{
 "artifacts": null,
 "category": "dynamics",
 "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.",
 "formulas_latex": [
  "$$r(t)=q_{1-\\alpha}+\\Gamma\\delta(t),\\qquad \\Gamma=\\widehat L+\\widehat L_{\\mathrm{pred}},\\qquad \\delta(t)=\\min_k|t-t_k|.$$",
  "$$\\delta_{\\max}=\\frac{\\varepsilon-q_{1-\\alpha}}{\\Gamma},\\qquad r(t)\\le\\varepsilon\\quad\\text{whenever}\\quad \\delta(t)\\le\\delta_{\\max}.$$",
  "$$\\max_k(t_{k+1}-t_k)\\le\\frac{2(\\varepsilon-q_{1-\\alpha})}{\\Gamma}\\quad\\Longrightarrow\\quad \\sup_{t\\in[0,T^*]}r(t)\\le\\varepsilon.$$"
 ],
 "id": 2835,
 "implementation": "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 \u003e 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.",
 "math_summary": "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)\u003c=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)\u003c=2(epsilon-q_(1-alpha))/Gamma.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "numerical-analysis"
 ],
 "ml_areas": [
  "world-model",
  "ssm",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2608.29559",
  "arxiv_url": "https://arxiv.org/abs/2608.29559",
  "summary_what_math_gives_to_ml": "The paper provides a constructive way to extend split-conformal coverage from randomly sampled trajectory points to an entire continuous-time path by combining a discrete conformal residual with a bound on trajectory regularity. Its key transferable asset is a deterministic gap-inflation rule: uncertainty between observations grows proportionally to the largest time gap and a Lipschitz or derivative bound. This can wrap neural ODEs, state-space models, and world models to produce safety-relevant continuous-time prediction tubes despite sparse or random observations. A second transfer is an adaptive sampling controller that refines solver or sensor time points whenever the conformal tube becomes too wide.",
  "title": "Conformal Prediction Regions for Continuous-Time Trajectories under Random Sampling",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 8,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "speedup",
  "accuracy"
 ],
 "title": "Coverage-Controlled Adaptive Time Sampling",
 "url": "https://synthcore.org/idea/2835/coverage-controlled-adaptive-time-sampling",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}