Velocity-Inferred Hamiltonian Networks: Symplectic Dynamics from Position-Only Observations

Conventional Hamiltonian Neural Networks (HNNs) require both positions q and momenta p for training, limiting their use when only positional data is available. We show that if the momentum–velocity relation p = M(q) v is invertible, the Hamiltonian H(q, p)—a scalar function that governs a system’s total energy—can be re-parameterized as H(q, v), where M(q) v is a positive-definite mass matrix encoding how momentum depends on position and velocity. Leveraging this, we train a standard HNN solely on position traces, estimate velocities via finite differences, and recover momenta through the learned map. We provide a formal proof of the substitution’s validity and show that it permits momentum inference using only position measurements. Experiments on spring–mass, pendulum, two-body, and chaotic three-body systems show that position-only training achieves stable long-horizon trajectories and near-exact energy preservation, matching full-state baselines. The method thus provides a practical route to data-driven discovery of Hamiltonian structure when momentum sensors are unavailable.