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In addition, based on the theoretical results concerning infinite-dimensional dynamical systems, we demonstrate that this reservoir should have some desirable properties called the echo state property and the separation property. By considering the projection of the system onto the inertial manifold, we describe the behavior of the reservoir as a system of finite-dimensional ordinary differential equations called an inertial form. Next, we present a finite-dimensional model that approximates the inertial form. Finally, we propose a temporally discretized model, which is more suitable for computer computation. We give the theoretical proofs of our theorems herein.<\/jats:p>","DOI":"10.1137\/20m134527x","type":"journal-article","created":{"date-parts":[[2021,7,22]],"date-time":"2021-07-22T10:53:40Z","timestamp":1626951220000},"page":"1320-1347","source":"Crossref","is-referenced-by-count":0,"title":["Reservoir Computing with an Inertial Form"],"prefix":"10.1137","volume":"20","author":[{"given":"Hirotada","family":"Honda","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,7,22]]},"reference":[{"key":"atypb2","unstructured":"R. Aiken,\n                      Stiff Computation\n                      , 1st ed., Oxford Universiy Press, New York, 1985."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/BF00250586"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1109\/72.279181"},{"key":"atypb5","unstructured":"Y. Bengio and P. Frasconi,\n                      Credit assignment through time: Alternatives to backpropagation\n                      , Advances in Neural Information Processing Systems 6, J. D. Cowan, G. Tesauro, and J. Alspector, eds., Morgan Kaufmann, San Mateo, CA, pp. 75-82."},{"key":"atypb6","first-page":"154","volume":"279","author":"Berezanskya L.","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160230107"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"P. Brockwell and R. A. Davis,\n                      Introduction to Time Series and Forecasting\n                      , 3rd ed., Springer, Cham, Switzerland, 2016.","DOI":"10.1007\/978-3-319-29854-2"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1109\/TSMC.1983.6313075"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1973-0313873-1"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.2307\/2373376"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761448"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1988220100931"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(88)90110-6"},{"key":"atypb15","first-page":"1","author":"Gibbons T.","year":"2010","journal-title":"NJ"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"P. Grassberger and I. Procaccia,\n                      Measuring the strangeness of strange attractors\n                      , Phys. 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Morton,\n                      Difference Methods for Initial-Value Problems\n                      , 2nd ed., Interscience Publishers, New York, 1967."},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1002\/wcs.1"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.51.980"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2008.03.038"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1137\/080721340"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2019.03.005"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0197704"},{"key":"atypb46","doi-asserted-by":"crossref","unstructured":"T. Yamane, H. Numata, J. B. H\u00e9roux, N. Kanazawa, S. Takeda, G. Tanaka, R. Nakane, A. Hirose, and D. 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