Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach

2026-03-23

AI Summary

This work proposes an efficient computational framework based on Physics-Informed Random Projection Neural Networks (PI-RPNN) to address ill-conditioned eigenvalue problems arising from stochastic projections in stability and bifurcation analysis of nonlinear partial differential equations (PDEs). The solution to the PDE is represented via a single-hidden-layer neural network with random weights, and a generalized eigenvalue problem is constructed using a collocation approach. Crucially, a matrix-free shifted inverse Krylov–Arnoldi method is introduced to directly compute dominant eigenpairs in the weight space, circumventing explicit inversion of the ill-conditioned collocation matrix. Theoretical analysis establishes that the PI-RPNN eigenvalue problem is almost surely regular and that the singular values of the collocation matrix decay exponentially. Without requiring additional PDE solves or training overhead, the method effectively filters out spurious near-zero modes and accurately captures the dominant spectral features governing stability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We address a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solution is sought in the function space spanned by physics-informed random projection neural networks (PI-RPNNs), and discretized via a collocation approach. These are single-hidden-layer networks with randomly sampled and fixed a priori hidden-layer weights; only the linear output layer weights are optimized, reducing training to a single least-squares solve. This linear output structure enables the direct and explicit formulation of the eigenvalue problem governing the linear stability of stationary solutions. This takes a generalized eigenvalue form, which naturally separates the physical domain interior dynamics from the algebraic constraints imposed by boundary conditions, at no additional training cost and without requiring additional PDE solves. However, the random projection collocation matrix is inherently numerically rank-deficient, rendering naive eigenvalue computation unreliable and contaminating the true eigenvalue spectrum with spurious near-zero modes. To overcome this limitation, we introduce a matrix-free shift-invert Krylov-Arnoldi method that operates directly in weight space, avoiding explicit inversion of the numerically rank-deficient collocation matrix and enabling the reliable computation of several leading eigenpairs of the physical Jacobian - the discretized Frechet derivative of the PDE operator with respect to the solution field, whose eigenvalue spectrum determines linear stability. We further prove that the PI-RPNN-based generalized eigenvalue problem is almost surely regular, guaranteeing solvability with standard eigensolvers, and that the singular values of the random projection collocation matrix decay exponentially for analytic activation functions.

Problem

• nonlinear PDEs
• stability analysis
• bifurcation
• random projection
• eigenvalue problem

Innovation

• Physics-Informed Neural Networks
• Random Projection
• Krylov-Arnoldi Method
• Linear Stability Analysis
• Generalized Eigenvalue Problem