Extracting Dynamical Models from Data

The FJet approach is introduced for determining the underlying model of a dynamical system. It borrows ideas from the fields of Lie symmetries as applied to differential equations (DEs), and numerical integration (such as Runge-Kutta). The technique can be considered as a way to use machine learning (ML) to derive a numerical integration scheme. The technique naturally overcomes the "extrapolation problem", which is when ML is used to extrapolate a model beyond the time range of the original training data. It does this by doing the modeling in the phase space of the system, rather than over the time domain. When modeled with a type of regression scheme, it's possible to accurately determine the underlying DE, along with parameter dependencies. Ideas from the field of Lie symmetries applied to ordinary DEs are used to determine constants of motion, even for damped and driven systems. These statements are demonstrated on three examples: a damped harmonic oscillator, a damped pendulum, and a damped, driven nonlinear oscillator (Duffing oscillator). In the model for the Duffing oscillator, it's possible to treat the external force in a manner reminiscent of a Green's function approach. Also, in the case of the undamped harmonic oscillator, the FJet approach remains stable approximately $10^9$ times longer than $4$th-order Runge-Kutta.