[2603.21568] Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach

Mathematics > Numerical Analysis arXiv:2603.21568 (math) [Submitted on 23 Mar 2026] Title:Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach Authors:Gianluca Fabiani, Michail E. Kavousanakis, Constantinos Siettos, Ioannis G. Kevrekidis View a PDF of the paper titled Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach, by Gianluca Fabiani and 3 other authors View PDF HTML (experimental) 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 numeri...