Inference of a time delay in stochastic systems

TL;DR

This work tackles the challenge of identifying a discrete time delay $\tau$ in stochastic, overdamped systems with delayed feedback. It introduces two complementary approaches: a PSD-based method that detects a delay-specific oscillation signature in the spectrum, and a perturbation-driven method that leverages a CNN to classify short trajectory responses to a strong perturbation. The PSD approach yields accurate delay estimates in linear and nonlinear regimes by exploiting oscillation periodicity $\Delta\omega \approx 2\pi/\tau$ and related peak locations, while the perturbation-CNN method enables real-time inference from short observation windows without requiring knowledge of the underlying force laws. Together, these methods extend the parameter-inference toolbox for stochastic delay systems and hold promise for applications ranging from colloidal transport under feedback to emergent collective behaviors such as flocking, with potential extensions to colored noise, multiple delays, and higher dimensions.

Abstract

Time delay is ubiquitous in many experimental and real-world situations. It is often unclear whether time delay plays a significant role in observed phenomena, and if it does, how long the time lag really is. This would be invaluable knowledge when analyzing and modeling such systems. Hitherto, no universal method is available by which the time delay can be inferred. To address this problem, we propose and demonstrate two different methods to infer time delay in overdamped Langevin systems with delayed feedback. In the first part, we focus on the power spectral density based on the positional data and use a characteristic signature of the time delay to infer the delay time. In limiting cases, we establish a direct relation of the observations made for nonlinear time-delayed feedback forces to analytical results obtained for the linear system. In other situations despite the absence of this direct relation, the characteristic signature remains and can be exploited by a semiautomatic method to infer the delay time. Furthermore, it may not always desirable or possible to observe a system for a long time to infer dependencies and parameters. Thus, in the second part, we propose for the first time a probing method combined with a neural network to infer the delay time, which requires only short observation time series. These proposed methods for inferring time delays in stochastic systems may prove to be valuable tools for gaining deeper insight into the role of delay across a wide range of applications -- from the behavior of individual colloidal particles under feedback control to emergent collective phenomena such as flocking and swarming.

Figures (17)

• Figure 1: Power spectral density (PSD) as a function of the frequency $\omega$ for a Brownian particle subject to linear time-delayed feedback \ref{['eq:linear_general']} at $\tau = 0.15\tau_B$. Solid/dashed lines: Analytical result \ref{['eq:psd_linear_ana']} (panel a) and the PSD multiplied by $\omega^2$ (panel b), for different values of $k_{a,b}$. Black solid curve represents a harmonically confined Brownian particle without feedback [i.e., $k_b = 0$ in Eq. \ref{['eq:psd_linear_ana']}. Dash-dotted grey vertical lines: Approximate inflection points at $\omega = 2\pi n/\tau$ with $n\in\mathbb{N}$. Dotted black vertical lines: Approximate maximum for $k_{a,b}>0$ at $\omega = 3\pi/(2\tau) + 2\pi n/\tau$ for $n>1$, where $n\in\mathbb{N}$.
• Figure 2: Power spectral density (PSD) as a function of the frequency $\omega$ for a Brownian particle subject to linear time-delayed feedback \ref{['eq:linear_general']} for $k_b = 0.5 k_a>0$. Solid lines: Analytical results \ref{['eq:psd_linear_ana']} (panel a) and the PSD multiplied by $\omega^2$ (panel b) for different values of the delay time $\tau$. Dash-dotted vertical lines: Approximate inflection points at $\omega = 2\pi n/\tau$ with $n\in\mathbb{N}$. Dotted vertical lines: Approximate maximum at $\omega = 3\pi/(2\tau) + 2\pi n/\tau$ for $n>1$, where $n\in\mathbb{N}$.
• Figure 3: The derivative of PSD \ref{['eq:psd_linear_ana']} [i.e., $\partial/\partial\omega [\omega^2{\rm PSD}(\omega)$] as a function of $\omega$. Here, $k_{a,b} > 0$ and $k_a\neq k_b$. Dotted horizontal line: Guide for the eye to indicate the zero-crossings of the blue solid curve. Dash-dotted vertical line: Inflection points of $\omega^2{\rm PSD}$ at $\omega = 2\pi n/\tau$, for $n \in \mathbb{N}$. Dotted vertical line: maxima of $\omega^2{\rm PSD}$ at $\omega = 3 \pi /(2\tau) + 2\pi n/\tau$.
• Figure 4: Comparison of analytical power spectral density (PSD) (solid grey curves) \ref{['eq:psd_linear_ana']} with numerically simulated noise-averaged PSD as a function of frequency $\omega$ for the linear system \ref{['eq:linear_general']} at $\tau = 0.15 \tau_B$. Panel a): $k_{a,b}>0$. Panel b): $k_{a,b}<0$, where we take $|k_b| < k_\mathrm{thresh}(\tau)$ to ensure finite feedback forces. Vertical dotted grey lines: Approximate analytical locations of maxima/minima for attractive/repulsive feedback. Numerical results are obtained by noise averaging over 400 noise realizations.
• Figure 5: Comparison of numerically simulated (colored points, obtained from averaging over 400 noise realizations) power spectral density (PSD) of nonlinear systems with the linear approximation (solid/dashed curves). Panel (a): $\tanh$ feedback force for various $\tau$; solid curves: Linear approximation with $k_a = k_b = 10k_\mathrm{B}T/\sigma$. Panel (b): $\tanh$ feedback force \ref{['eq:tanh_fb_force']} for different $A$; solid curves: Linear approximation with corresponding $k_a = k_b > 0$. Panel (c): Gaussian feedback force \ref{['eq:Gaussian_fb_force']}; solid (dashed) curves: Analytical result for the linear approximation with repulsive harmonic feedback ($k_a = k_b < 0$) and modified attractive harmonic feedback for the constant velocity state [see Appendix \ref{['sec:gaussian_linearized_const_vel']}, Eq. \ref{['eq:linear_const_vel']} for more details]. Panel (a): $B=1/\sigma$. Panels (b) and (c): $\tau = 0.15\tau_\mathrm{B}$. Vertical grey dotted lines: Approximate locations of maxima/minima according to the analytical result for the linear system in panels (b) and (c).