Predicting two-dimensional spatiotemporal chaotic patterns with optimized high-dimensional hybrid reservoir computing

TL;DR

This work tackles predicting high-dimensional spatiotemporal chaos when physics-based models are imperfect by extending hybrid reservoir computing with Local States to a 2D Barkley model on a $80\times80$ grid. It systematically compares input-hybrid, output-hybrid, and full-hybrid configurations against a reservoir-only baseline, showing that all hybrids outperform the pure RC, with the output-hybrid variant providing the best accuracy-CPU trade-off for small KBM errors. The study includes a hyper-parameter analysis and ensemble experiments, revealing how reservoir size, model mismatch, and local-state locality influence predictive performance. Overall, the results establish a scalable approach to forecasting complex spatiotemporal patterns in excitable media and offer practical guidance on configuring hybrid RC under computational constraints.

Abstract

As an alternative approach for predicting complex dynamical systems where physics-based models are no longer reliable, reservoir computing (RC) has gained popularity. The hybrid approach is considered an interesting option for improving the prediction performance of RC. The idea is to combine a knowledge-based model (KBM) to support the fully data-driven RC prediction. There are three types of hybridization for RC, namely full hybrid (FH), input hybrid (IH) and output hybrid (OH), where it was shown that the latter one is superior in terms of the accuracy and the robustness for the prediction of low-dimensional chaotic systems. Here, we extend the formalism to the prediction of spatiotemporal patterns in two dimensions. To overcome the curse of dimensionality for this very high-dimensional case we employ the local states ansatz, where only a few locally adjacent time series are utilized for the RC-based prediction. Using simulation data from the Barkley model describing chaotic electrical wave propagation in cardiac tissue, we outline the formalism of high-dimensional hybrid RC and assess the performance of the different hybridization schemes. We find that all three methods (FH, IH and OH) perform better than reservoir only, where improvements are small when the model is very inaccurate. For small model errors and small reservoirs FH and OH perform nearly equally well and better than IH. Given the smaller CPU needs for OH and especially the better interpretability of it, OH is to be favored. For large reservoirs the performance of OH drops below that of FH and IH. Generally, it maybe advisable to test the three setups for a given application and select the best suited one that optimizes between the counteracting factors of prediction performance and CPU needs.

Figures (13)

• Figure 1: Schematic illustration of the different hybrid methods considered in this study
• Figure 2: $U(t)$ given by the simulation with the Euler method at a randomly chosen instant
• Figure 3: Local States: the training and the prediction are done for the red point taking only the information of the red point itself and the orange points.
• Figure 4: Training and prediction sections in the ensemble experiment. $n_\text{T}=2, n_\text{P}=3$ in this study.
• Figure 5: Prediction by the reservoir-only, $U(t)$: (a)-(c), $V(t)$: (d)-(f), ground truth, prediction and error $e(t)=|\bm{y}_\text{t}(t) - \bm{y}_\text{r}(t)|$, from left to right, where $\bm{y}_\text{t}(t)$ is the ground truth, $\bm{y}_\text{r}(t)$ is the prediction.