A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs
This work addresses the challenge of model reduction for stiff dynamical systems, which is important for computational efficiency in fields like chemical kinetics, but it is incremental as it builds on existing PINN and GSPT frameworks.
The authors tackled the problem of approximating slow invariant manifolds (SIMs) for stiff ODE systems by developing a physics-informed neural network (PINN) method that simultaneously decomposes vector fields and provides explicit SIM functionals, achieving equivalent or higher accuracy than traditional methods like QSSA, PEA, and CSP, especially near boundaries.
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.