hamiltonian neural networks

**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.