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    Area of Science:

    • Computational Neuroscience
    • Machine Learning
    • Control Systems

    Background:

    • Recurrent neural networks (RNNs) are powerful for time-varying zero-finding problems (TVZFPs).
    • Existing RNNs lack a comprehensive design framework for predefined-time convergence.
    • A need exists for advanced RNNs with controllable convergence and improved accuracy.

    Purpose of the Study:

    • To present a comprehensive framework for designing adaptive arbitrarily predefined-time convergent RNNs (A-APTC-RNNs).
    • To develop RNNs with arbitrarily predefined convergence times and enhanced stability.
    • To improve the steady-state residual errors and enable automatic parameter determination.

    Main Methods:

    • Development of a novel piecewise evolution formula for arbitrary convergence time predefinition.
    • Integration of a proportional-integral-derivative (PID) regulatory mechanism for error reduction.
    • Implementation of a novel adaptive parameter initialization scheme for automatic model parameter determination.

    Main Results:

    • The proposed framework generates A-APTC-RNNs with arbitrarily predefined convergence times.
    • A-APTC-RNNs demonstrate lower steady-state residual errors due to the PID mechanism.
    • The adaptive initialization scheme allows A-APTC-RNNs to self-determine model parameters.
    • Theoretical analysis confirms the stability and arbitrarily predefined-time convergence (APTC) capabilities.

    Conclusions:

    • The developed comprehensive framework successfully generates A-APTC-RNNs.
    • These A-APTC-RNNs exhibit superior performance in various applications, including numerical simulations and robotic motion generation.
    • The A-APTC-RNNs represent a significant advancement in solving TVZFPs with enhanced control and accuracy.