Learning Beyond Experience: Generalizing to Unseen State Space with Reservoir Computing

TL;DR

The paper addresses the challenge of generalizing predictive models to unseen regions of state space in multistable dynamical systems when training data are limited. It proposes a multiple-trajectory training scheme for reservoir computers (RCs) that leverages disjoint time series to better sample transient dynamics and enable out-of-domain forecasting. Across Duffing, multi-well, magnetic pendulum, and Lorenz-like systems, RCs trained on one basin or attractor successfully infer and forecast behavior in unseen basins or attractors, including chaotic ones, using partial observations and without explicit structural priors. These results demonstrate data-efficient, model-free generalization capabilities of RCs and suggest a potential inductive bias from the regularized linear readout that supports extrapolation beyond training data. The work has practical implications for modeling complex multistable dynamics in domains where prior knowledge is scarce and data are limited.

Abstract

Machine learning techniques offer an effective approach to modeling dynamical systems solely from observed data. However, without explicit structural priors -- built-in assumptions about the underlying dynamics -- these techniques typically struggle to generalize to aspects of the dynamics that are poorly represented in the training data. Here, we demonstrate that reservoir computing -- a simple, efficient, and versatile machine learning framework often used for data-driven modeling of dynamical systems -- can generalize to unexplored regions of state space without explicit structural priors. First, we describe a multiple-trajectory training scheme for reservoir computers that supports training across a collection of disjoint time series, enabling effective use of available training data. Then, applying this training scheme to multistable dynamical systems, we show that RCs trained on trajectories from a single basin of attraction can achieve out-of-domain generalization by capturing system behavior in entirely unobserved basins.

Figures (12)

• Figure 1: A Reservoir Computer for Time Series Prediction.
• Figure 2: RC identification of an unseen attractor in the Duffing system.(a) We train an RC with ${N_r=200}$ nodes on ${N_{train}=10}$ fully-observed trajectories (gray lines) from one of the Duffing system's basins of attraction. (b) Then, we forecast the system's evolution from $36$ initial conditions (dots) and corresponding short test signals (thick lines, ${N_{test}=10}$ observations). Predictions are illustrated by thin lines. We color each trajectory according the true basin of its initial condition. The RC recovers system behavior in both the seen (blue) and unseen (pink) basins, and predicts the correct fixed point for all sample test signals.
• Figure 3: RCs can generalize to unseen basins even with sparse and restricted training data.(a) to (e) We train an RC of ${N_r=200}$ nodes on ${N_{train}=10}$ partially-observed trajectories ($x$ only) of the unforced Duffing system ($F_0=0$) from the basin of attraction $B(\boldsymbol{A}^-)$ (blue). Then we make predictions from short test signals (${N_{test}=10}$ observations each) from both basins of attraction. We draw the initial conditions, $(x_0,y_0)$, of the training signals from a random distribution that uniformly covers the intersection of $B(\boldsymbol{A}^-)$ with a square of side $2\Delta_0^{train}$ centered at the origin (dashed boxes), i.e. ${x_0\sim\mathcal{U}\left[-\Delta_0^{train},\Delta_0^{train}\right]}$ and ${y_0\sim\mathcal{U}\left[-\Delta_0^{train},\Delta_0^{train}\right]}$, subject to $(x_0,y_0)\in B(\boldsymbol{A}^-)$. (a) to (c): The RC-predicted basins for different values of $\Delta_0^{train}$, from ${\Delta_0^{train}=10}$ (a) to ${\Delta_0^{train}=4}$ (c). The blue/pink initial conditions are those that the RC correctly predicts belong to the seen/unseen basin. The initial conditions of test signals for which the RC-predicted trajectory converges to the incorrect attractor are yellow. White initial conditions indicate that the predicted trajectory converges to a spurious attractor. Crosses mark the true fixed point attractors and dots mark the training initial conditions. We use $y_0$ for plotting purposes only; the RC has access to and then predicts only the $x$-component of the Duffing system. (d) and (e): Example predictions for $\Delta_0^{train}=10$. The test signal ends at the vertical dashed line. (f) and (g) The RC still generalizes to unseen basins when a constant external forcing (${F_0=1}$) breaks the Duffing system's rotational symmetry, whether we train the RC on the smaller basin (f) or the larger basin (g).
• Figure 4: In many scenarios, RC performance in unseen basins is comparable to performance in the training basin. We train an RC of ${N_r=200}$ nodes to forecast partially-observed states of the Duffing system ($x$ only) and plot the fraction of short test signals for which the RC predicts the correct basin of attraction as we vary the number of training initial conditions, $N_{train}$, and the half-widths of the ranges of the training and test initial conditions, ${\Delta_0^{train}}$ and ${\Delta_0^{test}}$. (a) and (c): All training signals are from ${B(\boldsymbol{A}^-)}$. (b) and (d): Training signals are from both basins, ${B(\boldsymbol{A}^\pm)}$. In all cases, the training initial conditions are random, but the initial conditions of the test signals, belonging to both basins, form a ${50\times50}$ uniform grid. Each test signal consists of ${N_{test}=10}$ consecutive data points from the true system, starting from the specified initial condition. For all grid points, we plot the mean fraction correct calculated over ten random realizations of the RC's internal structure and of the training initial conditions. (a) and (b):${N_{train}=10}$. (c) and (d):${\Delta_0^{test}=10}$. Vertical lines mark the distance of the fixed points from the origin, $||\boldsymbol{A}^\pm||$, and diagonal lines mark ${\Delta_0^{test}=\Delta_0^{train}}$. Crosses and asterisks indicate the values of $N_{train}$, $\Delta_0^{train}$, and $\Delta_0^{test}$ that we use in \ref{['fig:Duffing_Basins']}a,d,e and in \ref{['fig:Duffing_Basins']}b-c, respectively
• Figure 5: RC identification of unseen attractors in a multi-well system with segregated basins. We train an RC of ${N_r=200}$ nodes on ${N_{train}=25}$ fully-observed trajectories (gray lines) from two (a, b) or just one (c) of the multi-well system's basins of attraction (left). Then, we forecast the system's evolution from $100$ initial conditions (dots) and corresponding short test signals (thick lines, ${N_{test}=5}$ observations) in two cases: when the inputs to RC are standardized (center), and when the inputs to the RC are not standardized (right). We color predictions (thin lines) orange if they go to an incorrect attractor, and according to their true basin otherwise.