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

Source: arXiv - 2511.22059v1

Overview

This paper digs into a core question for anyone using reservoir computing (RC): How does the wiring diagram of the reservoir affect its ability to predict complex, nonlinear dynamics? By systematically testing five different network topologies on four benchmark dynamical systems—from the classic Mackey‑Glass delay equation to a three‑dimensional turbulent shear flow—the authors show that symmetry in the reservoir’s connectivity can dramatically boost prediction accuracy, but only for certain types of dynamics.

Key Contributions

• Systematic topology study – Five distinct reservoir graphs are examined, separating the effects of connectivity pattern from edge‑weight distribution.
• Broad benchmark suite – Experiments cover four systems of increasing dynamical richness: Mackey‑Glass delay, two low‑dimensional thermal‑convection models, and a high‑dimensional shear‑flow model that transitions to turbulence.
• Direct vs. cross‑prediction analysis – The paper distinguishes between predicting the same system used for training (direct) and predicting a related system with different parameters (cross).
• Symmetry advantage quantified – Symmetric reservoirs consistently outperform asymmetric ones for the convection models, especially when the input dimension is lower than the system’s intrinsic degrees of freedom.
• Chaos‑dimensionality boundary – For the strongly chaotic shear‑flow case, topological symmetry has negligible impact, highlighting a regime where reservoir design matters less.

Methodology

1. Reservoir Construction – Each reservoir consists of a fixed, randomly initialized recurrent neural network (RNN) whose internal weights are not trained. The authors generate five topologies:

   • Fully random (no imposed symmetry)
   • Symmetric adjacency matrix with random weights
   • Symmetric adjacency + symmetric weight distribution
   • Sparse random, etc.
     The key is that the graph (who connects to whom) and the weight values can be toggled independently.

2. Training Procedure – Only the linear read‑out layer is trained via ridge regression, a standard RC practice. Input signals from the target dynamical system are fed into the reservoir, and the read‑out learns to map the high‑dimensional reservoir state to the next time step of the target.

3. Benchmark Systems –

   • Mackey‑Glass (delay differential equation) – a classic chaotic time series.
   • Low‑dimensional convection – two models (2‑D and 3‑D) representing thermal rolls.
   • Shear‑flow turbulence – a 3‑D Navier‑Stokes‑based model that exhibits a transition from laminar to turbulent flow.

4. Evaluation Metrics – Prediction horizon (how long the forecast stays within a tolerance), normalized root‑mean‑square error (NRMSE), and a “symmetry gain” factor comparing symmetric vs. asymmetric reservoirs.

5. Direct vs. Cross Prediction – Direct: train and test on the same parameter set; Cross: train on one parameter set and test on another (e.g., different Rayleigh numbers in convection).

Results & Findings

• Symmetric reservoirs excel for low‑dimensional convection – When the input vector (e.g., temperature field sampled at a few points) is smaller than the system’s true degrees of freedom, symmetric connectivity yields up to a 30 % increase in prediction horizon and a 20 % reduction in NRMSE compared with random topologies.
• Mackey‑Glass shows modest sensitivity – Symmetry improves performance but the gain is smaller than for convection, likely because the system’s effective dimensionality is already low.
• Shear‑flow turbulence is indifferent to symmetry – Across all five topologies, prediction horizons and errors are statistically indistinguishable. The authors attribute this to the high-dimensional chaotic attractor that overwhelms any structural advantage.
• Cross‑prediction benefits – Symmetric reservoirs maintain higher accuracy when extrapolating to unseen parameter regimes, suggesting better generalization for systems where the training data sparsely covers the state space.

Practical Implications

• Designing RC for physical simulators – Engineers building surrogate models for fluid dynamics (e.g., HVAC, aerospace) can deliberately enforce symmetric connectivity in the reservoir to squeeze more predictive power out of modestly sized networks.
• Resource‑constrained deployments – Since the reservoir weights stay fixed, imposing symmetry does not increase computational cost; it merely changes the wiring pattern. This is attractive for edge devices that need fast, low‑power forecasts of chaotic processes (e.g., real‑time weather or power‑grid stability).
• Guidance for hyper‑parameter search – Instead of blindly randomizing the reservoir, practitioners can start with a symmetric adjacency matrix and then fine‑tune sparsity or spectral radius, reducing the search space and training time.
• Hybrid modeling pipelines – For high‑dimensional turbulent flows, the study suggests that architectural tweaks (symmetry) won’t help; developers should combine RC with other techniques (e.g., physics‑informed neural networks or reduced‑order models) to handle the intrinsic chaos.

Limitations & Future Work

• Scope of topologies – Only five relatively simple graph families were examined; more exotic structures (e.g., small‑world, scale‑free, or learned adjacency matrices) could reveal additional gains.
• Fixed reservoir size – The experiments keep the reservoir size constant; scaling up or down might interact with symmetry in non‑linear ways.
• Single‑task focus – The study concentrates on short‑term prediction. Extending the analysis to control, classification, or long‑term statistical reconstruction would broaden applicability.
• Physical interpretability – While symmetry improves performance, the paper does not link specific symmetric patterns to physical invariances of the target systems; future work could explore physics‑guided reservoir designs.

Bottom line: If you’re deploying reservoir computing for low‑dimensional chaotic systems—think thermal convection, simple climate models, or sensor‑driven process control—consider wiring your reservoir symmetrically. For high‑dimensional turbulence, the payoff is minimal, and you’ll need to look beyond topology to tame the chaos.

Authors

• Shailendra K. Rathor
• Lina Jaurigue
• Martin Ziegler
• Jörg Schumacher

Paper Information

• arXiv ID: 2511.22059v1
• Categories: physics.flu-dyn, cs.NE, math.DS, nlin.CD
• Published: November 27, 2025
• PDF: Download PDF