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arXiv · 2408.15393

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

Gianluca Fabiani·Erik Bollt·Constantinos Siettos·Athanasios N. Yannacopoulos
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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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Gianluca Fabiani, Erik Bollt, Constantinos Siettos, Athanasios N. Yannacopoulos. 2024-08-27. Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs. https://arxiv.org/abs/2408.15393

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