Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning

TL;DR

This paper introduces a physics-informed machine learning method using neural networks to efficiently approximate slow invariant manifolds in singularly perturbed systems, enabling accurate reduced order models regardless of perturbation size.

Contribution

The paper develops a novel PIML approach employing neural networks to solve invariance equations for SIMs, outperforming traditional methods in accuracy and robustness.

Findings

Achieves high-accuracy SIM approximations comparable or superior to traditional methods.

Method remains effective regardless of the magnitude of the perturbation parameter.

Provides insights into computational costs of different derivative approximation techniques.

Abstract

We present a physics-informed machine-learning (PIML) approach for the approximation of slow invariant manifolds (SIMs) of singularly perturbed systems, providing functionals in an explicit form that facilitate the construction and numerical integration of reduced order models (ROMs). The proposed scheme solves a partial differential equation corresponding to the invariance equation (IE) within the Geometric Singular Perturbation Theory (GSPT) framework. For the solution of the IE, we used two neural network structures, namely feedforward neural networks (FNNs), and random projection neural networks (RPNNs), with symbolic differentiation for the computation of the gradients required for the learning process. The efficiency of our PIML method is assessed via three benchmark problems, namely the Michaelis-Menten, the target mediated drug disposition reaction mechanism, and the 3D Sel'kov model. We show that the proposed PIML scheme provides approximations, of equivalent or even higher accuracy, than those provided by other traditional GSPT-based methods, and importantly, for any practical purposes, it is not affected by the magnitude of the perturbation parameter. This is of particular importance, as there are many systems for which the gap between the fast and slow timescales is not that big, but still ROMs can be constructed. A comparison of the computational costs between symbolic, automatic and numerical approximation of the required derivatives in the learning process is also provided.