An Echo State Network Imparts a Curve Fitting

Manjunath, G.

doi:10.1109/tnnls.2021.3099091

Cited by 8 publications

(3 citation statements)

References 29 publications

(56 reference statements)

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“…It can be established that the USP implies the existence of a function h in a more straightforward way (e.g. [19]), but in lemma 5 we show that the existence of a universal semi-conjugacy is equivalent to the USP. We also observe that h( ← u k ) = x k , where x k is the value of the solution {x n } at the kth instant for any input ū whose left-infinite segment is ← u k .…”

Section: Introductionmentioning

confidence: 80%

Embedding information onto a dynamical system

Manjunath

2022

Nonlinearity

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The celebrated Takens’ embedding theorem concerns embedding an attractor of a dynamical system in a Euclidean space of appropriate dimension through a generic delay-observation map. The embedding also establishes a topological conjugacy. In this paper, we show how an arbitrary sequence can be mapped into another space as an attractive solution of a nonautonomous dynamical system. Such mapping also entails a topological conjugacy and an embedding between the sequence and the attractive solution spaces. This result is not a generalisation of Takens embedding theorem but helps us understand what exactly is required by discrete-time state space models widely used in applications to embed an external stimulus onto its solution space. Our results settle another basic problem concerning the perturbation of an autonomous dynamical system. We describe what exactly happens to the dynamics when exogenous noise perturbs continuously a local irreducible attracting set (such as a stable fixed point) of a discrete-time autonomous dynamical system.

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Section: Introductionmentioning

confidence: 80%

Embedding information onto a dynamical system

Manjunath

2022

Nonlinearity

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“…Simply put, the first category focuses on the theoretical aspects of RC, aiming to drive efficient model design and ensure reliable RC applications. This includes works like reservoir memory machines [319], consistency capacity [320] and curve fitting abilities [321] analysis. The second category delves into the exploration of novel model designs and applications of RC, with an aim to enhance computational performance and efficiency in tasks related to pattern recognition.…”

Section: Physical Rc and Extremely Efficient Hardwarementioning

confidence: 99%

A Survey on Reservoir Computing and its Interdisciplinary Applications Beyond Traditional Machine Learning

Zhang

Vargas

2023

IEEE Access

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Reservoir computing (RC), first applied to temporal signal processing, is a recurrent neural network in which neurons are randomly connected. Once initialized, the connection strengths remain unchanged. Such a simple structure turns RC into a non-linear dynamical system that maps lowdimensional inputs into a high-dimensional space. The model's rich dynamics, linear separability, and memory capacity then enable a simple linear readout to generate adequate responses for various applications. RC spans areas far beyond machine learning, since it has been shown that the complex dynamics can be realized in various physical hardware implementations and biological devices. This yields greater flexibility and shorter computation time. Moreover, the neuronal responses triggered by the model's dynamics shed light on understanding brain mechanisms that also exploit similar dynamical processes. While the literature on RC is vast and fragmented, here we conduct a unified review of RC's recent developments from machine learning to physics, biology, and neuroscience. We first review the early RC models, and then survey the state-of-the-art models and their applications. We further introduce studies on modeling the brain's mechanisms by RC. Finally, we offer new perspectives on RC development, including reservoir design, coding frameworks unification, physical RC implementations, and interaction between RC, cognitive neuroscience and evolution.

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“…Although there is no theoretical framework that guarantees the existence of a post-processing function, it is approximated typically by a linear regression. These methods often give good short-term predictions but and fail to learn the attractor for many chaotic systems [12,13,14,15]. .…”

mentioning

confidence: 99%

Universal set of Observables for the Koopman Operator through Causal Embedding

Manjunath

Clercq

2021

Preprint

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Obtaining repeated measurements from physical and natural systems for building a more informative dynamical model of such systems is engraved in modern science. Results in reconstructing equivalent chaotic dynamical systems through delay coordinate mappings, Koopman operator based data-driven approach and reservoir computing methods have shown the possibility of finding model equations on a new phase space that is relatable to the dynamical system generating the data. Recently, rigorous results that point to reducing the functional complexity of the map that describes the dynamics in the new phase have made the Koopman operator based approach very attractive for data-driven modeling. However, choosing a set of nonlinear observable functions that can work for different data sets is an open challenge. We use driven dynamical systems comparable to that in reservoir computing with the causal embedding property to obtain the right set of observables through which the dynamics in the new space is made equivalent or topologically conjugate to the original system. Deep learning methods are used to learn a map that emerges as a consequence of the topological conjugacy. Besides stability, amenability for hardware implementations, causal embedding based models provide long-term consistency even for maps that have failed under previously reported data-driven or machine learning methods.

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An Echo State Network Imparts a Curve Fitting

Cited by 8 publications

References 29 publications

Embedding information onto a dynamical system

Embedding information onto a dynamical system

A Survey on Reservoir Computing and its Interdisciplinary Applications Beyond Traditional Machine Learning

Universal set of Observables for the Koopman Operator through Causal Embedding

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