A comparative study of physics-informed neural network models for learning unknown dynamics and constitutive relations

TL;DR

This paper compares discrete and continuous physics-informed neural network models for learning unknown system dynamics and constitutive relations, demonstrating higher accuracy when learning constitutive relations alone.

Contribution

It introduces a comparative analysis of discrete and continuous PINN approaches for unknown dynamics and constitutive relations, highlighting their effectiveness and applicability.

Findings

Higher accuracy when learning constitutive relations compared to full dynamics

Discrete and continuous PINN methods effectively model unknown system behaviors

Structural information improves neural network model performance

Abstract

We investigate the use of discrete and continuous versions of physics-informed neural network methods for learning unknown dynamics or constitutive relations of a dynamical system. For the case of unknown dynamics, we represent all the dynamics with a deep neural network (DNN). When the dynamics of the system are known up to the specification of constitutive relations (that can depend on the state of the system), we represent these constitutive relations with a DNN. The discrete versions combine classical multistep discretization methods for dynamical systems with neural network based machine learning methods. On the other hand, the continuous versions utilize deep neural networks to minimize the residual function for the continuous governing equations. We use the case of a fedbatch bioreactor system to study the effectiveness of these approaches and discuss conditions for their applicability. Our results indicate that the accuracy of the trained neural network models is much higher for the cases where we only have to learn a constitutive relation instead of the whole dynamics. This finding corroborates the well-known fact from scientific computing that building as much structural information is available into an algorithm can enhance its efficiency and/or accuracy.