https://doi.org/10.1140/epjs/s11734-026-02245-7
Regular Article
Trajectory diversity and output matrix convergence in reservoir computing
1
Lobachevsky State University of Nizhny Novgorod, 23 Gagarin ave, 603950, Nizhny Novgorod, Russia
2
Center for Biomedical Technology, Universidad Politecnica de Madrid, Campus de Montegancedo, Pozuelo de Alarcon, 28223, Madrid, Spain
a
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Received:
23
July
2025
Accepted:
4
March
2026
Published online:
11
March
2026
Abstract
In this work, we investigate the training dynamics of reservoir computing (RC) in the context of trajectory prediction tasks. Specifically, we test the hypothesis that as the number of training trajectories increases, the resulting trained output weight matrices exhibit convergence. This phenomenon is demonstrated through numerical experiments on two representative chaotic systems: the Rössler, the Li–Sprott and the Lorenz-96 models. Our results suggest that the output weight matrices tend to converge toward a common projection from the reservoir states to the system states. This insight highlights the importance of robust generalization in RC, particularly for chaotic systems, where small perturbations can lead to significant divergence in system behavior. The findings offer practical implications for optimizing the training process in RC, especially for real-world applications involving complex dynamical systems.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
