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 {\em 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 the jet space of the variables.) This approach is shown to accurately replicate the dynamics for the examples of the harmonic oscillator, the pendulum, and the Duffing oscillator; the underlying differential equation is also accurately recovered in 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 this approach (named "FJet") is similar to the model resulting from a Taylor series expansion of the Runge-Kutta (RK) numerical integration scheme. This analogy confers the advantage of explicitly revealing the appropriate functions to use in the modeling, as well as revealing the error estimate of the updates. Thus, this new approach can be thought of as a way to determine the coefficients of an RK scheme by machine learning. Finally, it is shown in the undamped harmonic oscillator example that the stability of the updates is stable for $10^9$ times longer than with $4$th-order RK.