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    Poisson neural networks (PNNs) learn autonomous system trajectories using structured neural networks. These networks accurately model complex systems like particle motion and the nonlinear Schrödinger equation.

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    Area of Science:

    • Dynamical systems theory
    • Machine learning
    • Computational physics

    Background:

    • Poisson systems are fundamental in classical mechanics.
    • Learning system dynamics from data is a key challenge.
    • Existing methods may struggle with complex, nonlinear trajectories.

    Purpose of the Study:

    • To introduce Poisson neural networks (PNNs) for learning Poisson systems.
    • To extend the Darboux-Lie theorem's framework to autonomous system trajectories.
    • To develop a data-driven approach for modeling complex dynamics.

    Main Methods:

    • Utilizing structured neural networks with incorporated physical priors.
    • Approximating coordinate transformations, extended symplectic maps, and their inverses.
    • Applying PNNs to diverse simulation tasks.

    Main Results:

    • PNNs accurately learn Poisson systems and trajectories from data.
    • Demonstrated effectiveness on challenging problems: particle motion in electromagnetic potential, nonlinear Schrödinger equation, and two-body problem observations.
    • Successful approximation of the three key maps derived from the Darboux-Lie theorem.

    Conclusions:

    • PNNs offer a powerful, data-driven method for modeling complex autonomous systems.
    • The approach extends theoretical results to practical, data-intensive applications.
    • PNNs show significant potential for advancing scientific simulation and prediction.