Forced Hamiltonian neural networks on manifolds and Lie groups

Develop analogous Generalized Forced Hamiltonian Neural Network and Parametric Generalized Forced Hamiltonian Neural Network architectures on manifolds, in particular on Lie groups, to extend the structure-preserving learning framework to those settings.

Background

The paper develops GFHNNs and PGFHNNs for deterministic, time-dependent, and stochastic forced Hamiltonian systems on the vector space configuration space Q ≃ R^n. Its universal approximation results and numerical constructions are formulated in canonical coordinates on cotangent bundles of such vector spaces.

The authors explicitly identify the development of analogous architectures on manifolds, especially Lie groups, as an open direction. Such an extension would broaden the framework to geometric systems with non-Euclidean configuration spaces and could support applications such as robotics, where Lie-group structure is intrinsic. The cited literature indicates that related variational and geometric neural-network methods on Lie groups provide a basis for this future development, but the corresponding forced Hamiltonian neural-network architectures are not developed in the paper.