# Coverage-Controlled Adaptive Time Sampling

- ID: 2835
- Canonical URL: https://synthcore.org/idea/2835/coverage-controlled-adaptive-time-sampling
- API JSON: https://synthcore.org/api/idea/2835.json
- API Markdown: https://synthcore.org/api/idea/2835.md
- Verification status: unverified
- Source: [arXiv:2608.29559](https://arxiv.org/abs/2608.29559)
- Category: dynamics
- Solves: stability, speedup, accuracy
- ML areas: world-model, ssm, inference-speedup
- Math tags: dynamical-systems, control-theory, numerical-analysis
- Ratings: usefulness 7/10; difficulty 5/10; novelty 8/10

## Idea description

Use the conformal regularity inflation law as a controller for observation placement or neural-ODE solver refinement. Sample or evaluate the learned dynamics more densely only where the predicted continuous-time uncertainty exceeds a prescribed safety radius, rather than using a uniform time grid.

## Mathematical statement

Let q_(1-alpha) be the conformal residual quantile and Gamma=L_hat+L_hat_pred the sum of an upper bound on true trajectory regularity and a bound on predictor regularity. The tube radius at time t is r(t)=q_(1-alpha)+Gamma delta(t), where delta(t) is the distance to the nearest anchor time. To enforce r(t)<=epsilon, every time must be within delta_max=(epsilon-q_(1-alpha))/Gamma of an anchor. For an interval with endpoint anchors, the worst nearest-anchor distance is half its length, so a sufficient maximum-gap condition is max_k(t_(k+1)-t_k)<=2(epsilon-q_(1-alpha))/Gamma.

## Key formulas

- $$r(t)=q_{1-\alpha}+\Gamma\delta(t),\qquad \Gamma=\widehat L+\widehat L_{\mathrm{pred}},\qquad \delta(t)=\min_k|t-t_k|.$$
- $$\delta_{\max}=\frac{\varepsilon-q_{1-\alpha}}{\Gamma},\qquad r(t)\le\varepsilon\quad\text{whenever}\quad \delta(t)\le\delta_{\max}.$$
- $$\max_k(t_{k+1}-t_k)\le\frac{2(\varepsilon-q_{1-\alpha})}{\Gamma}\quad\Longrightarrow\quad \sup_{t\in[0,T^*]}r(t)\le\varepsilon.$$

## Implementation notes

Use this as an inference-time controller for neural-ODE integration, latent-dynamics rollout, sensor acquisition, or teacher-forcing time selection. First train a base predictor normally and calibrate q, L_hat, and L_hat_pred as in the conformal tube method. Choose a safety threshold epsilon larger than q. Start with the available irregular observation times or with the endpoints of a rollout interval. For each interval [t_k,t_(k+1)], use the conservative radius q+Gamma(t_(k+1)-t_k)/2. If it exceeds epsilon, insert a midpoint, evaluate or integrate the model there, and repeat. A priority-queue implementation can always split the interval with the largest current bound. Pseudocode is: while max_interval_length > 2*(epsilon-q)/Gamma, split the longest interval; after each split, recompute local bounds if state-dependent slopes are used. The paper supplies the linear gap-inflation law and regularity-based guarantee; neural predictor slopes are estimated from adjacent solver evaluations. The first cheap experiment is a neural ODE pendulum or Lorenz predictor with fixed end-time accuracy, comparing uniform RK evaluation with adaptive splitting under randomly sampled observations. The predicted sharp signature is a refinement threshold h_c=2(epsilon-q)/Gamma: the maximum tube radius crosses epsilon when the largest gap crosses h_c, while adaptive sampling uses fewer evaluations than a uniform grid and preserves dense-grid coverage near the target level.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.