Stochastic modeling of deterministic laser chaos using generator extended dynamic mode decomposition

TL;DR

This work addresses how to extract stochastic descriptions from deterministic, time-delayed laser chaos to support reinforcement learning tasks. It employs a generator-extended dynamic mode decomposition (gEDMD) within the Koopman framework to estimate drift and diffusion terms from short-term cross-correlation data, casting the dynamics as $d\mathbf{X}(t)=\mathbf{b}(\mathbf{X}(t))dt+\sigma(\mathbf{X}(t))d\mathbf{W}(t)$. Key findings show that low-pass filtered data preserve essential leader-laggard statistics, including peak shifts and power-law behavior in switching-time distributions, and that the resulting stochastic models perform comparably to the original chaotic system in a two-armed bandit task. The work demonstrates principled coarse-graining of deterministic chaos into practical stochastic dynamics, with implications for photonic RL hardware and data-driven modeling.

Abstract

Recently, chaotic phenomena in laser dynamics have attracted much attention to its applied aspects, and a synchronization phenomenon, leader-laggard relationship, in time-delay coupled lasers has been used in reinforcement learning. In the present paper, we discuss the possibility of capturing the essential stochasticity of the leader-laggard relationship; in nonlinear science, it is known that coarse-graining allows one to derive stochastic models from deterministic systems. We derive stochastic models with the aid of the Koopman operator approach, and we clarify that the low-pass filtered data is enough to recover the essential features of the original deterministic chaos, such as peak shifts in the distribution of being the leader and a power-law behavior in the distribution of switching-time intervals. We also confirm that the derived stochastic model works well in reinforcement learning tasks, i.e., multi-armed bandit problems, as with the original laser chaos system.

Figures (9)

• Figure 1: Problem settings. The main aim here is to construct stochastic models from data generated by two lasers mutually coupled with the time-delay $\tau$. The original data stems from deterministic descriptions, and we investigate whether the data generated by the constructed stochastic models has similar features to the original one.
• Figure 2: Sample trajectories of optical intensities for Laser 1 (a) and Laser 2 (b). The vertical axis is drawn using arbitrary unit. The initial optical frequency detuning $\Delta f_\mathrm{ini}$ is set to $1$ GHz.
• Figure 3: Examples of trajectories of short-term cross-correlation values $C_1(t)$ and $C_2(t)$. These cross-correlation values are evaluated from the optical intensities generated by the same settings as Fig. \ref{['fig_intensity']}.
• Figure 4: Training trajectory data for the gEDMD algorithm. (a) Original trajectory. (b) Extracted dataset within the high-density cluster region. (c) Preprocessed dataset by the high-pass filter with $5.0 \times 10^7$ Hz. (d) Preprocessed dataset by the low-pass filter with $5.0 \times 10^6$ Hz. The orange-colored rectangle in (a) indicates the high-density cluster region depicted in (b).
• Figure 5: The distribution of being the leader. The distribution evaluated by the original trajectory is (a). By contrast, (b), (c), and (d) are evaluated by the artificial dataset generated by the stochastic models constructed from the trajectories in Figs. \ref{['fig_training']}(b), \ref{['fig_training']}(c), and \ref{['fig_training']}(d), respectively. The blue lines correspond to the histograms, and the orange curves are obtained from the kernel density estimation. The vertical gray lines correspond to $C_1-C_2 = 0.0$.