Designing Poisson Integrators Through Machine Learning

The paper states (via Theorem 9 and Proposition 11) that Lagrangian bisections in a symplectic groupoid induce Poisson diffeomorphisms and that a generalized Hamilton–Jacobi solution L ⊂ R^2×G with (p_t + H∘sou)|_L constant recovers the Hamiltonian flow φ_t^H up to an initial Poisson diffeomorphism; if the initial slice induces the identity, one recovers the exact flow. These statements appear explicitly (local HJ formulation and Proposition 11) and the ML loss (Eq. (3)) is given to approximate solutions, ensuring Poisson structure whenever time slices are bisections . The model’s solution supplies a detailed proof sketch using standard symplectic-groupoid facts (sou Poisson, tar anti-Poisson, dual-pair property), which the paper cites but does not prove . Thus, the paper’s claims and the model’s argument align; the model gives a complementary proof not fully written in the paper. Minor caveat: both rely on an implicit bisection/nondegeneracy hypothesis for time slices of L, which the paper mentions contextually but does not spell out as an explicit assumption.