symplectic neural networks

**Symplectic Neural Networks** are **neural network architectures that preserve the symplectic structure of Hamiltonian dynamics** — ensuring that the learned dynamics conserve energy and phase-space volume, which is critical for accurate long-term prediction of physical systems. **How Symplectic Networks Work** - **Symplectic Structure**: Hamiltonian systems preserve the symplectic 2-form $omega = dp wedge dq$. - **Symplectic Integrators**: Use integration schemes (leapfrog, Störmer-Verlet) that preserve this structure exactly. - **Network Design**: Compose symplectic maps (shear transformations) to build a neural network that is inherently symplectic. - **Separable Hamiltonians**: $H(q,p) = T(p) + V(q)$ structure enables efficient symplectic layers. **Why It Matters** - **Energy Conservation**: Standard neural ODE solvers accumulate energy errors — symplectic networks conserve energy by construction. - **Long-Term Prediction**: Symplectic structure ensures bounded errors over long integration times. - **Physics-Informed**: Embeds fundamental physics (conservation laws) directly into the architecture. **Symplectic Networks** are **physics-preserving neural dynamics** — architectures that maintain the fundamental conservation laws of Hamiltonian mechanics.