A physics-informed neural network method for the approximation of slow
  invariant manifolds for the general class of stiff systems of ODEs
Dimitrios G. Patsatzis Lucia Russo Constantinos Siettos
We present a physics-informed neural network (PINN) approach for the
discovery of slow invariant manifolds (SIMs), for the most general class of
fast/slow dynamical systems of ODEs. In contrast to other machine learning (ML)
approaches that construct reduced order black box surrogate models using simple
regression, and/or require a priori knowledge of the fast and slow variables,
our approach, simultaneously decomposes the vector field into fast and slow
components and provides a functional of the underlying SIM in a closed form.
The decomposition is achieved by finding a transformation of the state
variables to the fast and slow ones, which enables the derivation of an
explicit, in terms of fast variables, SIM functional. The latter is obtained by
solving a PDE corresponding to the invariance equation within the Geometric
Singular Perturbation Theory (GSPT) using a single-layer feedforward neural
network with symbolic differentiation. The performance of the proposed
physics-informed ML framework is assessed via three benchmark problems: the
Michaelis-Menten, the target mediated drug disposition (TMDD) reaction model
and a fully competitive substrate-inhibitor(fCSI) mechanism. We also provide a
comparison with other GPST methods, namely the quasi steady state approximation
(QSSA), the partial equilibrium approximation (PEA) and CSP with one and two
iterations. We show that the proposed PINN scheme provides SIM approximations,
of equivalent or even higher accuracy, than those provided by QSSA, PEA and
CSP, especially close to the boundaries of the underlying SIMs.