Adaptive control for multi-scale stochastic dynamical systems with stochastic next generation reservoir computing

TL;DR

This work introduces a data-driven stochastic next-generation reservoir computing (S-NGRC) controller for adaptive, closed-loop control of multi-scale stochastic systems. By learning system dynamics with stochastic NG-RC and deriving a control law that enforces convergence under noise via an extended stochastic LaSalle theorem, the framework provides theoretical guarantees and practical robustness. The approach is validated on a stochastic Van der Pol oscillator under additive and multiplicative noise and applied to epilepsy EEG data to demonstrate seizure-suppression capabilities, including governing-law learning through a Kramers–Moyal network and a three-phase control protocol. The results indicate rapid, robust tracking across time scales and noise regimes, with potential for real-time, data-driven decision making in engineering and neuroscience, while outlining directions to improve realism, safety, and joint amplitude–frequency control.

Abstract

The rapid advancement of neuroscience and machine learning has established data-driven stochastic dynamical system modeling as a powerful tool for understanding and controlling high-dimensional, spatio-temporal processes. We introduce the stochastic next-generation reservoir computing (NG-RC) controller, a framework that integrates the computational efficiency of NG-RC with stochastic analysis to enable robust event-triggered control in multiscale stochastic systems. The asymptotic stability of the controller is rigorously proven via an extended stochastic LaSalle theorem, providing theoretical guarantees for amplitude regulation in nonlinear stochastic dynamics. Numerical experiments on a stochastic Van-der-Pol system subject to both additive and multiplicative noise validate the algorithm, demonstrating its convergence rate across varying temporal scales and noise intensities. To bridge theoretical insights with real-world applications, we deploy the controller to modulate pathological dynamics reconstructed from epileptic EEG data. This work advances a theoretically guaranteed scalable framework for adaptive control of stochastic systems, with broad potential for data-driven decision making in engineering, neuroscience, and beyond.

Figures (14)

• Figure 1: Stochastic NG-RC framework diagram (take 2D sde as an example).
• Figure 2: Phase portrait of the original, perturbed, and desired trajectories under low-intensity noise without multiple time scales. (Red) Original trajectory. (Yellow) A simplified desired trajectory used to illustrate the essence of the control problem (not the actual desired trajectory used in the experiment). (Purple) Perturbed trajectory, which serves as training data for the stochastic NG-RC controller to learn the influence of the external input $u$ on the stochastic dynamical system. Blue stars indicate the initial points of the three trajectories after discarding the first 500 sample points. Black stars indicate their respective termination points."
• Figure 3: Training results for stochastic NG-RC in low-intensity noise scenarios without multiple time scales. The Fig.consists of three subplots. The left panel displays the phase portrait of the original, perturbed, and designed desired trajectories under low-intensity noise without multiple time scales. Blue stars indicate the starting points of the trajectories after discarding the first 500 sample points, while black stars mark their termination points. The top-right panel shows the randomly generated perturbation signals $u_x, u_y$ and the bottom-right panel presents the stepwise prediction error of the stochastic NG-RC model over 500 test samples.
• Figure 4: Phase-space tracking performance of the stochastic NG-RC controller in low-intensity noise scenarios without multiple time scales. a) Grey curve: Original Van der Pol (VDP) system trajectory without control. b) Yellow marker "X": Initial control activation point(1.2× the original amplitude). c) Dark red dashed line: Controlled trajectory converging toward desired trajectory (red). d) Blue and green markers "X": Trigger events for trajectory contraction (0.8× and 0.5× amplitude scaling, respectively).
• Figure 5: Control results of the conventional NG-RC at $\sigma_1 = 1,\sigma_2 = 2,\epsilon = 0.5$. The controller diverges at the 302th step after the start of control.

Theorems & Definitions (6)

• Proposition 1
• Theorem 1: Kolmogorov-Centsov theoremkaratzas1991brownian
• Lemma 1
• proof
• Lemma 2
• proof