Geometry-Consistent Latent Particle Rollouts

1. Integration point: implement this inside a latent state-space model or neural ODE used for partially observed rollout, planning, or filtering. The neural dynamics network outputs $f_\theta(x_k)$ and optionally a diagonal diffusion $G_\theta(x_k)$; a decoder outputs $h_\phi(x_k)$ and its Jacobian with respect to the latent state. Do not modify the training target initially: use the projection only for inference-time particle proposals. 2. Pseudocode: for every particle, compute $H=J_xh_\phi(x_k)$, estimate $\dot y=(y_{k+1}-y_k)/\Delta t$ or set $\dot y=0$ for static observations, compute $f_{gc}=f_\theta+QH^\top(HQH^\top+\tau R)^{-1}(\dot y-Hf_\theta)$, sample $x_{k+1}$ from the Gaussian proposal, evaluate the nominal transition density, proposal density, and observation likelihood, then update the log weight with the displayed importance ratio. Resample only when ESS $=(\sum_iw_i)^2/\sum_iw_i^2$ falls below $0.5N$. 3. Computed from the mechanism: the projection, co-state, proposal density, and importance correction. Estimated empirically: Jacobians by autodiff, $Q$ from the learned diffusion or a diagonal tunable matrix, and $\dot y$ by finite differences. 4. First cheap experiment: use a 4D Lorenz-63 or Lorenz-96 system with only two observed coordinates, train a small MLP latent dynamics model on nominal trajectories, and test filtering under injected drift perturbations. Compare bootstrap PF, projected PF without correction, and geometry-consistent PF with correction at equal particle count. The key falsifiable signature is that RMSE and ESS degradation under perturbation correlate with $\|P\delta f\|$, not $\|\delta f\|$: perturbations parallel to the rows of $H$ should be substantially suppressed, while tangent-space perturbations should remain. Across perturbation directions, regress squared filtering error against both quantities; the $\|P\delta f\|^2$ model should explain more variance. Also verify $\|Hf_{gc}-\dot y\|$ decreases approximately monotonically as $\tau$ decreases until numerical conditioning fails, and that corrected and unprojected filters have statistically indistinguishable posterior means when proposal variance is sufficiently large.