Topology-Aware Learning of Hamiltonian Structures for Correcting Imperfect Hamiltonian Simulation Models on Contractible Manifolds

Many scientific simulations based on classical mechanics are designed using energy functions but may use imperfect coordinates, inertia, or material parameters. In general observation coordinates, Hamiltonian dynamics can be written with an invertible skew-symmetric matrix field $\Omega(x)$ and an energy $H(x)$. We study simulator correction for Hamiltonian systems when both objects are uncertain. The key linear-algebraic obstruction is that invertible skew-symmetric matrices form two disconnected sets, distinguished by the sign of the Pfaffian. A single continuously trained correction cannot move between the two sets without passing through a singular matrix. We introduce a method, which maintains one neural matrix branch for each Pfaffian sign, shares one Hamiltonian correction between the branches, fits the equation without differentiating through $\Omega^{-1}$, and rejects updates that approach singularity. The same construction extends to contractible manifolds after choosing a smooth state-dependent basis. Our method identifies the correct branch in all ten curved-representation trials and is valid and sign-correct in all $20/20$ mass--spring trials. For heterogeneous wave simulators, the method jointly corrects inertia and potential energy: median rollout error decreases from $0.361$ to $0.0043$ on a one-dimensional rod and from $0.307$ to $0.0308$ on a two-dimensional square-domain discretization. These results separate continuous parameter discrepancy from a discrete error in simulator geometry.