  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.
