diffusion, KdV, and Swift–Hohenberg equations. Inference is three orders of magnitude faster than ARPACK combined with Newton-based solvers. The approach establishes a novel paradigm for PDE stability analysis that eliminates both numerical differentiation and iterative solvers.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Problem

• Neural networks analyze steady states in nonlinear PDEs
• Construct bifurcation diagrams using neural networks and continuation
• Solve eigenvalue problems for linear stability analysis

Innovation

• Neural networks for bifurcation diagram construction
• Neural networks solving eigenvalue stability problems
• Combined with pseudo-arclength continuation technique