Learning systems of ordinary differential equations with Physics-Informed Neural Networks: the case study of enzyme kinetics

Abstract

Physics Informed Neural Networks (PINNs) are a type of function approximators that use both data-driven supervised neural networks to learn the model of the dynamics of a physical system, and mathematical equations of the physical laws governing that system. PINNs have the benefit of being data-driven to train a model, but also of being able to assure consistency with the physics, and to extrapolate accurately beyond the range of data that currently accessible. As a result, PINNs can provide models that are more reliable while using less data. Specifically, the PINNs objective is to learn the solutions of a systems of equations using supervised learning on the available data and incorporating the knowledge of physical laws and constraints into the training process. However, solving single differential equations with a PINN may be relatively simple, solving systems of coupled differential equations may not be so simple. In this study, I present a neural network model specialized in solving differential equations of enzyme kinetics that has the main characteristic of being a demonstrative simple case of coupled equations system. The study focuses mainly on the theoretical aspects of the definition of a physics-informed loss function and shows a case study that highlights the challenges still to be overcome in solving systems of coupled differential equations.