Direct Poisson neural networks: learning non-symplectic mechanical systems

Abstract In this paper, we present neural networks learning mechanical systems that are both symplectic
(for instance particle mechanics) and non-symplectic (for instance rotating rigid body). Mechanical
systems have Hamiltonian evolution, which consists of two building blocks: a Poisson bracket and an
energy functional. We feed a set of snapshots of a Hamiltonian system to our neural network models
which then find both the two building blocks. In particular, the models distinguish between symplectic
systems (with non-degenerate Poisson brackets) and non-symplectic systems (degenerate brackets).
In contrast with earlier works, our approach does not assume any further a priori information about
the dynamics except its Hamiltonianity, and it returns Poisson brackets that satisfy Jacobi identity.
Finally, the models indicate whether a system of equations is Hamiltonian or not.