Surrogate-Based Reliability

Replacing an expensive limit-state function with an adaptively refined surrogate makes rare-failure probability estimation affordable.

The bottleneck

Estimating a small failure probability by Monte Carlo may need millions of model evaluations, infeasible for expensive simulations. Surrogate-based reliability trains a fast emulator of the performance function, then samples the emulator abundantly. The challenge is ensuring the emulator is accurate exactly where it matters: near the failure boundary.

Learning where it counts

Kronos motion — state estimation

Global accuracy is wasteful; only the sign of the performance function near g = 0 determines failure classification. Active-learning schemes add training points where the surrogate is both uncertain and close to the limit state, refining the boundary with few expensive runs.

Learning functions

EFF (expected feasibility function): prioritizes points near the boundary with high predictive variance
U-function: selects points where the probability of misclassifying the sign is highest
Both drive adaptive Kriging methods such as AK-MCS

AK-MCS in outline

Adaptive Kriging Monte Carlo Simulation generates a large candidate sample, classifies each point with the current Kriging surrogate, adds the most informative candidate to the training set, retrains, and repeats until the failure classification stabilizes. The expensive model is called only for the added points, often a few dozen to a few hundred.

Cautions

The surrogate's own uncertainty must be propagated into the probability estimate; a confidently wrong emulator gives a confidently wrong probability. Very small probabilities may leave too few candidate failure samples, requiring combination with importance sampling or subset simulation. Always validate the final estimate against a modest number of true model runs near the identified boundary.