Symplectic Neural Networks Based on Dynamical Systems

Benjamin K Tapley

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arXiv · 2408.09821

Symplectic Neural Networks Based on Dynamical Systems

Benjamin K Tapley

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Abstract

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.

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Benjamin K Tapley. 2024-08-19. Symplectic Neural Networks Based on Dynamical Systems. https://arxiv.org/abs/2408.09821

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