Controlling Dynamical Systems into Unseen Target States Using Machine Learning

We present a novel, model-free, and data-driven methodology for controlling complex dynamical systems into previously unseen target states, including those with significantly different and complex dynamics. Leveraging a parameter-aware realization of next-generation reservoir computing (NGRC), our approach accurately predicts system behavior in unobserved parameter regimes, enabling control over transitions to arbitrary target states utilizing a new prediction evaluation and selection scheme. Crucially, this includes states with dynamics that differ fundamentally from known regimes, such as shifts from periodic to intermittent or chaotic behavior. The method’s parameter awareness facilitates non-stationary control with which control scenarios are generated and evaluated on the basis of predefined control objective. In addition to proving the method for transient-free control to extrapolated chaotic target states over transition times, we demonstrate the method’s effectiveness on a nonlinear power system model. Our method successfully navigates transitions even in scenarios where system collapse is observed frequently, while ensuring fast transitions and avoiding prolonged transient behavior. By extending the applicability of machine learning-based control mechanisms to previously inaccessible target dynamics, the methodology opens the door to new control applications while maintaining exceptional efficiency.

💡 Research Summary

The paper introduces a model‑free, data‑driven control framework that can steer complex dynamical systems into previously unseen target states, even when those states exhibit fundamentally different dynamics such as a transition from periodic to intermittent or chaotic behavior. The cornerstone of the approach is a parameter‑aware next‑generation reservoir computing (NGRC) architecture. Unlike traditional reservoir computers that rely on randomly generated recurrent networks, NGRC builds a deterministic library of monomials from the input time series and its time‑shifted copies, explicitly encoding the dependence on system parameters. This enables accurate prediction of system trajectories in both interpolated and extrapolated regions of the parameter space using only a few short training recordings.

The authors demonstrate the method on two benchmark problems. First, they train NGRC on four periodic Lorenz trajectories (ρ = 100.0, 100.1, 100.2, 100.3) each consisting of 5 000 time steps. Despite being trained exclusively on periodic data, the model successfully predicts the weakly chaotic regime at ρ = 99.5, a region never observed during training. Control is realized by applying a force proportional to the difference between the current state and the NGRC‑predicted target state. To avoid transient bursts that typically occur during abrupt parameter changes, the authors introduce a running correlation dimension (rcd) metric. Candidate control trajectories are generated by the NGRC for various transition times; those whose rcd never exceeds the maximum rcd of the target chaotic attractor are selected as admissible control strategies. Across a wide range of transition durations, the transient probability is reduced to zero, and statistical climate measures (correlation dimension and largest Lyapunov exponent) of the controlled systems match those of the true chaotic system within statistical error.

The second case study concerns a small‑scale power‑system model (Dobson et al.) where the reactive power demand Q₁ serves as a bifurcation parameter. NGRC is trained on seven distinct Q₁ values, allowing interpolation and extrapolation across the parameter range. The authors identify a narrow periodic window just before the voltage‑collapse threshold (Q_target ≈ 2.989788) and aim to move the system from an initial periodic operating point (Q_initial ≈ 2.98940) to this window. Uncontrolled simulations show that 35 % of trajectories collapse, 37 % experience prolonged chaotic transients (>12.5 s), and only 28 % transition quickly. By generating non‑stationary candidate trajectories with NGRC and evaluating them using spectral entropy—a measure that remains constant for periodic dynamics—the authors select trajectories that reach the target without exceeding a predefined entropy threshold. Applying the corresponding control forces to 250 initial conditions eliminates both collapse and long transients, achieving fast, reliable convergence to the desired periodic regime.

A key contribution is the “prediction evaluation and selection scheme,” which formalizes the process of turning NGRC‑generated candidate trajectories into actionable control policies. The scheme requires the user to define quantitative control objectives (e.g., minimize transition time, keep rcd below a threshold, maintain low spectral entropy) and then automatically filters NGRC outputs to satisfy these criteria. This decouples the heavy lifting of learning system dynamics from the design of control actions, dramatically reducing the amount of training data (10–100× less than traditional reservoir computing) and computational overhead compared with deep‑learning or reinforcement‑learning based controllers.

Overall, the work demonstrates that a parameter‑aware NGRC can serve as an efficient digital twin, enabling safe, fast, and transient‑free navigation to target states that were never observed during training. The methodology extends the applicability of machine‑learning‑based control to regimes previously considered inaccessible, offering a promising tool for future micro‑grid stability, aerospace vehicle guidance, and biomedical device regulation where system models are uncertain, data are scarce, and rapid adaptation is essential.