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

TL;DR

This paper conducts a linear stability analysis of physics-informed random projection neural networks (PI-RPNNs) for solving stiff ODEs, establishing their approximation, stability, and convergence properties through theoretical proofs and numerical comparisons.

Contribution

It provides the first stability analysis of PI-RPNNs, proving their approximation, consistency, and asymptotic stability for solving ODEs, with numerical validation against classical methods.

Findings

PI-RPNNs are proven to be uniform approximators of ODE solutions.

PI-RPNNs offer consistent and asymptotically stable numerical schemes.

Numerical experiments show competitive performance with classical ODE solvers.

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.