Adaptive Sampling and Active Learning

Active learning chooses the next expensive evaluation where it most improves the surrogate, building accurate models from few runs.

Sampling where it helps

A fixed space-filling design spends runs uniformly, but a response is often smooth in some regions and complex in others. Adaptive sampling, or active learning, uses the current surrogate to decide where the next expensive evaluation will most reduce uncertainty or improve a target quantity, then retrains and repeats.

Acquisition criteria

Kronos motion — active learning

Maximum predictive variance: reduce overall emulator uncertainty (exploration)
Integrated variance reduction: minimize average posterior variance
Expected improvement: for optimization, balance exploring and exploiting
Boundary-focused (U-function, EFF): refine a failure surface for reliability

The loop

Start from a small initial design, fit the surrogate, evaluate the acquisition function over candidate points, run the true model at the selected point, add it to the training set, and iterate until an accuracy or budget criterion is met. Batch variants select several points at once for parallel evaluation.

Goal-oriented adaptation

The acquisition function should match the goal. Reducing global error, finding an optimum, and estimating a rare-failure probability call for different criteria; using a global-accuracy criterion to estimate a tail probability wastes runs in irrelevant regions. Aligning acquisition with purpose is the main lever on efficiency.

Cautions

Adaptive sampling relies on the surrogate's uncertainty estimate to guide itself, so a miscalibrated surrogate can steer sampling away from important regions and reinforce its own blind spots. Seeding with an adequate space-filling design, occasionally adding exploratory points, and validating against held-out truth guard against this self-confirming failure.