Active-learning methods and literature#

Use active learning when true limit-state evaluations are expensive enough to justify fitting and sequentially enriching a surrogate. Install the al extra, begin with Active learning for reliability analysis, and reserve true-model evaluations for independent validation. A final probability interval is conditional on the frozen surrogate and does not measure its approximation bias.

The workflow combines a surrogate, a reliability estimator, a learning function and a stopping criterion. Choose these explicitly using the table below and the composition example. See Troubleshooting reliability analyses when a run stops at its budget.

Literature basis#

Two complementary reviews guide PySTRA’s coverage:

[TeixeiraNogalOConnor2021] surveys adaptive response surfaces, polynomial chaos, support vector machines and Kriging, including the choices of experimental design, enrichment and stopping. Its journal year is 2021; the article was available online in 2020.
[Moustapha2022] organizes active-learning reliability into four components: surrogate model, reliability estimator, learning function and stopping criterion. Its benchmark compares 39 strategies on 20 problems. The findings are conditional on those methods/problems, rather than a universal ranking. The journal year is 2022; its preprint appeared in 2021.

The second review’s Section 5 and Table 2 provide a useful reason to prioritize surrogate-assisted subset simulation and PC-Kriging. They also distinguish learning-point selection from stopping based on the probability/reliability estimate. Simply implementing a learning function does not reproduce the complete algorithm that originally introduced it.

PySTRA targets a carefully validated implementation of established and selected modern structural reliability methods, with practical code calibration and structural assessment tools.

Choose the components for your analysis#

Component	Implemented	Limitations
Surrogates	Kriging; adaptive sparse Hermite PCE with selected-support bootstrap; explicit dense OLS option; sequential PC-Kriging	Adaptive response-surface workflows; SVM classification
Reliability estimation	Independent final MC; replicated subset simulation with adaptive enrichment and independent final sampling; active Gaussian-mixture importance sampling with explicit centers	Automatic proposal adaptation and true-model correction workflows
Learning functions	Explicit scalar/ensemble selection interfaces; U, EFF and FBR voting	Clustered batch selection
Stopping	Explicit policies for learning thresholds, beta bands, beta stability and combined criteria; separate final sampling precision; bootstrap probability-range stopping; weighted sampling diagnostics; explicit budget/exhaustion status	Bounds accounting for model-selection bias

Standalone FORM, SORM, crude Monte Carlo, importance sampling, line sampling and subset simulation already exist. Having these solvers does not mean that active learning can compose every standalone solver. ActiveLearning now composes a replicated subset estimator through EnrichmentEstimator. The existing standalone subset solver remains a comparison baseline, with its original documented CoV limitation. ImportanceSamplingEstimator now composes explicit Gaussian-mixture proposals with active learning and independent weighted final estimation.

The fixed-pool Kriging/U loop is an AK-MCS-style implementation. EFF is provided as a learning function; this is not a complete EGRA reproduction. Likewise, the PCE implementation reproduces specified UQLab sparse-selection routines and fast-bootstrap behavior, with documented numerical differences, but uses PySTRA’s single-point enrichment and stopping contract. Dense quadratic PCE can span quadratic response surfaces in normal coordinates, but does not supply a classical adaptive response-surface reliability algorithm.

Composing the implemented methods#

The public import path remains pystra.active_learning. Internally, surrogates, learning functions, estimation, stopping and results have separate modules. For example, with a PySTRA model and limit_state:

from pystra.active_learning import (
    ActiveLearning, KrigingSurrogate, MonteCarloEstimator, UFunction,
    AllCriteria, LearningThreshold, BetaBounds, BetaStability,
)

analysis = ActiveLearning(
    model,
    limit_state,
    surrogate=KrigingSurrogate(seed=7),
    estimator=MonteCarloEstimator(n_samples=100_000),
    learning_function=UFunction(threshold=2),
    stopping_criterion=AllCriteria(criteria=(
        LearningThreshold(),
        BetaBounds(consecutive=2),
        BetaStability(consecutive=2),
    )),
    rng=7,
)
result = analysis.run()

Surrogate predicts mean/spread from row-wise independent standard normal points. LearningFunction.select chooses an available candidate and returns a LearningDecision. StoppingCriterion inspects an immutable fit history and separately accepts or rejects final sampling precision. Its policies are stateless across runs. ReliabilityEstimator.estimate receives a batched surrogate predictor and a run-owned random generator. It returns a ReliabilityEstimate identifying the sampling method, dependence and uncertainty calculation. It cannot call the true limit state through this interface. Weighted or correlated sampling must provide its own uncertainty calculation; the runner does not recompute an IID binomial CoV.

The concise settings surrogate="kriging", learning_function="u", n_estimation and target_cov remain available for the default workflow. An explicit component cannot be combined with its shortcut settings: configure sample size on the estimator, threshold on the learning function and sampling precision on the stopping policy. This avoids silently ignored options.

result.history records the exploratory probability and the probabilities obtained by classifying mean + 2 std and mean - 2 std. The fixed MC workflow uses sample proportions; adaptive estimators supply their own measure-correct probabilities. Transforming these endpoints gives an ascending beta_band. BetaBounds and BetaStability follow Eqs. (2) and (3) of [Moustapha2022], including default three-consecutive-test requirements. Setting both to two consecutive tests reproduces the review’s combined stopping rule; the example above adds the learning threshold as a third requirement. At beta zero or unresolved infinite endpoints, the relevant beta test cannot pass. For negative beta the relative denominator uses its absolute value.

These are surrogate sensitivity diagnostics, not confidence intervals for the true failure probability. Bootstrap spread in particular need not be Gaussian and can miss common model bias. A stable but inaccurate surrogate can satisfy a beta-stability test. result.estimate keeps final sampling diagnostics separate from these bands, and every default policy still requires adequate final sampling precision before reporting convergence.

Active Kriging with subset simulation#

Use estimator=SubsetSimulationEstimator() to resample nested conditional populations after every surrogate fit, using the nested-event construction of [AuBeck2001]. Its default settings are 2000 samples per level, intermediate probability 0.1, four independent replications, pCN proposal scale 0.5 and at most 12 levels. These are practical starting settings, not the review’s much larger benchmark configuration. Set n_samples and n_replications on the estimator to improve sampling precision; n_candidates only caps the unique points passed to learning-point selection. The default initial design increases to max(30, 5*n_variables) for adaptive estimators. A larger design helps establish disconnected failure surfaces.

Exploration uses mean - 2 std to build the subsets. Until the terminal level, every conditioning threshold is positive. Failure under mean + 2 std, mean and mean - 2 std is therefore nested inside the same conditioning event. The three probabilities use the same product of intermediate conditional probabilities and the same final conditional population. This preserves their ordering without treating pooled conditional samples as draws from the prior. All generated level states are eligible for the enrichment pool, with duplicate states removed and uniform subsampling if the pool cap is exceeded. Previously observed points cannot be evaluated again. This is a documented optimistic-band variant of the modular framework in [Moustapha2022], not a reproduction of all its benchmark settings or of Bayesian subset simulation.

The Gaussian-preserving pCN proposal [CotterEtAl2013] is reversible for the independent standard normal prior. Acceptance only tests membership in the current subset. Initial seeds are included in each chain, and ties in an adaptive threshold retain their actual conditional fraction. Nondecreasing positive thresholds terminate with stalled; exceeding the level budget returns max_levels. Incomplete exploration cannot satisfy the stopping policy, but the runner can continue adding observations to improve the surrogate. Independent final-estimator failure produces estimation_failed when surrogate stopping had otherwise succeeded.

A separate exploration seed is restarted after each fit, providing common random numbers for convergence comparisons. The final estimate starts fresh replications on the frozen surrogate mean, independently of enrichment. Each SubsetRun records levels, acceptance rates, probability, the number of surrogate samples (n_samples) and a within-chain CoV approximation. result.estimate.sampling_cov uses the larger standard error from independent full-run replication and the aggregate within-chain approximation. Replication reflects variation from chain ancestry and adaptive levels that an IID calculation omits. This is still an estimated sampling error, particularly noisy with few replications; it does not bound bias or guarantee discovery of all failure regions. No binomial interval is assigned to subset samples.

The Active learning with subset simulation tutorial demonstrates rare and disconnected failure events, diagnostics, independent validation and visible budget failure. The default U-threshold stopping policy is available, but combined beta-band and beta-stability criteria often avoid continued enrichment of remote points whose contribution to failure probability is negligible.

PC-Kriging, bootstrap voting and active importance sampling#

PCKrigingSurrogate selects a sparse Hermite trend, fits its coefficients by GLS for each correlation candidate and retains the universal prediction variance, including trend uncertainty. Anisotropic Matérn 5/2 correlation is the default; Gaussian correlation and fixed length scales are explicit options. Likelihood gradients are analytic and scaled per observation. Optimization failure invalidates the fit. This implements sequential PC-Kriging, not the optimal variant’s search along every LAR support. It requires only NumPy/SciPy.

PCESurrogate.predict_replicates(points) exposes actual bootstrap responses, shape (n_points, n_replicates), with consistent replicate columns between calls. FBRLearning minimizes absolute safe/failure vote imbalance divided by the number of replicates; zero response counts as failure. It requires an EnsembleSurrogate. EnsembleLearningFunction is the corresponding extension interface. Scalar U/EFF interfaces remain unchanged.

The named learning_function="fbr" defaults to BootstrapBounds: the min/max bootstrap probability range divided by the full-design estimate must be at most 0.1 for two consecutive fits. These are actual replicate probability estimates on the fixed normal pool, not mean/spread approximations. The range can exclude the full-design estimate and is not a confidence interval for total error. LearningStep.bootstrap_probability_band records it separately. BootstrapBounds rejects weighted or conditional enrichment: such estimators do not yet supply replicate probability estimates. FBR selection can still use their candidate pools with an explicit alternative stopping policy.

ImportanceSamplingEstimator accepts row-wise normal-coordinate proposal centers, common spherical scale and a default 0.1 defensive fraction of the target normal distribution. Centers are fixed during a run. A converged FORM design_point_u can supply one center; known separated modes require appropriate additional centers. The proposal cannot guarantee discovery of unknown modes. Learning adapts the surrogate, while both exploratory and final probabilities use ordinary likelihood weighting. Final sampling uses fresh draws.

For samples from proposal \(q\), define \(Z_i=I_i\phi(u_i)/q(u_i)\). The probability estimate is \(\bar Z\) and its sampling variance is \(s_Z^2/N\). Weights are not self-normalized. No binomial interval is reported. Diagnostics retain effective sample/failure counts, mean/max weights, standard error and the raw probability. Out-of-range estimates remain explicitly invalid, with the exposed point clipped to a probability endpoint, infinite CoV and the raw value preserved. Sensitivity bands are clipped diagnostics. Zero/all observed failures and inadequate effective failure counts remain incomplete. A zero defensive fraction requires adequate proposal tails for finite sampling variance. There is no true-model correction factor in this workflow.

See PC-Kriging, bootstrap voting and importance sampling for composed examples and independent checks, and download the method provenance and UQLab comparison for exact sources, variants and regeneration.

Scope of later work#

System-specific active learning, clustered batch enrichment and automatic proposal adaptation are possible later additions. An adaptive quadratic response-surface workflow also fits classical assessment practice. These are outside the three additions above. SVM reliability needs classification-specific learning contracts; a generic neural-network or general UQ suite is outside the current scope.

Acceptance evidence#

Use the same published/analytic problems and multiple seeds across methods. The suite includes normal-sum, lognormal-beam and four-branch benchmarks, rare linear and two-tail active problems, and a replicated subset check of an analytic probability near 2.9e-7 in both 2 and 20 dimensions. Measure probability error, missed-region/classification error, true model evaluations, sampling error and explicit nonconvergence. These cases do not reproduce the full 20-problem review benchmark.

See Active learning for reliability analysis for implemented methods, numerical references and limitations, and Active-learning reliability for their formulation.

The The 23-bar truss benchmark and The hat-function benchmark tutorials add a ten-variable structural model and a curved cubic limit state. Their Literature benchmark definitions and provenance record published settings, source provenance, independent references and discrepancies in the source comparisons.

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