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    This study introduces trainable delay neural networks (TrTDNNs) to learn complex dynamics in time delay systems. These networks can simultaneously identify system nonlinearities and time delays from data.

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    Area of Science:

    • Control Systems Engineering
    • Machine Learning
    • Dynamical Systems Theory

    Background:

    • Time delay systems are prevalent in various scientific and engineering fields.
    • Learning the nonlinear dynamics of these systems, especially with unknown delays, remains challenging.
    • Traditional neural networks often struggle to model temporal dependencies inherent in time-delay systems.

    Purpose of the Study:

    • To establish a continuous-time framework connecting time delay systems and time delay neural networks (TDNNs).
    • To introduce a novel TDNN architecture with trainable delays (TrTDNN) for simultaneous learning of dynamics and delays.
    • To develop training algorithms for TrTDNNs to effectively learn from trajectory data.

    Main Methods:

    • Developed the concept of Time Delay Neural Networks with Trainable Delays (TrTDNNs).
    • Constructed training algorithms for simultaneous identification of nonlinearities and time delays.
    • Utilized continuous-time perspective for modeling and learning.

    Main Results:

    • Successfully demonstrated the ability of TrTDNNs to learn nonlinear dynamics from trajectory data.
    • Validated the proposed techniques on autonomous systems using simulation data.
    • Applied and validated the methods on real experimental data for connected automated vehicle (CAV) longitudinal dynamics.

    Conclusions:

    • TrTDNNs offer a powerful approach for modeling and learning complex dynamics in time delay systems.
    • The simultaneous learning of nonlinearities and time delays is feasible and effective.
    • The approach shows promise for applications in autonomous systems and connected vehicles.