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    Area of Science:

    • Computational neuroscience
    • Machine learning theory
    • Dynamical systems

    Background:

    • Reservoir computing (RC) offers a powerful framework for processing time-series data.
    • Echo state networks (ESNs) and trigonometric state-affine systems are prominent RC architectures.
    • Understanding their theoretical approximation capabilities is crucial for advanced applications.

    Purpose of the Study:

    • To establish the universal approximation properties of three major reservoir computer families.
    • To analyze these properties under L^p-type criteria with stochastic discrete-time semi-infinite inputs.
    • To validate the applicability of these systems in complex machine learning scenarios.

    Main Methods:

    • Mathematical proofs demonstrating universal approximation.
    • Analysis using L^p-type criteria for input and filter properties.
    • Focus on reservoir systems with linear readout maps.

    Main Results:

    • Linear reservoir systems with polynomial or neural network readouts are proven to be universal approximators.
    • Trigonometric state-affine systems and echo state networks with linear readouts also exhibit universal approximation properties.
    • Universality holds without requiring input boundedness or fading memory properties.

    Conclusions:

    • The demonstrated universality confirms the suitability of these reservoir computing models for supervised machine learning.
    • Linear readouts are key for handling high-dimensional data and large datasets effectively.
    • These findings advance the theoretical understanding and practical application of reservoir computing.