Nonseparable Symplectic Neural Networks
This work addresses a computational challenge in scientific machine learning for predicting complex systems like fluid dynamics, though it appears incremental as it builds on existing symplectic methods.
The paper tackled the problem of predicting nonseparable Hamiltonian systems, which are common in fluid dynamics and quantum mechanics but rarely addressed in machine learning, by proposing Nonseparable Symplectic Neural Networks (NSSNNs) that embed symplectic priors from limited data, resulting in long-term, accurate predictions for chaotic flows.
Predicting the behaviors of Hamiltonian systems has been drawing increasing attention in scientific machine learning. However, the vast majority of the literature was focused on predicting separable Hamiltonian systems with their kinematic and potential energy terms being explicitly decoupled while building data-driven paradigms to predict nonseparable Hamiltonian systems that are ubiquitous in fluid dynamics and quantum mechanics were rarely explored. The main computational challenge lies in the effective embedding of symplectic priors to describe the inherently coupled evolution of position and momentum, which typically exhibits intricate dynamics. To solve the problem, we propose a novel neural network architecture, Nonseparable Symplectic Neural Networks (NSSNNs), to uncover and embed the symplectic structure of a nonseparable Hamiltonian system from limited observation data. The enabling mechanics of our approach is an augmented symplectic time integrator to decouple the position and momentum energy terms and facilitate their evolution. We demonstrated the efficacy and versatility of our method by predicting a wide range of Hamiltonian systems, both separable and nonseparable, including chaotic vortical flows. We showed the unique computational merits of our approach to yield long-term, accurate, and robust predictions for large-scale Hamiltonian systems by rigorously enforcing symplectomorphism.