Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913962949345280 |
|---|---|
| author | Fabiani, Gianluca Bollt, Erik Siettos, Constantinos Yannacopoulos, Athanasios N. |
| author_facet | Fabiani, Gianluca Bollt, Erik Siettos, Constantinos Yannacopoulos, Athanasios N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15393 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs Fabiani, Gianluca Bollt, Erik Siettos, Constantinos Yannacopoulos, Athanasios N. Numerical Analysis Machine Learning Dynamical Systems 65L20, 68T07, 65L04, 37N30 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. |
| title | Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs |
| topic | Numerical Analysis Machine Learning Dynamical Systems 65L20, 68T07, 65L04, 37N30 |
| url | https://arxiv.org/abs/2408.15393 |