Construction of ODE systems from time series data by a highly flexible modelling approach

In this paper, a down-to-earth approach to purely data-based modelling of unknown dynamical systems is presented. Starting from a classical, explicit ODE formulation $y' = f(t,y)$ of a dynamical system, a method determining the unknown right-hand side $f(t,y)$ from some trajectory data $y_{k}(t_{j})$, possibly very sparse, is given. As illustrative examples, a semi-standard predator-prey model is reconstructed from a data set describing the population numbers of hares and lynxes over a period of twenty years, and a simple damped pendulum system with a highly non-linear right-hand side is recovered from some artificial but very sparse data.