Slow Invariant Manifolds of Fast-Slow Systems of ODEs with Physics-Informed Neural Networks

We present a physics-informed neural network (PINN) approach using symbolic differentiation for the discovery of slow invariant manifolds (SIMs), for the general class of fast-slow dynamical systems of ODEs. In contrast to other machine learning approaches that construct reduced order black-box surrogate models using simple regression, and/or require a priori knowledge of the fast and slow variables per se, 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 numerical framework is assessed via three benchmark problems. We also provide a comparison with other GSPT methods, namely the quasi steady state approximation (QSSA), the partial equilibrium approximation (PEA), and computational singular perturbation (CSP) with one and two iterations. We show that the proposed PINN scheme provides SIM approximations of equivalent or even higher numerical accuracy than those provided by QSSA, PEA, and CSP, especially close to the boundaries of the underlying SIMs.