Adaptive Sampling

Adaptive sampling adds training points sequentially where a surrogate is weakest, balancing exploration of unseen regions against refinement of complex ones.

Beyond one-shot designs

A fixed space-filling design spends its budget before it knows where the response is difficult. Adaptive sampling instead grows the training set in stages: build a surrogate, identify where it needs help, evaluate the expensive model there, and refit. Each stage uses everything learned so far to place the next points.

Exploration versus exploitation

Kronos motion — uncertainty

Two forces compete. Exploration seeks under-sampled regions to guard against surprises far from existing points. Exploitation refines regions where the surrogate already sees strong nonlinearity or high error. A good adaptive criterion blends both, often by combining a distance-based space-filling term with a local error or variance term.

Common criteria

Predictive variance - refine where a Gaussian process is uncertain
Cross-validation error - refine where leave-one-out predictions fail
Gradient or curvature - refine where the response changes fastest
Expected improvement - refine near suspected optima for optimization tasks
Maximin distance - explore the largest empty gaps

Contour and boundary refinement

Many engineering questions concern a threshold: where does a stress exceed a limit, or a plasma cross a stability boundary. Contour-oriented adaptive sampling concentrates points along the level set of interest, resolving that boundary sharply while ignoring regions far from it, which is far more efficient than uniform refinement.

Stopping rules

Adaptive sampling stops when a convergence metric plateaus - cross-validation error, maximum predictive variance, or the change in a quantity of interest between iterations falls below a tolerance - or when the evaluation budget is exhausted. A clear stopping rule prevents both under-training and wasted runs.

Practical value

Adaptive sampling is the engine behind efficient global optimization and reliability analysis with surrogates. In Kronos design work it places full-fidelity runs of the machines where operating limits and nonlinear response make them most valuable, so a limited simulation budget yields a surrogate that is accurate exactly where decisions are made.