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A Local Gaussian Process-Based Active Learning Method for Efficient Failure Probability Estimation

This paper proposes an efficient reliability analysis method that integrates local Gaussian processes, a novel adaptive learning function, and active subspaces to overcome the limitations of global optimization-based approaches in estimating failure probabilities for highly nonlinear, high-dimensional rare-event problems.

Original authors: Junfeng Zhao, Luoyi Lu, Yuxuan Zheng, Hongdan Zheng, Pei Yin, Xiaofei Guan

Published 2026-06-25

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CC BY 4.0

Original authors: Junfeng Zhao, Luoyi Lu, Yuxuan Zheng, Hongdan Zheng, Pei Yin, Xiaofei Guan

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an engineer trying to figure out the odds of a bridge collapsing. The bridge is made of thousands of parts, and the math to predict a failure is incredibly complex. Running a full simulation for every possible scenario is like trying to count every grain of sand on a beach to find the one that's slightly different—it takes too long and costs too much money.

This paper introduces a smarter, faster way to solve this problem. Here is how it works, broken down into simple concepts:

1. The Problem: The "Global Map" is Too Heavy

Traditional methods try to build one giant, perfect map of the entire bridge to find where it might break. This is like trying to draw a single, ultra-detailed map of the whole world on one piece of paper. It's so detailed that your computer gets overwhelmed, and it takes forever to update the map every time you learn something new. Also, most of the map (the safe parts of the bridge) doesn't need that much detail; you only really care about the cracks.

2. The Solution: A "Local Neighborhood" Approach

Instead of one giant map, the authors propose splitting the problem into small neighborhoods.

The Analogy: Imagine you are a detective trying to find a specific type of rare flower. Instead of sending one detective to survey the entire country (which is slow and tiring), you send a team of local detectives. Each detective only looks at their own small neighborhood.
How it works: The computer splits the data into small chunks and builds a simple, fast model for each chunk. This is much faster because the computer doesn't have to juggle the whole world at once; it just focuses on one small area at a time. This handles "wiggly" or complex shapes (non-linearities) much better than a single giant model.

3. The Strategy: "Smart Sampling" (Active Learning)

Once the computer has these local maps, it needs to decide where to look next. It shouldn't just guess randomly. The paper introduces a special "learning function" (a rule for picking the next spot to test) that combines two ideas:

Uncertainty (The Fog): "Where is the map most blurry?" The computer looks for areas where it is least sure about the answer.
The Edge (The Cliff): "Where is the danger zone?" The computer specifically looks near the line where the bridge is safe versus where it fails.

The Creative Metaphor:
Imagine you are painting a picture of a cliff edge in the fog.

Old Way: You paint the whole sky and the whole ground, even though you only care about the cliff.
New Way: You have a smart brush. At first, it paints broadly to see where the fog is thickest (exploration). As you get closer to the cliff edge, the brush gets smaller and more precise, focusing entirely on the exact line where the ground drops off (refinement).

The paper's method uses a "weighting system" that starts by exploring the whole area to find the danger zone, and then gradually shifts to focusing intensely on the edge of that zone to get the exact failure probability.

4. Handling Big Problems: The "Shrinking Lens"

For problems with hundreds of variables (like a bridge with thousands of sensors), the computer can still get confused. The authors add a tool called "Active Subspace."

The Analogy: Imagine trying to describe a 3D object using a 2D shadow. The "Active Subspace" is like finding the perfect angle to shine a light so the shadow captures all the important details while ignoring the noise. It simplifies the problem so the local neighborhood models can handle it easily.

5. The Results: Faster and Just as Accurate

The authors tested this method on five different engineering problems, from simple math puzzles to complex concrete beam structures and high-dimensional models.

Speed: Their method was significantly faster (sometimes 50-60% faster) than the traditional "global map" method.
Accuracy: It found the failure probability just as accurately, or even better, especially for very rare events.
Efficiency: It needed far fewer computer simulations to get the right answer, saving time and money.

Summary

In short, this paper says: "Don't try to solve the whole puzzle at once. Break it into small pieces, focus your energy on the edges where the danger lies, and use a smart strategy to zoom in only where it matters." This makes predicting engineering failures much cheaper and faster without losing accuracy.

Technical Summary: A Local Gaussian Process-Based Active Learning Method for Efficient Failure Probability Estimation

1. Problem Statement

Reliability analysis is critical for engineering design and safety assessment, yet it faces significant challenges when estimating the failure probability of complex systems. In many practical scenarios, the limit state function $g(x)$ relies on computationally expensive numerical simulations (e.g., finite element analysis), and system responses often exhibit non-stationarity or locally strong nonlinearities.

Traditional methods struggle to balance efficiency and accuracy:

Asymptotic approximations (e.g., FORM/SORM) may lose accuracy with multiple most probable points or strong nonlinearities.
Moment-based methods suffer from ill-posed inverse problems and distributional biases.
Stochastic simulation methods (e.g., Monte Carlo Simulation) are robust but converge slowly for rare events, requiring prohibitive numbers of expensive model evaluations.

While surrogate models (e.g., global Gaussian Process Regression) have been adopted to reduce costs, they face scalability issues in high-dimensional or non-stationary problems due to the $O(n^3)$ complexity of covariance matrix inversion and reduced local prediction accuracy. Furthermore, standard active learning strategies often generate redundant samples far from the failure boundary, wasting computational resources.

2. Methodology

The paper proposes LGP-AL, an active learning reliability analysis method that integrates Local Gaussian Processes (LGP) with a novel Active Learning strategy.

2.1 Local Gaussian Process (LGP) Surrogate

Instead of constructing a single global model, the method partitions the training dataset into $m$ distinct subdomains using K-means clustering. Independent Gaussian Process Regression (GPR) sub-models are constructed within each subdomain.

Assumption: Replaces the global stationarity assumption with local stationarity, allowing the model to better capture non-stationary responses and strong local nonlinearities.
Scalability: By decomposing the global problem into $m$ balanced local subsets, the training complexity is reduced from $O(n^3)$ to approximately $O(n^3/m^2)$ .
Stability: To prevent instability caused by frequent re-partitioning, a partition structure inheritance mechanism is employed. Re-clustering occurs only when the relative change in the failure probability estimate exceeds a prescribed threshold; otherwise, new samples are assigned to existing subsets via a nearest-neighbor rule.

2.2 Active Learning Strategy (UBAF)

To guide the sequential sampling process, the authors construct a joint acquisition function called the Uncertainty-Boundary Acquisition Function (UBAF). This function combines two criteria:

Upper-Bound Posterior Variance Contribution (UPVC): Derived from the upper bound of the posterior variance of the failure probability, this metric quantifies the contribution of each input point to the global uncertainty of the failure probability estimate. It drives global exploration.
Boundary Proximity Degree (BPD): Defined as $-|u_i|$ (where $u_i$ is the posterior mean), this metric measures the distance of a candidate point to the estimated failure boundary. It drives local refinement.

The UBAF is formulated as a weighted sum:
$UBAF(x) = (1 - w) \cdot \widehat{UPVC}(x) + w \cdot \widetilde{BPD}(x)$
where $w$ is an adaptive weight that increases with the iteration count. This ensures the sampling strategy transitions from global uncertainty-driven exploration in early iterations to boundary-focused refinement in later stages. The weight of BPD is capped at 0.5 to prevent premature clustering.

2.3 High-Dimensional Extension

For high-dimensional inputs, the method integrates the Active Subspace Method. This dimensionality reduction technique projects the high-dimensional input space onto a low-dimensional active subspace based on the gradient of the model response, preserving dominant input-output relationships while reducing the dimensionality for the LGP framework.

3. Key Contributions

The paper outlines three primary contributions:

Local Modeling Framework: Partitioning the training set to construct local Gaussian process models, replacing global stationarity with local stationarity to better capture non-stationary structures while significantly reducing kernel matrix inversion costs.
Novel Active Learning Criterion: Formulating a sampling criterion (UBAF) that combines the upper bound of posterior variance with failure boundary proximity. This directs samples to high-uncertainty and near-boundary regions, improving sign discrimination of the limit state function.
Dimensionality Reduction Integration: Incorporating the active subspace method to maintain stable approximation in high-dimensional settings, enhancing the scalability of the LGP framework.

4. Numerical Results

The method was validated on five numerical examples: an iso-probability contour problem, a four-branch system, a reinforced concrete beam segment, a bracket model, and a high-dimensional linear problem.

Efficiency: Across all examples, LGP-AL demonstrated significantly reduced training times compared to global Gaussian Process-based active learning (GP-AL). For instance, in the four-branch system, LGP-AL reduced training time by approximately 63% (8.9 min vs. 19.4 min). In the reinforced concrete beam example, GP-AL failed to identify the failure domain (estimating zero probability), whereas LGP-AL achieved a relative error of $1.90 \times 10^{-2}$ .
Accuracy: The proposed method maintained high estimation accuracy. In the four-branch system, LGP-AL achieved a relative error of $1.12 \times 10^{-2}$ compared to $2.58 \times 10^{-1}$ for GP-AL.
Sampling Strategy: Comparisons showed that the UBAF strategy outperformed both Latin Hypercube Sampling (LHS) and a strategy using only UPVC. UBAF effectively concentrated samples near the failure boundary, leading to faster convergence and lower relative errors.
High-Dimensional Performance: In the high-dimensional linear problem (dimensions $d=100, 500, 1000$ ), the combination of active subspaces and LGP maintained relative errors within $10^{-1}$ with training times ranging only from 29 to 35 seconds, demonstrating robustness against increasing dimensionality.

5. Significance and Claims

The paper claims that the proposed LGP-AL method offers a robust and scalable solution for failure probability estimation in complex engineering systems. Its significance lies in:

Reducing Computational Burden: By decomposing the global problem and utilizing local stationarity, the method significantly lowers the computational cost of model training and covariance matrix inversion.
Improving Sampling Efficiency: The adaptive UBAF criterion ensures that expensive limit state function evaluations are focused on critical regions (high uncertainty and near the failure boundary), minimizing redundant sampling.
Handling Complexity: The method effectively addresses challenges posed by non-stationary responses, strong local nonlinearities, and high-dimensional inputs.

The authors modestly acknowledge limitations, noting that the method may still face computational challenges in scenarios involving extremely small failure probabilities or ultra-high-dimensional systems. They suggest future work could incorporate low-rank matrix approximations, inducing point sparsification, and multi-fidelity modeling to further enhance scalability and convergence.