Visible-Time Drift Training

(1) Integration point: replace the ordinary one-step drift loss in a neural SDE, latent-state world model, or continuous-time sequence model. Let the network be $f_\psi(y,t,\theta)$, predicting drift from the projected observation, time, and optional censoring-region parameters. Do not train on intervals whose endpoints are not both visible, because boundary projections must not be interpreted as physical states. (2) Pseudocode: for each trajectory and adjacent pair $(Y_k,\delta_k),(Y_{k+1},\delta_{k+1})$, set $m_k=\delta_k\delta_{k+1}$, $\Delta Y_k=Y_{k+1}-Y_k$, $\widetilde Y_k=(Y_k+Y_{k+1})/2$, and accumulate $m_k\rho(\Delta Y_k-f_\psi(\widetilde Y_k,t_k,\theta_k)\Delta t_k)$. Divide by $\sum_k m_k\Delta t_k$, backpropagate, and optionally add a diffusion covariance head trained on the same mask. Use Huber loss $\rho$ initially for robustness. (3) Computed from the paper: the projection observation rule, visibility mask, and restriction of stochastic estimating equations to visible time. Estimated empirically: $f_\psi$, diffusion covariance, and any correction for unequal visibility rates. If visibility is available only at sample points, use consecutive-visible intervals conservatively; if censoring switches rapidly, subdivide intervals or use midpoint visibility. (4) First experiment: simulate 2D Ornstein–Uhlenbeck and nonlinear double-well diffusions, censor them with moving disks, rectangles, and nonsmooth polytopes, and compare an MLP drift model trained with all projected increments against this visible-time loss. Evaluate drift MSE on uncensored states, rollout error, and stability near boundaries. The expected signal is lower drift bias and more stable rollouts at the same number of observed samples, especially for moving or nonsmooth boundaries.