Learning momenta for Hamiltonian-based models

There has been increasing interest in methodologies that incorporate physics priors into neural network architectures to enhance their modeling capabilities. Among these methodologies, Hamiltonian neural networks (HNNs) and their variants have gained considerable attention. These architectures explicitly encode Hamiltonian mechanics both in their structure and loss function, resulting in desirable properties and additional modeled information. A crucial limitation of HNNs is their reliance on data on the target system's generalized momenta. This severely hampers their applicability when the analytical form of the generalized momenta is unknown, which frequently is the case in real-world applications. The goal of the present work is to develop a general extension of Hamiltonian-based architectures that addresses this issue by simultaneously learning the system's generalized momenta, Hamiltonian structure, and dynamics from data. The proposed architecture learns the generalized momenta when their analytical form is unknown, and only (potentially non-sequential) data on the state and its derivatives is available. A dissipative mass-spring system, a double pendulum, and a rolling disk under nonholonomic constraints are considered as canonical examples to assess the performance of the proposed architecture.