Inference of Hamiltonian and Lie-Poisson structures from vector field samples
Abstract
We present a method for the data-driven inference of Hamiltonian and Lie-Poisson dynamical systems from vector-field samples, in which the Hamiltonian and the underlying Poisson tensor are estimated jointly rather than assuming the structure is known. The Hamiltonian is represented as a linear combination of random feature maps with fixed random internal weights, so its estimation is a linear regression problem for any given Poisson structure. We treat three cases of increasing generality: a known Poisson tensor, an unknown constant invertible (symplectic) structure matrix, and Lie-Poisson and affine Poisson structures, with structure matrix linear in the state. In the constant case, the Hamiltonian and structure matrix are recovered jointly from an eigenvalue problem that avoids the trivial zero solution due to the scaling symmetry of Hamiltonian systems. For Lie-Poisson and affine Poisson systems we alternate between an initial Hamiltonian estimate from a generalized eigenvalue problem for an approximate first integral, and a constrained nonlinear least-squares fit of the structure constants that enforces the Jacobi identity and, for affine structures, the cocycle condition, via continuation in a penalty parameter. Orthogonal matching pursuit selects a compact, well-conditioned subset of random features. Because the learned Hamiltonian is a sum of single-feature terms, the model admits an explicit Poisson/symplectic splitting integrator, which we use to validate the learned systems via long-time Poincare sections. We demonstrate the approach on the four- and five-dimensional Lorenz-86 model and a six-dimensional Kirchhoff rigid-body system, recovering Hamiltonians, structure constants and Casimir invariants to high accuracy, and showing the inferred Poincare sections converge to the correct invariant-tori topology as the number of random features increases.
Related
- Butcher series for Hamiltonian Poisson integrators through symplectic groupoids
- Equivariant Poisson structures on moment-angle manifolds and their quotients
- Extending structures for the linear framework of Universal Algebra with applications to transposed Poisson algebras
- An outline of shifted Poisson structures and deformation quantisation in derived differential geometry
- Affinization of algebraic structures: Poisson algebras
BibTeX
@misc{frank2026inference,
title = {{Inference of Hamiltonian and Lie-Poisson structures from vector field samples}},
author = {Jason E. Frank and Georg A. Gottwald},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04350},
eprint = {2610.04350},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04350},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Inference of Hamiltonian and Lie-Poisson structures from vector field samples
AU - Jason E. Frank
AU - Georg A. Gottwald
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04350
UR - https://arxiv.org/abs/2610.04350
AB - We present a method for the data-driven inference of Hamiltonian and Lie-Poisson dynamical systems from vector-field samples, in which the Hamiltonian and the underlying Poisson tensor are estimated jointly rather than assuming the structure is known. The Hamiltonian is represented as a linear combination of random feature maps with fixed random internal weights, so its estimation is a linear regression problem for any given Poisson structure. We treat three cases of increasing generality: a known Poisson tensor, an unknown constant invertible (symplectic) structure matrix, and Lie-Poisson and affine Poisson structures, with structure matrix linear in the state. In the constant case, the Hamiltonian and structure matrix are recovered jointly from an eigenvalue problem that avoids the trivial zero solution due to the scaling symmetry of Hamiltonian systems. For Lie-Poisson and affine Poisson systems we alternate between an initial Hamiltonian estimate from a generalized eigenvalue problem for an approximate first integral, and a constrained nonlinear least-squares fit of the structure constants that enforces the Jacobi identity and, for affine structures, the cocycle condition, via continuation in a penalty parameter. Orthogonal matching pursuit selects a compact, well-conditioned subset of random features. Because the learned Hamiltonian is a sum of single-feature terms, the model admits an explicit Poisson/symplectic splitting integrator, which we use to validate the learned systems via long-time Poincare sections. We demonstrate the approach on the four- and five-dimensional Lorenz-86 model and a six-dimensional Kirchhoff rigid-body system, recovering Hamiltonians, structure constants and Casimir invariants to high accuracy, and showing the inferred Poincare sections converge to the correct invariant-tori topology as the number of random features increases.
ER -