Forecasting the Forced van der Pol Equation with Frequent Phase Shifts Using Reservoir Computing

TL;DR

The paper addresses forecasting circadian-clock-like dynamics in a nonautonomous system that experiences frequent phase shifts. It employs reservoir computing (RC) to learn from data generated by a forced van der Pol oscillator driven by a phase-shifting external input, training on multiple phase-shift levels and testing on unseen shifts. RC demonstrates robust predictive ability across a range of phase-shift configurations, with higher accuracy when training data exhibit richer dynamics, and it remains effective even with partial observability. The work suggests a practical, data-driven route to anticipate circadian misalignment under shift work and informs potential individualized scheduling strategies, while highlighting avenues for extending RC to more complex or real-world data.

Abstract

We tested the performance of reservoir computing (RC) in predicting the dynamics of a certain non-autonomous dynamical system. Specifically, we considered a van del Pol oscillator subjected to periodic external force with frequent phase shifts. The reservoir computer, which was trained and optimized with simulation data generated for a particular phase shift, was designed to predict the oscillation dynamics under periodic external forces with different phase shifts. The results suggest that if the training data have some complexity, it is possible to quantitatively predict the oscillation dynamics exposed to different phase shifts. The setting of this study was motivated by the problem of predicting the state of the circadian rhythm of shift workers and designing a better shift work schedule for each individual. Our results suggest that RC could be exploited for such applications.

Figures (6)

• Figure 1: Variable $v$ of the van der Pol equation subjected to external drive $P_n(t)$, to which the phase shift of $n$ hours is applied at every 4 d. (a) $n = -7$, (b) $n = 0$, and (c) $n = 7$.
• Figure 2: Basic RC structure. The input $\bm{x}_t$ is mapped onto $\bm{u}_t$ in the hidden layer by the matrix $W_{\text{in}}$. During the training and testing periods, the RC is fed with the true data $\bm{x}_t$ of the dynamical system at every step. During the forecasting phase, the reservoir computer updates autonomously using its output as the input for the new step. Here, we inject the true value of the external drive $P_n(t)$ into the input assuming that $P_n(t)$ is accessible at all times, including the forecast period.
• Figure 3: Prediction by RC of the data with $n = 7$ during the testing period. Only $v$ and $P_n(t)$ are shown.
• Figure 4: Time series of the prediction and true $v_i$ for $n=7$ and $m=10$ during the forecast period. Only $v$ and $P_n(t)$ are shown.
• Figure 5: Error of the forecast, $E^{(n\to m)}$, for $n = -7, 0, 7$ and $m \in \{-11,10,\cdots,12\}$, where the input was (a) $\bm{x} = (v, w, P_n)$ and (b) $\bm{x} = (v, P_n)$.