Extracting Dynamical Models from Data

TL;DR

FJet addresses the challenge of extracting governing dynamics from time-series data by learning updates in phase space rather than raw states. It builds a regression model for tangent-space updates from jet-space features, then derives the governing differential equation by extrapolating the time-step to zero, providing uncertainty quantification. The method is demonstrated on a damped harmonic oscillator, a damped pendulum, and a Duffing oscillator, successfully recovering the underlying DEs and delivering extrapolations competitive with, or superior to, RK-based baselines. A key contribution is a principled procedure to construct the feature space X_n from Runge-Kutta expansions, enabling robust, interpretable dynamics with potential extensions to forcing and conserved quantities.

Abstract

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 $0.1$).

Figures (18)

• Figure 1: This figure is meant to illustrate how an initial set of features ($X_{init)}$ leads to a DE, which in turn leads to a set of features ($X_n$) which are each associated with an error estimate of ${\cal O}(\epsilon^n)$. Having gone through these steps, it is suggested that one may then repeat the process beginning with one of the $X_n$ to see if it leads to the same DE. If it does, then it may be said to be self-consistent.
• Figure 2: Three plots for the harmonic oscillator, using a noise level of $\sigma=0.2$. The left plot corresponds to the (clockwise) path taken by updates of the system, starting from an initial condition of $(u,\dot{u})=(1,0)$. The light gray circle is added as a visual reference, and corresponds to the path that would have been taken in the undamped, noiseless case. The center and right plots correspond to the changes in $u$ and $\dot{u}$, as defined in Eq. \ref{['eqn:Delta_defn']}.
• Figure 3: Plots of the $\epsilon$-dependence of the coefficients in the model $h(X)$ (cf. Eq. \ref{['eqn:HO_mapping']}). The intersection of the thin vertical and horizontal gray lines indicate the correct value at $\epsilon=0$. Note the different scales for the error for each plot.
• Figure 4: The dependence of $b_1$ and $b_2$ as a function of $\omega_0$ and $\gamma$ is plotted using blue dots. As a visual guide, the exact curves for $-\omega_0^2$ and $-2\gamma$ are added as orange curves to the left and right plots, respectively.
• Figure 5: A comparison of the extrapolation of the exact solution to that generated by FJet with $\sigma = 0.2$.