Particular-Integral Latent Reduction

Integrate this at the latent dynamics module of a neural ODE, state-space model, or Hamiltonian world model. Let an encoder produce $z\in\mathbb R^d$, split the latent state into candidate gauge coordinates $q_g\in\mathbb R^k$ and retained coordinates $u\in\mathbb R^{d-k}$, and add a constraint head $f_\theta(z)\in\mathbb R^k$. The vector field is $v_\theta(z)$, implemented by the existing ODE or SSM transition network. Use automatic differentiation to compute the Jacobian-vector product $J_{f_\theta}(z)v_\theta(z)$ without forming the full Jacobian. Pseudocode: sample $z$; compute $f=f_\theta(z)$ and $v=v_\theta(z)$; compute $g=\mathrm{jvp}(f_\theta,z,v)$; predict $A=A_\theta(z)$, preferably diagonal or low-rank; set $r=g-Af$; add $\lambda\|r\|^2$ to the task loss; during rollout initialize or project $z$ to $f(z)=0$ using one Newton or least-squares correction $z\leftarrow z-J_f(z)^\dagger f(z)$; integrate only $u$ if the ablation confirms that the vector field is independent of $q_g$ on the manifold. The paper supplies the invariant-manifold and presymplectic reduction principle; $A_\theta$, the constraint coordinates, and the projection tolerance are learned or estimated empirically. Start with a 2D or 4D latent neural ODE on damped and undamped harmonic oscillators, then test a small MuJoCo trajectory prediction task. Compare an unconstrained latent ODE, a standard energy-regularized model, and this closure model at equal parameter count. Measure long-horizon rollout error, violation $\|f(z_t)\|$, number of function evaluations, and performance after deleting $k$ gauge coordinates. Success is lower constraint drift and rollout error, or at least a 20% reduction in retained latent dimension or function evaluations at equal prediction accuracy.