Understanding how training shapes the geometry of recurrent network dynamics is a central problem in time-series modeling. We study the emergence of low-dimensional dominant manifolds in the training of Reservoir Computing (RC) networks for temporal forecasting tasks. For a general linear continuous-time reservoir in the infinite-data limit, we show that the training data generate an invariant subspace of the trained reservoir, whose dimension equals the number of dominant modes. We then specialize to a simplified diagonal linear reservoir, where we link the dominant eigenvalues and eigenvectors to the spectrum of a backward Dynamic Mode Decomposition matrix, which yields a finite-dimensional approximation of the backward-time Koopman operator of the system generating the training data. We illustrate the emergence of these dominant modes during training in simulation, and discuss how the analysis may be extended to nonlinear RC via tangent dynamics and differential p-dominance.
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China（电子科技大学基础与前沿科学研究所）；Potsdam Institute for Climate Impact Research（波茨坦气候影响研究所）；Department of Physics, Humboldt University Berlin（柏林洪堡大学物理系）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究所）；School of Physics, University of Electronic Science and Technology of China（电子科技大学物理学院）；Key Laboratory of Quantum Physics and Photonic Quantum Information, Ministry of Education, University of Electronic Science and Technology of China（电子科技大学教育部量子物理与光子量子信息重点实验室）；Non-classical Information Science Basic Discipline Research Center of Sichuan Province, University of Electronic Science and Technology of China（电子科技大学四川省非经典信息科学基础学科研究中心）
Reservoir Computing meets Recurrent Kernels and Structured Transforms
Ecole Normale Supérieure（巴黎高等师范学院）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université de Paris（巴黎大学）；Collège de France（法兰西公学院）；LightOn（LightOn公司）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
School of Electrical, Computer, and Energy Engineering, Arizona State University（亚利桑那州立大学电气、计算机与能源工程学院）；Department of Electrical and Computer Engineering, Colorado State University（科罗拉多州立大学电气与计算机工程系）；DEVCOM Army Research Office（DEVCOM陆军研究办公室）；Department of Physics, Arizona State University（亚利桑那州立大学物理系）
Discrete-time signatures and randomness in reservoir computing
University of Vienna（维也纳大学）；Ludwig-Maximilians-Universität München（路德维希-马克西米利安慕尼黑大学）；Universität Konstanz（康斯坦茨大学）；Universität Sankt Gallen（圣加仑大学）；Centre National de la Recherche Scientifique (CNRS)（法国国家科学研究中心）；ETH Zürich（苏黎世联邦理工学院）
Wyant College of Optical Sciences, University of Arizona（亚利桑那大学怀恩特光学科学学院）；School of Electrical Engineering and Telecommunications, University of New South Wales（新南威尔士大学电气工程与电信学院）
Insect-Wing Structured Microfluidic System for Reservoir Computing
School of Computing, University of Nebraska-Lincoln（计算机学院，内布拉斯加大学林肯分校）；Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln（机械与材料工程学院，内布拉斯加大学林肯分校）
Harnessing Synthetic Active Particles for Physical Reservoir Computing
Peter Debye Institute for Soft Matter Physics, Leipzig University（莱比锡大学彼得·德拜软物质物理研究所）；Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI) Dresden/Leipzig（德累斯顿/莱比锡可扩展数据分析与人工智能中心（ScaDS.AI））
ChemReservoir -- An Open-Source Framework for Chemically-Inspired Reservoir Computing
Department of Mathematics and Computer Science（数学与计算机科学系）；University of Southern Denmark（南部丹麦大学）；Algorithmic Cheminformatics（算法化学信息学）；Faculty of Technology, Bielefeld University（技术学院，比勒菲尔德大学）
Auto-Encoded Reservoir Computing for Turbulence Learning
Delft University of Technology（代尔夫特理工大学）；Technical University of Munich（慕尼黑工业大学）；University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）；Imperial College London（伦敦帝国学院）
Reservoir Computing Approach for Gray Images Segmentation
Institute of Information；Communication Technologies Bulgarian Academy of Sciences Sofia, Bulgaria ORCID 0000-0002-0447-9667；Communication Technologies Bulgarian Academy of Sciences, Sofia, Bulgaria
