Organizations: Division of Environmental and Agricultural Engineering, Institute of Agriculture, Tokyo University of Agriculture and Technology, Tokyo, Japan · Amity Institute of Integrative Sciences and Health, Amity University Haryana - Gurugram, Manesar, 122 413 Haryana, India · Graduate School of Engineering, Nagoya Institute of Technology, Nagoya 466 8555, Japan · Department of Physics, Central University of Rajasthan, Ajmer, 305 817 Rajasthan, India
Reliable prediction of vehicle dynamics is essential for smart driving applications such as autonomous control and advanced driver-assistance systems. Off-road vehicles used in agricultural and construction settings are particularly prone to nonlinear behavior, including bifurcations and chaotic motion arising from intermittent loss of tire--road contact. Predicting such dynamics is challenging because it requires resolving both smooth nonlinearities and the discontinuous switching associated with contact loss. In this work, we investigate the feasibility of reservoir computing (RC) -- specifically an echo state network (ESN) -- for data-driven prediction of a jumping quarter-car model. The reservoir is trained on time-series data from a small number of points and evaluated on its ability to reconstruct bifurcation diagrams, phase-space attractors, and time trajectories across periodic and chaotic regimes. The trained reservoir qualitatively reproduces the period-doubling route to chaos, captures the geometric structure of periodic and chaotic attractors. These results demonstrate that reservoir computing is a feasible data-driven predictor of nonlinear dynamics in a practical, non-smooth vehicle system.
Figures & tables
Figure 1: Schematic representation of the quarter-car model incorporating a jumping (contact-loss) nonlinearity. The sprung mass ms and unsprung mass mu are coupled through a linear suspension spring–damper pair ( ks , cs ), while the unsprung mass interacts with the road profile d(t)=d0sinωt through a tire modeled as a unilateral spring of stiffness kt . The tire stiffness is switched to zero whenever the wheel loses contact with the road surface, giving rise to the piecewise-smooth governing dynamics.
Figure 2: Schematic diagram of the echo state network (ESN) architecture employed as the reservoir-computing predictor. The input layer supplies the instantaneous state vector of the quarter-car model together with the road excitation amplitude d0 to a fixed, sparsely and randomly connected reservoir of nonlinear units. Only the linear readout mapping from the reservoir state to the predicted output is trained, via ridge regression, while the internal reservoir weights remain untrained throughout.
Figure 3: Bifurcation diagrams of the tire deflection Zu as a function of the road excitation amplitude d0 , comparing (a) the ground-truth response obtained by direct numerical integration of the governing equations with (b) the response reconstructed by the trained echo state network. Vertical dashed lines in panel (a) mark the three operating points ( d0=0.0165 , 0.0185 , and 0.0205 ) used to train the reservoir. The predicted diagram in panel (b) qualitatively reproduces the period-doubling route to chaos observed in the ground truth, including extrapolation beyond the training range into the chaotic regime.
Figure 4: Comparison of ground-truth and reservoir-predicted phase-space attractors, projected onto the (Zs,Z˙s) and (Zu,Z˙u) planes, for three representative dynamical regimes: period-1 motion at d0=0.015 (a, d), period-2 motion at d0=0.02 (b, e), and chaotic motion at d0=0.025 (c, f). Panels (a)–(c) show the ground-truth attractors obtained from direct simulation; panels (d)–(f) show the corresponding attractors reconstructed from the trained reservoir’s autonomous output. Close qualitative agreement in loop structure and, for the chaotic case, folded-sheet topology, indicates that the reservoir captures the underlying geometric organization of the flow rather than merely its instantaneous amplitude.
Figure 5: Time-domain forecasts of the suspension stroke Zs generated by the trained echo state network (blue) superimposed on the corresponding ground-truth trajectories (orange), for (a) period-1 motion at d0=0.015 , (b) period-2 motion at d0=0.02 , and (c) chaotic motion at d0=0.025 . The reservoir sustains accurate long-horizon tracking of the periodic waveforms in (a) and (b), whereas in the chaotic case (c) the predicted and ground-truth trajectories diverge after a comparatively short horizon, consistent with the exponential error growth expected from sensitive dependence on initial conditions.
We study how the degree of nonlinearity in the input data affects the optimal design of reservoir computers, focusing on how closely the model's nonlinearity should align with that of the data. By reducing minimal RCs to a single tunable nonlinearity parameter, we explore how the predictive performance varies with the degree of nonlinearity in the reservoir. To provide controlled testbeds, we generalize to the fractional Halvorsen system, a novel chaotic system with fractional exponents. Our experiments reveal that the prediction performance is maximized when the reservoir's nonlinearity matches the nonlinearity present in the data. In cases where multiple nonlinearities are present in the data, we find that the correlation dimension of the predicted signal is reconstructed correctly when the smallest nonlinearity is matched. We use this observation to propose a method for estimating the minimal nonlinearity in unknown time series by sweeping the reservoir exponent and identifying the transition to a successful reconstruction. Applying this method to both synthetic and real-world datasets, including financial time series, we demonstrate its practical viability. Finally, we transfer these insights to classical RC by augmenting traditional architectures with fractional, generalized reservoir states. This yields performance gains, particularly in resource-constrained scenarios such as physical reservoirs, where increasing reservoir size is impractical or economically unviable. Our work provides a principled route toward tailoring RCs to the intrinsic complexity of the systems they aim to model.
Davide Prosperino, Haochun Ma, Christoph Räth
Ludwig-Maximilians-Universität München, Faculty of Physics, Geschwister-Scholl-Platz 1, 80539 Munich, Germany · Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institute for Frontier Materials on Earth and in Space, Linder Höhe, 51147 Cologne, Germany
Reservoir computing has emerged as an efficient machine learning framework for predicting time series generated by dynamical systems. In contrast to other machine and deep learning approaches, a reservoir computing trains only the output layer via linear regression, leaving the reservoir (recurrent layer) untrained. This simplification makes reservoir computers easier to train and more amenable to experimentation. However, because current reservoirs consist of networks of randomly connected nodes and require the optimization of numerous hyperparameters, a framework that precisely explains how reservoir computing operates and how it can be optimized remains missing. Here, we propose a frequency-based reservoir inspired by the brain's oscillatory dynamics and its hierarchy of timescales. The frequency-based reservoir can be interpreted as an ensemble of independent oscillatory units, each processing a portion of the input's frequency content. This allows us to understand the reservoir's internal behavior by modeling it as a single unit driven by an external input. Borrowing from the theory of a nonlinear oscillator forced by complex periodic inputs, we found that units of the frequency-based reservoir selectively amplify and store specific input frequencies, which are then used for prediction. The frequency-based reservoir performs as well as or better than equivalent random reservoirs. Furthermore, the frequency-based approach can be optimized to improve short-term prediction, a property that random reservoirs lack. Finally, we show that the frequency-based reservoir can also predict complex spatiotemporal dynamics. Our results show that reservoir computing can be designed using brain properties and theoretical insights borrowed from the physics of forced nonlinear oscillators.
Arthur S Powanwe
Department of Mathematics, Western University, London, ON, Canada
Time series prediction is a common application of reservoir computing. When the training and testing time series data contains multiple dynamical regimes, because an underlying parameter is changing, or the data in fact consists of multiple distinct systems, simple application of the reservoir computing principle produces high prediction errors. Here, we propose a HyperReservoir as an extended model of reservoir computing especially designed for such cases. The HyperReservoir combines a main reservoir with a smaller context reservoir, where the latter modulates the output weights of the former. This structure resembles the hypernetworks from deep neural network literature. However, in contrast, HyperReservoirs retain the simple training via linear regression of standard reservoir computing. We compare the proposed architecture with a conventional ESN, in which context acts at the input, and a full-matrix Conceptor, in which context modulates the reservoir state space. We evaluate all three models on time-series prediction tasks based on Lorenz and Rössler systems, including for varying bifurcation parameters and time sampling scales. We find that the HyperReservoir achieves the lowest mean test error in all three tasks, and particularly outperforms conceptors on data that is sampled from the same attractor but at different time scales.
Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan