Symplectic Neural Networks Based on Dynamical Systems
This work provides a novel framework for symplectic neural networks, addressing a domain-specific need in computational physics and dynamical systems.
The authors tackled the problem of designing neural networks that preserve symplectic structure in Hamiltonian systems, resulting in SympNets that achieve orders of magnitude better accuracy with lower training costs compared to existing architectures.
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.