Deep Neural Networks with Symplectic Preservation Properties

TL;DR

The paper tackles learning the flow map of unknown Hamiltonian systems while strictly preserving the underlying symplectic structure. It introduces SymplectoNet, an invertible neural network composed of symplectic coupling blocks (q-shearing, p-shearing, and symplectic stretching) that guarantee the output is a symplectomorphism, with explicit inverses. By extending the architecture to parameterized families, including time-dependent cases, the approach yields a potential Hamiltonian interpretation $H(q,p,t)$ and enables structure-preserving learning and control. The preliminary experiments on a polar nonlinear mapping demonstrate accurate recovery of symplectic mappings and highlight practical considerations around singularities and domain boundaries, supporting the viability of explicit symplecticity in neural network design for Hamiltonian systems.

Abstract

We propose a deep neural network architecture designed such that its output forms an invertible symplectomorphism of the input. This design draws an analogy to the real-valued non-volume-preserving (real NVP) method used in normalizing flow techniques. Utilizing this neural network type allows for learning tasks on unknown Hamiltonian systems without breaking the inherent symplectic structure of the phase space.

Figures (4)

• Figure 1: A diagram of the transformation (\ref{['NlCG4CGPyQTqtKx5RvwW']})
• Figure 2: A diagram of the composition transformation
• Figure 3: The diagram expression of $\operatorname{pSh}_{G} \circ \operatorname{St}_{\Phi } \circ \operatorname{qSh}_{F}$
• Figure 4: Numerical experiment results of symplectomorphism neural network fitting the symplectomorphism (\ref{['rvSHvBdfv8WquGXG0ixV']}). (a): The result of (\ref{['rvSHvBdfv8WquGXG0ixV']}) with $( q,p) \in [ 0,1] \times [ 0,1]$. Blue dots: true data; Orange stars: predicted results. Note that most of the error comes from data near $q=0$ because there is a singularity there; (b): The loss decay of (a); (c): The result of (\ref{['rvSHvBdfv8WquGXG0ixV']}) with $( q,p) \in [ 1/2,3/2] \times [ 0,3\pi / 2]$. Blue dots: true data; Orange stars: predicted results. Note that most of the error comes from data near $q=0$ because there is a singularity there; (d): The loss decay of (c);