  We describe a method for the identification of models for dynamical systems
from observational data. The method is based on the concept of symbolic
regression and uses genetic programming to evolve a system of ordinary
differential equations (ODE). The novelty is that we add a step of
gradient-based optimization of the ODE parameters. For this we calculate the
sensitivities of the solution to the initial value problem (IVP) using
automatic differentiation. The proposed approach is tested on a set of 19
problem instances taken from the literature which includes datasets from
simulated systems as well as datasets captured from mechanical systems. We find
that gradient-based optimization of parameters improves predictive accuracy of
the models. The best results are obtained when we first fit the individual
equations to the numeric differences and then subsequently fine-tune the
identified parameter values by fitting the IVP solution to the observed
variable values.
