Symplectic Latent Rollouts

1. Integration point: use this as the transition operator in a latent world model or state-space model. An encoder maps observations $x_t$ to $z_t=(q_t,p_t)$, the Hamiltonian MLP takes $(q,p,c)$, and a decoder maps predicted latent states back to observations. Use two or four leapfrog substeps per observation interval; train with reconstruction loss, one-step latent matching, and multi-step rollout loss. 2. Pseudocode: encode $z_0=E_\phi(x_0)$; for each rollout step compute $g_q,g_p=\nabla_qH_\theta,\nabla_pH_\theta$; set $p\leftarrow p-hg_q/2$, $q\leftarrow q+hg_p$, recompute $g_q$ at the new $(q,p)$, then set $p\leftarrow p-hg_q/2$; decode every predicted $z$. Backpropagate through the integrator and minimize $L=L_{recon}+\lambda_1L_{one-step}+\lambda_TL_{rollout}$. 3. Computed from the mechanism: the Hamiltonian gradients and symplectic update are autodifferentiated quantities; estimate energy drift, Jacobian symplectic error, and prediction error empirically. Physical energy need not be known because $H_\theta$ is a learned diagnostic. 4. First cheap experiment: train a 4- or 8-dimensional latent model on simulated pendulum trajectories or a small N-body dataset, comparing an MLP transition, explicit-Euler Hamiltonian transition, and leapfrog Hamiltonian transition at equal parameter count. Sweep step size $h$ and rollout horizon $T$. The prediction is that leapfrog $|\Delta H_k|$ remains bounded and approximately oscillatory, with typical amplitude scaling as $O(h^2)$ for a smooth system, while explicit Euler exhibits secular energy drift and becomes unstable at a smaller step size. Also verify that $\|(D\Phi_h)^TJD\Phi_h-J\|_F$ is near autodiff and floating-point error for the leapfrog map.