Structure-preserving dimensionality reduction for learning Hamiltonian dynamics

Abstract

Structure-preserving data-driven learning algorithms have recently received high attention, e.g., the development of the symplecticity-preserving neural networks SympNets for learning the flow of a Hamiltonian system. The preservation of structural properties by neural networks has been shown to produce qualitatively better long-time predictions. Learning the flow of high-dimensional Hamiltonian dynamics still poses a great challenge due to the increase in neural network model complexity and, thus, the significant increase in training time. In this work, we investigate dimensionality reduction techniques of training datasets of solutions to Hamiltonian dynamics, which can be well modeled in a lower-dimensional subspace. For learning the flow of such Hamiltonian dynamics with symplecticity-preserving neural networks SympNets, we propose dimensionality reduction with the proper symplectic decomposition (PSD). PSD was originally proposed to obtain symplectic reduced-order models of Hamiltonian systems. We demonstrate the proposed purely data-driven approach by learning the nonlinear localized discrete breather solutions in a one-dimensional crystal lattice model. Considering three near-optimal PSD solutions, i.e., cotangent lift, complex SVD, and dimension-reduced nonlinear programming solutions, we find that learning the SPD-reduced Hamiltonian dynamics is not only more computationally efficient compared to learning the whole high-dimensional model, but we can also obtain comparably qualitatively good long-time predictions. Specifically, the cotangent lift and nonlinear programming PSD solutions demonstrate significantly enhanced long-term prediction capabilities, outperforming the approach of learning Hamiltonian dynamics with non-symplectic proper orthogonal decomposition (POD) dimensionality reduction.