Necessary and Sufficient Reservoir Condition for Universal Reservoir Computing

Sugiura, Shuhei; Ariizumi, Ryo; Asai, Toru; Azuma, Shun-ichi

doi:10.3390/math13213440

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Necessary and Sufficient Reservoir Condition for Universal Reservoir Computing

¹

Department of Mechanical and Aerospace Engineering, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan

²

Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Koganei 184-8588, Japan

³

Department of Mechanical Engineering, College of Engineering, Chubu University, Kasugai 487-8501, Japan

⁴

Department of Informatics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(21), 3440; https://doi.org/10.3390/math13213440

Submission received: 18 September 2025 / Revised: 20 October 2025 / Accepted: 27 October 2025 / Published: 28 October 2025

(This article belongs to the Special Issue Machine Learning: Mathematical Foundations and Applications)

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Abstract

We discuss necessary and sufficient conditions for universal approximation using reservoir computing. Reservoir computing is a machine learning method used to train a dynamical system model by tuning only the static part of the model. The universality is the ability of the model to approximate any dynamical system with any precision. In the previous studies, we provided two sufficient conditions for the universality. We employed the universality definition that has been discussed since the earliest studies on reservoir computing. In this present paper, we prove that these two conditions and the universality are equivalent to one another. Using this equivalence, we show that a universal model must have a “pathological” property that can only be achieved or approached by chaotic reservoirs.

Keywords: machine learning; reservoir computing; neural network; nonlinear dynamical system

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MDPI and ACS Style

Sugiura, S.; Ariizumi, R.; Asai, T.; Azuma, S.-i. Necessary and Sufficient Reservoir Condition for Universal Reservoir Computing. Mathematics 2025, 13, 3440. https://doi.org/10.3390/math13213440

AMA Style

Sugiura S, Ariizumi R, Asai T, Azuma S-i. Necessary and Sufficient Reservoir Condition for Universal Reservoir Computing. Mathematics. 2025; 13(21):3440. https://doi.org/10.3390/math13213440

Chicago/Turabian Style

Sugiura, Shuhei, Ryo Ariizumi, Toru Asai, and Shun-ichi Azuma. 2025. "Necessary and Sufficient Reservoir Condition for Universal Reservoir Computing" Mathematics 13, no. 21: 3440. https://doi.org/10.3390/math13213440

APA Style

Sugiura, S., Ariizumi, R., Asai, T., & Azuma, S.-i. (2025). Necessary and Sufficient Reservoir Condition for Universal Reservoir Computing. Mathematics, 13(21), 3440. https://doi.org/10.3390/math13213440

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Necessary and Sufficient Reservoir Condition for Universal Reservoir Computing

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