On Dominant Manifolds in Reservoir Computing Networks

• URL: http://arxiv.org/abs/2604.05967v1
• Date: Tue, 07 Apr 2026 15:00:30 GMT
• Title: On Dominant Manifolds in Reservoir Computing Networks
• Abstract summary: We study the emergence of low-dimensional dominant manifold in the training of Reservoir Computing networks for temporal forecasting tasks.<n>For a simplified linear and continuous-time reservoir model, we link the dimensionality and structure of the dominant modes directly to the intrinsic dimensionality and information content of the training data.<n>In particular, for training data generated by an autonomous dynamical system, we relate the dominant modes of the trained reservoir to approximations of the Koopman eigenfunctions of the original system.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 simplified linear and continuous-time reservoir model, we link the dimensionality and structure of the dominant modes directly to the intrinsic dimensionality and information content of the training data. In particular, for training data generated by an autonomous dynamical system, we relate the dominant modes of the trained reservoir to approximations of the Koopman eigenfunctions of the original system, illuminating an explicit connection between reservoir computing and the Dynamic Mode Decomposition algorithm. We illustrate the eigenvalue motion that generates the dominant manifolds during training in simulation, and discuss generalization to nonlinear RC via tangent dynamics and differential p-dominance.

Related papers

• Scalar Representations of Neural Network Training Dynamics [1.9583341377413463]
  Training in artificial neural networks can be viewed as a trajectory evolving through a high-dimensional loss landscape.<n>We investigate whether such a scalar embedding provides a meaningful low-dimensional representation of neural network training dynamics.
  arXiv  Detail & Related papers  (2026-06-29T14:40:03Z)
• Disordered Dynamics in High Dimensions: Connections to Random Matrices and Machine Learning [52.26396748560348]
  We provide an overview of high dimensional dynamical systems driven by random matrices.<n>We focus on applications to simple models of learning and generalization in machine learning theory.
  arXiv  Detail & Related papers  (2026-01-03T00:12:32Z)
• Neural Koopman prior for data assimilation [7.875955593012905]
  We use a neural network architecture to embed dynamical systems in latent spaces. We introduce methods that enable to train such a model for long-term continuous reconstruction. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques.
  arXiv  Detail & Related papers  (2023-09-11T09:04:36Z)
• Identifying Equivalent Training Dynamics [3.793387630509845]
  We develop a framework for identifying conjugate and non-conjugate training dynamics. By leveraging advances in Koopman operator theory, we demonstrate that comparing Koopman eigenvalues can correctly identify a known equivalence between online mirror descent and online gradient descent. We then utilize our approach to: (a) identify non-conjugate training dynamics between shallow and wide fully connected neural networks; (b) characterize the early phase of training dynamics in convolutional neural networks; (c) uncover non-conjugate training dynamics in Transformers that do and do not undergo grokking.
  arXiv  Detail & Related papers  (2023-02-17T22:15:20Z)
• Forecasting through deep learning and modal decomposition in two-phase concentric jets [2.362412515574206]
  This work aims to improve fuel chamber injectors' performance in turbofan engines. It requires the development of models that allow real-time prediction and improvement of the fuel/air mixture.
  arXiv  Detail & Related papers  (2022-12-24T12:59:41Z)
• Decomposed Linear Dynamical Systems (dLDS) for learning the latent components of neural dynamics [6.829711787905569]
  We propose a new decomposed dynamical system model that represents complex non-stationary and nonlinear dynamics of time series data. Our model is trained through a dictionary learning procedure, where we leverage recent results in tracking sparse vectors over time. In both continuous-time and discrete-time instructional examples we demonstrate that our model can well approximate the original system.
  arXiv  Detail & Related papers  (2022-06-07T02:25:38Z)
• Linearization and Identification of Multiple-Attractors Dynamical System through Laplacian Eigenmaps [8.161497377142584]
  We propose a Graph-based spectral clustering method that takes advantage of a velocity-augmented kernel to connect data-points belonging to the same dynamics. We prove that there always exist a set of 2-dimensional embedding spaces in which the sub-dynamics are linear, and n-dimensional embedding where they are quasi-linear. We learn a diffeomorphism from the Laplacian embedding space to the original space and show that the Laplacian embedding leads to good reconstruction accuracy and a faster training time.
  arXiv  Detail & Related papers  (2022-02-18T12:43:25Z)
• KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics [84.18625250574853]
  We present KalmanNet, a real-time state estimator that learns from data to carry out Kalman filtering under non-linear dynamics. We numerically demonstrate that KalmanNet overcomes nonlinearities and model mismatch, outperforming classic filtering methods.
  arXiv  Detail & Related papers  (2021-07-21T12:26:46Z)
• Dynamic Mode Decomposition in Adaptive Mesh Refinement and Coarsening Simulations [58.720142291102135]
  Dynamic Mode Decomposition (DMD) is a powerful data-driven method used to extract coherent schemes. This paper proposes a strategy to enable DMD to extract from observations with different mesh topologies and dimensions.
  arXiv  Detail & Related papers  (2021-04-28T22:14:25Z)
• An Ode to an ODE [78.97367880223254]
  We present a new paradigm for Neural ODE algorithms, called ODEtoODE, where time-dependent parameters of the main flow evolve according to a matrix flow on the group O(d) This nested system of two flows provides stability and effectiveness of training and provably solves the gradient vanishing-explosion problem.
  arXiv  Detail & Related papers  (2020-06-19T22:05:19Z)
• Liquid Time-constant Networks [117.57116214802504]
  We introduce a new class of time-continuous recurrent neural network models. Instead of declaring a learning system's dynamics by implicit nonlinearities, we construct networks of linear first-order dynamical systems. These neural networks exhibit stable and bounded behavior, yield superior expressivity within the family of neural ordinary differential equations.
  arXiv  Detail & Related papers  (2020-06-08T09:53:35Z)
• Kernel and Rich Regimes in Overparametrized Models [69.40899443842443]
  We show that gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. We also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
  arXiv  Detail & Related papers  (2020-02-20T15:43:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.