Symplectic Neural Networks Based on Dynamical Systems
<|reference_start|>Symplectic Neural Networks Based on Dynamical Systems: We present and analyze a framework for designing symplectic neural networks (SympNets) based on geometric integrators for Hamiltonian differential equations. The SympNets are universal approximators in the space of Hamiltonian diffeomorphisms, interpretable and have a non-vanishing gradient property. We also give a representation theory for linear systems, meaning the proposed P-SympNets can exactly parameterize any symplectic map corresponding to quadratic Hamiltonians. Extensive numerical tests demonstrate increased expressiveness and accuracy -- often several orders of magnitude better -- for lower training cost over existing architectures. Lastly, we show how to perform symbolic Hamiltonian regression with SympNets for polynomial systems using backward error analysis.<|reference_end|>
@article{tapley2024symplectic,
    title={Symplectic Neural Networks Based on Dynamical Systems},
    author={Benjamin K Tapley},
    journal={arXiv preprint arXiv:2408.09821},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.09821},
    primaryClass={cs.LG cs.CE cs.NA math.NA physics.comp-ph}
}