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

arXiv: https://arxiv.org/abs/2603.21568 · HTML: https://mlsanity.com/paper/2603.21568/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 91 references that are in the map.
- First posted on arXiv 23 Mar 2026.

## Builds on (most cited first, 30 of 33)

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- [Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning](https://mlsanity.com/paper/2309.07946/) (2023)
- [Enabling Local Neural Operators to perform Equation-Free System-Level Analysis](https://mlsanity.com/paper/2505.02308/) (2025)
- [Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.12360/) (2024)
- [Koopman-informed recurrent neural networks](https://mlsanity.com/paper/2410.23467/) (2024)
- [Gradient-Free Training of Recurrent Neural Networks using Random Perturbations](https://mlsanity.com/paper/2405.08967/) (2024)
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- [Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations](https://mlsanity.com/paper/2407.19707/) (2024)
- [Discrete-Time Nonlinear Feedback Linearization via Physics-Informed Machine Learning](https://mlsanity.com/paper/2303.08884/) (2023)

## Abstract (arXiv)

We present a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solutions are sought in the function space spanned by physics-informed random projection neural networks (PI-RPNNs), and discretized via a collocation approach. These are shallow, 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 explicit generalized eigenvalue formulation governing the linear stability of stationary solutions, separating the physical domain interior dynamics from the algebraic constraints imposed by boundary conditions (BCs), 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, rigorously explaining the observed ill-conditioning.