Modern predictive modeling increasingly calls for a single learned dynamical substrate to operate across multiple regimes. From a dynamical-systems viewpoint, this capability decomposes into the storage of multiple attractors and the selection of the appropriate attractor in response to contextual cues. In reservoir computing (RC), multi-attractor learning has largely been pursued using large, randomly wired reservoirs, on the assumption that stochastic connectivity is required to generate sufficiently rich internal dynamics. At the same time, recent work shows that minimal deterministic reservoirs can match random designs for single-system chaotic forecasting. Under which conditions can minimal topologies learn multiple chaotic attractors? In this paper, we find that minimal architectures can successfully store multiple chaotic attractors. However, these same architectures struggle with task switching, in which the system must transition between attractors in response to external cues. We test storage and selection on all 28 unordered system pairs formed from eight three-dimensional chaotic systems. We do not observe a robust dependence of multi-attractor performance on reservoir topology. Over the ten topologies investigated, we find that no single one consistently outperforms the others for either storage or cue-dependent selection. Our results suggest that while minimal substrates possess the representational capacity to model coexisting attractors, they may lack the robust temporal memory required for cued transitions.
Figures & tables
Figure 1: Minimal deterministic reservoir topologies. Each panel shows the nonzero entries (magenta) of the 10×10 reservoir matrix W . All the nonzero weights have the same magnitude and sign. Zero entries are shown in light gray. Panels correspond to the following structures: (a) delay line (DL), (b) delay line with feedback connections (DLFB), (c) simple cycle (SC), (d) cycle with jumps (CJ), (e) self-loop cycle (SLC), (f) self-loop feedback cycle (SLFB), (g) self-loop delay line with backward connections (SLDB), (h) self-loop with forward connections (SLFC), (i) forward connections (FC), and (j) double cycle (DC).
Figure 2: Approaches for multi-attractor learning in reservoir computing. Panel (a) shows the approach for attractor storage in an echo state network (ESN) using the blending technique (BT). Two chaotic trajectories are provided in parallel by concatenating their state vectors into a single input, ubt(t)=[uχ1(t);uχ2(t)] . This joint input drives the ESN, and a single linear readout is trained to produce a concatenated output trajectory that reconstructs both systems simultaneously (right time series and attractors; colors distinguish the two systems). Panel (b) illustrates cue-dependent selection using the parameter-aware (PA) approach. In addition to the state input, a scalar cue βi identifying the target attractor is injected through the bias term in Eq. ( 8 ). The ESN is then run with βi held fixed so that the autonomous dynamics are expected to converge to, and reproduce, the attractor associated with that cue. In both panels, the light pink boxes indicate untrained matrices, while the light violet box indicates the trained readout.
Figure 3: Effect of prediction deterioration on metric changes. In this figure, we provide Lorenz attractor overlays, illustrating how reconstruction quality maps to the discrete state–space Kullback–Leibler divergence (KLD). Grey curves show the reference Lorenz trajectory. In panel (a), the comparison trajectory (pink) is generated by the same Lorenz equations but using different initial conditions; both the standard discrete KLD (dKLD) and the penalized variant (KLD) remain small. Panels (b–e) show progressively degraded ESN forecasts (violet). As prediction quality deteriorates and forecasted trajectories increasingly deviate from the true state space, the standard dKLD exhibits relatively low sensitivity. On the other hand, the penalized KLD increases with the degree of prediction failure, providing a clearer indication of reconstruction failure. Metric values reported below each panel are in log10 scale.
Figure 4: Multi-attractor learning accuracy across minimal reservoir topologies . Panel (a) shows the results of the blending technique (BT). The violin plots indicate the distributions of log10 Kullback-Leibler divergence (KLD) over all 28 unordered system pairs for each reservoir topology (ordered by increasing median). Panel (b) illustrates the results for the parameter-aware (PA) approach. The panel maintains the same topology ordering and visualization used in BT. In both panels, violins show the distribution across system pairs, and the black dot marks the median log10KLD for that topology; the apparent extension below zero is due to violin smoothing and does not imply negative KLD values. (c) Representative system pair reconstruction example for Lorenz–SprottS using the self-loop delay line with backward connections (SLDB) topology in the BT setting. Dashed curves show the reference attractors and solid curves the ESN-generated attractors, with different colors distinguishing the two systems. The accuracy for the system pair corresponds to log10KLD=0.692 .
Figure 5: Performance of minimal reservoirs across multi-attractor learning tasks and system pairs . (a) Heat map of prediction quality for each combination of reservoir initializer and chaotic system pair under the blending technique (BT) approach, quantified as the log 10 of the mean KLD over runs. Panel (b) reports the same results as Panel (a), but for the parameter-aware (PA) training scheme. In both panels, darker tiles indicate lower error (better forecasts), indicating how individual minimal initializers generalize across different blended systems and how their performance patterns change between training schemes.
Figure 6: Overall multifunctionality ranking of minimal reservoirs . Multifunctionality score for each reservoir initializer, defined as the difference between the fraction of blended system pairs where the initializer ranks in the top three and the fraction where it ranks in the bottom three (higher is better). Points show the score per initializer, with horizontal lines indicating the distance to zero (vertical dashed line). Initializers at the top with positive scores (e.g., SLFB, CJ, SLDB) are both frequently among the best and rarely among the worst across tasks, while those with negative scores (e.g., SLC) tend to perform poorly more often than well.
Figure 7: System–specific performance of minimal reservoirs . (a) Fraction of times each chaotic system appears in the top-5 best performing blended pairs for a given initializer (columns). Darker colors indicate that the corresponding initializer tends to perform especially well on tasks involving that system. (b) Mean prediction quality for each system–initializer combination, measured as the log 10 of the average Kullback-Leibler divergence (KLD) over all blended pairs that contain that system. Lower values (clearer tiles) correspond to better performance. Rows correspond to individual chaotic systems and columns to the minimal reservoir initializers (abbreviations as in the main text).
Figure 8: Chaotic systems. Attractors for the eight chaotic systems used in this work: (a) Aizawa, (b) Chen, (c) Chua, (d) GenesioTesi, (e) Halvorsen, (f) Lorenz, (g) Rössler, and (h) SprottS.
Biological systems maintain function in fluctuating environments by transforming past stimulation into internal dynamical states that support future-oriented responses. Reservoir computing provides a computational analogue, but standard formulations often treat the recurrent substrate as a fixed random network and train only the readout. Here we ask how the substrate itself changes when reservoir architecture is placed under evolutionary selection for prediction. Using the Kuramoto--Sivashinsky equation as a testbed for spatiotemporal chaos, we evolved reservoirs over five construction hyperparameters: size, connectivity degree, spectral radius, input scaling, and readout regularization. Evolution reduced prediction error at the population level, extended the low-error forecast horizon, and organized the design space along a diminishing-return size--efficiency frontier. Structural analyses showed that evolved reservoirs remained within a conserved stochastic-block-model-like spectral envelope while refining low-eigenvalue modes, locking modularity to an intermediate band, and pruning connection cost within that band. Pareto analysis showed that elite reservoirs occupied a horizontal floor in the cost--modularity plane, indicating that accuracy and efficiency were achieved jointly rather than through a simple trade-off. These findings show that evolutionary optimization does not merely improve prediction, but exposes interpretable structural constraints on the recurrent substrate: it stabilizes a task-suitable dynamical class and refines the architectural degrees of freedom most relevant for prediction. Evolutionary reservoir computing therefore provides a bio-inspired framework for studying how predictive demands shape adaptive dynamical networks.
Nima Dehghani
Department of Physics, MIT · McGovern Institute for Brain Research, MIT
We introduce chaos-controlled Reservoir Computing (cc-RC) for living neural cultures: dynamically rich substrates of unique potential for adaptive computation. To account for intrinsic biological variability, cc-RC combines: (i) pre-training identification of each culture's dynamical signature and phase-portrait attractor; (ii) low-power optical chaos control to stabilize spontaneous and stimulus-evoked activity; (iii) readout training within this controlled regime. Across hundreds of neural samples, cc-RC enables robust learning and pattern classification, improving both accuracy and model longevity by approximately 300% over standard RC. We further propose Knowledge Transplant (KT), for which the reservoir map learned by an expert culture is transplanted to an attractor-equivalent student culture, reducing training time to minutes while improving performance. By enabling cross-substrate, reusable learned models, KT paves the way for knowledge accumulation and sharing across neural populations, transcending biological lifespan limits.
Seung Hyun Kim, Zhi Dou, Gaurav Upadhyay +6
Mechanical Science and Engineering, University of Illinois Urbana-Champaign · Electrical and Computer Engineering, University of Illinois Urbana-Champaign · Coordinated Science Laboratory, University of Illinois Urbana-Champaign +6
We propose a dual-channel reservoir-computing scheme for inferring the dynamics of two distinct chaotic systems with a single machine. By augmenting a standard reservoir with a system-label channel and a parameter-control channel, the machine can be trained from time series collected from a few sampled states of the two systems. We show that the trained machine not only predicts the short-time evolution of the sampled states, but also reproduces the long-term statistical properties of unseen states, thereby enabling reconstruction of the bifurcation diagrams of both systems from partial observations. The effectiveness of the scheme is demonstrated for the Lorenz and Rössler systems in numerical simulations and for the Chua and Rossler circuits in experiments. Functional-network analysis further shows that the two target systems are encoded by distinct dynamical patterns in the reservoir. These results extend multifunctional and parameter-aware reservoir computing, and provide a route to data-driven inference of multiple nonlinear systems using a single machine.
Jianmin Guo, Yao Du, Yizhen Yu +2
School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China · College of Physics and Electronic Information Engineering, Qinghai Normal University, Xining 810008, China · School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China