  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.
