An efficient, accurate, and interpretable machine learning method for computing probability of failure

TL;DR

This work tackles estimating the probability of failure for complex systems by reframing it as a boundary-aware classification task. It introduces the Penalized Profile SVM on the Gabriel edited set (PPSVMG), built from Gabriel boundary points and adaptive POF--Darts sampling to preserve the true boundary geometry while delivering interpretable, locally linear surrogates. The method combines cluster-based local SVMs with a gradient-alignment penalty, ensuring robustness on small training sets and improved boundary fidelity. Convergence results show that Gabriel neighbors concentrate near the true boundary with increasing samples, and experiments across synthetic and Lotka–Volterra problems demonstrate competitive accuracy and reduced surrogate variance relative to direct Monte Carlo methods. The approach offers a practical, interpretable framework for reliability assessment in high-dimensional, physics-informed settings.

Abstract

We introduce a novel machine learning method called the Penalized Profile Support Vector Machine based on the Gabriel edited set for the computation of the probability of failure for a complex system as determined by a threshold condition on a computer model of system behavior. The method is designed to minimize the number of evaluations of the computer model while preserving the geometry of the decision boundary that determines the probability. It employs an adaptive sampling strategy designed to strategically allocate points near the boundary determining failure and builds a locally linear surrogate boundary that remains consistent with its geometry by strategic clustering of training points. We prove two convergence results and we compare the performance of the method against a number of state of the art classification methods on four test problems. We also apply the method to determine the probability of survival using the Lotka--Volterra model for competing species.

Figures (23)

• Figure 1: An example of a nonlinear decision boundary for $Q\geq 3.75$ the Brusselator model with fixed $x_3=1.65$.
• Figure 2: The surrogate classification boundary along with the classification results computed using the PSVM with MagKmeans clustering approach described in Section \ref{['App:PSVM']} on one of the test problems described in Section \ref{['sec:PSVMG']}. The classification of points is indicated using stars and circles. The true decision boundary is colored dark red and the region of failure is shaded.
• Figure 3: Examples of sample points and their corresponding spheres produced through POF--Darts sampling for the Brusselator model. Samples in negative and positive regions are colored red and green respectively.
• Figure 4: Example of the performance of PSVMG on a circular decision boundary.
• Figure 5: Decomposition of $\Lambda$ by classification from the PSVMG model for a circular decision boundary with labels colored in blue and orange.

Theorems & Definitions (2)

• Theorem 4.1
• Theorem 4.2