Extracting Dynamical Models from Data

The problem of determining the underlying dynamics of a system when only given data of its state over time has challenged scientists for decades. In this paper, the approach of using machine learning to model the updates of the phase space variables is introduced; this is done as a function of the phase space variables. (More generally, the modeling is done over functions of the jet space.) This approach (named FJet) allows one to accurately replicate the dynamics, and is demonstrated on the examples of the damped harmonic oscillator, the damped pendulum, and the Duffing oscillator; the underlying differential equation is also accurately recovered for each example. In addition, the results in no way depend on how the data is sampled over time (i.e., regularly or irregularly). It is demonstrated that a regression implementation of FJet is similar to the model resulting from a Taylor series expansion of the Runge-Kutta (RK) numerical integration scheme. This identification confers the advantage of explicitly revealing the function space to use in the modeling, as well as the associated uncertainty quantification for the updates. Finally, it is shown in the undamped harmonic oscillator example that the stability of the updates is stable $10^9$ times longer than with $4$th-order RK (with time step $0.1$).