Hamiltonian Neural Koopman Operator

Jingdong Zhang, Qunxi Zhu, Wei Lin

Abstract:

Recently, physics-informed learning, a class of deep learning framework that incorporates the physics priors and the observational noise-perturbed data into the
neural network models, has shown outstanding performances in learning physical
principles with higher accuracy, faster training speed, and better generalization ability. Here, for the Hamiltonian mechanics and using the Koopman operator theory,
we propose a typical physics-informed learning framework, named as Hamiltonian
Neural Koopman Operator (HNKO) to learn the corresponding Koopman operator
automatically satisfying the conservation laws. We analytically investigate the
dimension of the manifold induced by the orthogonal transformation, and use a
modified auto-encoder to identify the nonlinear coordinate transformation that is
required for approximating the Koopman operator. Taking the Kepler problem
as an example, we demonstrate that the proposed HNKO in robustly learning the
Hamiltonian dynamics outperforms the representative methods developed in the
literature. Our results suggest that feeding the prior knowledge of the underlying
system and the mathematical theory appropriately to the learning framework can
reinforce the capability of the deep learning.