Hamiltonian Neural Networks (HNNs) are neural networks that learn to predict the dynamics of physical systems by learning the Hamiltonian function — instead of directly predicting derivatives, HNNs learn $H(q, p)$ and derive the dynamics from Hamilton's equations, automatically conserving energy.

How HNNs Work

• Network: A neural network $H_ heta(q, p)$ approximates the system's Hamiltonian (total energy).
• Hamilton's Equations: $dot{q} = partial H / partial p$, $dot{p} = -partial H / partial q$ — dynamics derived from the learned $H$.
• Training: Train on observed trajectory data by minimizing the error between predicted and observed derivatives.
• Conservation: Energy $H$ is automatically conserved along the learned trajectories.

Why It Matters

• Physical Inductive Bias: Encodes the Hamiltonian structure — the most fundamental formulation of conservative mechanics.
• Generalization: HNNs generalize better to unseen initial conditions and longer time horizons than standard neural ODEs.
• Data Efficiency: Physical prior reduces the data needed to learn accurate dynamics.

HNNs are learning energy instead of forces — a physics-informed architecture that discovers the Hamiltonian and derives correct, energy-conserving dynamics.