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. We propose several SympNets that are universal approximators in the space of Hamiltonian diffeomorphisms, the best of which are referred to as P-SympNets, which have a number of interesting properties including a representation theory for linear systems, meaning they can exactly parameterize any symplectic map corresponding to quadratic Hamiltonians. Our models are able to achieve better test loss using fewer parameters for many symplectic data sets than existing one, including MSE test set losses at nearly machine-precision for polynomial Hamiltonian data sets with small time-steps. We also show how to extend the framework to derive conformal-SympNets, which account for linear dissipation in the system. Furthermore, we show how to perform symbolic Hamiltonian regression for polynomial systems using backward error analysis to accurately identify the true Hamiltonian.