Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs

2024/08/27 by Gianluca Fabiani, Erik M. Bollt, Fabiani, Gianluca +5
Computer Science · Physics and Astronomy · #37N30 #65L04 #65L20 #68T07 #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2408.15393

openalex publication_date 2024/08/27 · openalex created_date 2024/09/21 · openalex updated_date 2026/07/28

Abstract

We present a linear stability analysis of physics-informed random projection neural networks (PI-RPNNs), for the numerical solution of the initial value problem (IVP) of (stiff) ODEs. We begin by proving that PI-RPNNs are uniform approximators of the solution to ODEs. We then provide a constructive proof demonstrating that PI-RPNNs offer consistent and asymptotically stable numerical schemes, thus convergent schemes. In particular, we prove that multi-collocation PI-RPNNs guarantee asymptotic stability. Our theoretical results are illustrated via numerical solutions of benchmark examples including indicative comparisons with the backward Euler method, the midpoint method, the trapezoidal rule, the 2-stage Gauss scheme, and the 2- and 3-stage Radau schemes.

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