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

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C297-C322, August 2024.
Abstract. We present a physics-informed machine learning (PIML) approach for the approximation of slow invariant manifolds of singularly perturbed systems, providing functionals in an explicit form that facilitates the construction and numerical integration of reduced-order models (ROMs). The proposed scheme solves the 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 and random projection neural networks, 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 because 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. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://epubs.siam.org/doi/suppl/10.1137/23M1602991/suppl_file/131735_1_supp_551502_s5k7wy_sc.pdf and https://epubs.siam.org/doi/suppl/10.1137/23M1602991/suppl_file/SISC_PIML_SIMs_SP-main.zip.