Casimir invariants in the learned Poisson bracket

Determine whether training the Metriplector architecture results in the emergence of non-trivial Casimir invariants for the learned antisymmetric Poisson tensor J, beyond the energy conservation guaranteed by skew-symmetry of J, thereby identifying conserved quantities of the Hamiltonian flow other than energy.

Background

Metriplector employs a learned K×K antisymmetric Poisson tensor J to implement the Hamiltonian (reversible) channel of the metriplectic dynamics. In classical Hamiltonian systems, Casimir invariants are functions conserved by any Hamiltonian flow induced by a given Poisson structure. Establishing the presence of such invariants in the learned J would deepen understanding of the algebraic structure the architecture acquires during training and could provide additional conservation principles guiding computation.

References

The skew-symmetry of $J$ guarantees energy conservation ($dH/dt = 0$ along the Hamiltonian flow); whether the architecture additionally learns non-trivial Casimir invariants remains an open question for future investigation.

— Metriplector: From Field Theory to Neural Architecture  (2603.29496 - Oprisa et al., 31 Mar 2026) in Section 2.6, Algebraic Structure: Lie Groups and the Poisson Bracket