Microstructure characterization in thermal barrier coatings using Sparse Polynomial Chaos Classifiers and Global Sensitivity Analysis

Polynomial Chaos Expansion (PCE) surrogates have been routinely used for modeling explicit representations between inputs of physical systems and output quantities of interest. Their use in classification problems however, has been quite limited, as they are not suitable for predicting categorical or discrete variables. In addition, PCEs enable analytical computation of the Sobol Sensitivity Indices, resulting in a very efficient approach to quantifying the significance of each input parameter on the model outputs. However, rigorous and systematic Global Sensitivity Analysis methods barely exist for classifiers. We present a novel approach for training soft-max classifiers with built-in Polynomial Chaos expansions that enhances sparsity in the PCE representation making them suitable for classification problems in the small-data regime. This is achieved by introducing a generalization of the well-known Variational Relevance Vector Machine framework for multi-class problems. The proposed classifier also allows for efficient computational schemes for variance-based global sensitivity analysis using Monte Carlo sampling. We demonstrate the performance of our proposed method on two numerical examples, namely a modified Ishigami function with discrete output and a challenging experimental design problem in thermal-barrier coating for Industrial Gas Turbine blades, where the presence of specific pattern characteristics is required to ensure component durability.