Prediction performance of random reservoirs with different topology for nonlinear dynamical systems with different number of degrees of freedom

TL;DR

This work systematically probes how reservoir topology symmetry affects reservoir computing predictions across four nonlinear dynamical systems of varying complexity. By decoupling connectivity from edge weights and testing five topologies, it shows that symmetry improves cross-prediction performance when input dimensions are smaller than the system's degrees of freedom, particularly for convection-like models; however, highly chaotic, high-dimensional systems exhibit weak sensitivity to topology. The findings emphasize the role of memory-enabled cross-prediction in RC and offer concrete guidelines for choosing symmetric topologies in low-input scenarios, while highlighting limits and the potential need for plastic RC architectures to further boost performance. Overall, the study advances understanding of how structural properties of RC networks shape learning of complex dynamics and informs practical RC design for spatiotemporal systems.

Abstract

Reservoir computing (RC) is a powerful framework for predicting nonlinear dynamical systems, yet the role of reservoir topology$-$particularly symmetry in connectivity and weights$-$remains not adequately understood. This work investigates how the structure of the network influences the performance of RC in four systems of increasing complexity: the Mackey-Glass system with delayed-feedback, two low-dimensional thermal convection models, and a three-dimensional shear flow model exhibiting transition to turbulence. Using five reservoir topologies in which connectivity patterns and edge weights are controlled independently, we evaluate both direct- and cross-prediction tasks. The results show that symmetric reservoir networks substantially improve prediction accuracy for the convection-based systems, especially when the input dimension is smaller than the number of degrees of freedom. In contrast, the shear-flow model displays almost no sensitivity to topological symmetry due to its strongly chaotic high-dimensional dynamics. These findings reveal how structural properties of reservoir networks affect their ability to learn complex dynamics and provide guidance for designing more effective RC architectures.

Figures (13)

• Figure 1: Reservoir matrices $W$ for five different random network topologies, namely, random--asymmetric (R-A), random symmetrized--asymmetric (RS-A), random symmetrized--symmetric (RS-S), Watts-Strogatz--asymmetric (WS-A), and Watts-Strogatz--symmetric (WS-S). We display examples with $N=32$ reservoir nodes for a better visibility. In the subsequent analysis of the dynamical systems $N=1024$ in most cases. A filled square at position $(i,j)$ stands for an active connection from node $i$ to node $j$, i.e., $i\to j$, one at $(j,i)$ for $j\to i$. The color of the square stands for the magnitude of the weight, i.e., $w_{ij}\in \mathbb{R}$ for $i\to j$ and $w_{ji}\in \mathbb{R}$ for $j\to i$. The diagonal is indicated in all plots for better visual guidance with respect to symmetry. Bottom-right: node-degree distribution of network topologies R-A, RS-A, and WS-A with 1024 nodes each. The distributions of incoming and outgoing degrees (in, out) are different for asymmetric connection network R-A while they are same for remaining symmetric connection network topologies. The networks RS-S (WS-S) and RS-A (WS-A) are not shown as they have same node-degree distribution owing to shared connection matrices.
• Figure 2: A representative weight distribution $f(W_{ij})$ of reservoir matrix $W$ for the five network configurations, namely, R-A, RS-A, RS-S, WS-A and WS-S with $N=1024$ nodes and reservoir density $D_r = 0.008$ after setting the spectral radius (here, $\rho = 1$). Inset: The weight distributions of the same networks before setting the spectral radius. The pronounced peak at $W_{ij} = 0$ is due to small reservoir density $D_r = 0.008$.
• Figure 3: Sketch of the two-dimensional Rayleigh-Bénard convection configuration for both Lorenz models. We indicate the boundary conditions and the counter-rotating flow circulations in the form of convection rolls in the middle of the layer.
• Figure 4: Optimal time step size for sampling the input data of the (a) Mackey--Glass equation (MG), (b) Lorenz 63 (L63) model, (c) 8-dimensional Lorenz-type (L8) model, and (d) Galerkin model of the three-dimensional plane shear flow (SF) for reservoir networks with node numbers $N=1024$. Each case for each application, and each topology is obtained as an ensemble median of 50 different random reservoir network initializations. The legend in (c) applies to all panels.
• Figure 5: Reconstruction of the attractor of Lorenz 63 system with respect to output modes $A_1$ and $B_2$ for reservoir computing time step (a) $\Delta t = 0.01$ and (b) $\Delta t = 0.05$. The input mode is $B_1$. Solid blue line shows ground truth (GT) of $A_1$ and $B_1$, and solid orange line represents their prediction (Pr).