Polynomial chaos vs kernel machine learning methods for uncertainty quantification: A comparative study and benchmarking with the hybrid PCE-GPR approach

In the past two decades, the application of surrogate models to expedite uncertainty quantification (UQ) received wide attention in the applied physics and engineering communities. In particular, extensive literature focused on high-dimensional problems and training efficiency with the aim of addressing the so-called "curse of dimensionality" and achieving satisfactory accuracy even with small experimental designs. Besides popular methods based on polynomial chaos expansion (PCE), recent works explored the application of machine learning regression methods to UQ, particularly kernel-based methods such as support vector regression and Gaussian process regression (GPR). Promising performance and superior accuracy were highlighted especially for small dataset sizes. In this context, the present paper provides a twofold contribution. First, it extends the literature survey and benchmarks by comparing PCE and kernel-based machine learning methods. Second, it introduces an open source toolbox that implements "PCE-GPR", a recently proposed hybrid method that combines the advantages of PCE and GPR methods and is suitable for accurate high-dimensional UQ. The analysis shows that kernel techniques tend to outperform PCE in high-dimensional settings and in small-data scenarios, at the price of increased computational cost. Moreover, for most test cases, PCE-GPR is shown to significantly outperform all of the state-of-the-art methods considered. The comparisons are based on popular benchmarks for surrogate modeling and UQ, simulations in various fields of engineering, as well as stochastic boundary value problems with high-dimensional random fields and up to 145 uncertain parameters.