Stability Analysis of Physics-Informed Neural Networks for Stiff Linear Differential Equations

Document detail

ID

oai:arXiv.org:2408.15393

Topic

Mathematics - Numerical Analysis Computer Science - Machine Learnin... Mathematics - Dynamical Systems 65L20, 68T07, 65L04, 37N30

Author

Fabiani, Gianluca Bollt, Erik Siettos, Constantinos Yannacopoulos, Athanasios N.

Year

2024

listing date

9/4/2024

Keywords

analysis linear schemes numerical

Metrics

Abstract

We present a stability analysis of Physics-Informed Neural Networks (PINNs) coupled with random projections, for the numerical solution of (stiff) linear differential equations.

For our analysis, we consider systems of linear ODEs, and linear parabolic PDEs.

We prove that properly designed PINNs offer consistent and asymptotically stable numerical schemes, thus convergent schemes.

In particular, we prove that multi-collocation random projection PINNs guarantee asymptotic stability for very high stiffness and that single-collocation PINNs are $A$-stable.

To assess the performance of the PINNs in terms of both numerical approximation accuracy and computational cost, we compare it with other implicit schemes and in particular backward Euler, the midpoint, trapezoidal (Crank-Nikolson), the 2-stage Gauss scheme and the 2 and 3 stages Radau schemes.

We show that the proposed PINNs outperform the above traditional schemes, in both numerical approximation accuracy and importantly computational cost, for a wide range of step sizes.

Fabiani, Gianluca,Bollt, Erik,Siettos, Constantinos,Yannacopoulos, Athanasios N., 2024, Stability Analysis of Physics-Informed Neural Networks for Stiff Linear Differential Equations