Polynomial chaos expansion (PCE) is a versatile tool widely used in uncertainty quantification and machine learning, but its successful application depends strongly on the accuracy and reliability of the resulting PCE-based response surface. High accuracy typically requires high polynomial degrees, demanding many training points especially in high-dimensional problems through the curse of dimensionality. So-called sparse PCE concepts work with a much smaller selection of basis polynomials compared to conventional PCE approaches and can overcome the curse of dimensionality very efficiently, but have to pay specific attention to their strategies of choosing training points. Furthermore, the approximation error resembles an uncertainty that most existing PCE-based methods do not estimate. In this study, we develop and evaluate a fully Bayesian approach to establish the PCE representation via joint shrinkage priors and Markov chain Monte Carlo. The suggested Bayesian PCE model directly aims to solve the two challenges named above: achieving a sparse PCE representation and estimating uncertainty of the PCE itself. The embedded Bayesian regularizing via the joint shrinkage prior allows using higher polynomial degrees for given training points due to its ability to handle underdetermined situations, where the number of considered PCE coefficients could be much larger than the number of available training points. We also explore multiple variable selection methods to construct sparse PCE expansions based on the established Bayesian representations, while globally selecting the most meaningful orthonormal polynomials given the available training data. We demonstrate the advantages of our Bayesian PCE and the corresponding sparsity-inducing methods on several benchmarks.
翻译:在不确定性量化和机器学习中广泛使用多元混乱扩大(PCE)是一个多用途工具,在不确定性量化和机器学习中广泛使用,但其成功应用在很大程度上取决于基于PCE的反应表面的准确性和可靠性。 高准确性通常要求高多元度,要求许多高维问题的培训点,尤其是高维问题的培训点,这通过维度诅咒。 与常规PCE方法相比,所谓的稀疏多的PCE概念在基础多语种选择小得多,可以非常有效地克服维度的诅咒,但必须特别关注其选择培训点的战略。 此外,近似误差类似于大多数基于PCE的现有方法没有估计的不确定性。 在本研究中,我们制定和评估一种完全的Bayesian方法,通过共同缩小前期和Markov 链 Monte Carlo 来建立PCE代表点。 所建议的Bayesian PCE 模型直接旨在解决上述两个挑战:实现微弱的PCE代表点代表点,并估算PCEE本身的不确定性。 先前通过联合压缩的常规化使Bayesian正规化战略允许使用更高的多级培训点,因为大多数基于PCEEEEBCE的训练点,而我们现有的多级标准选择方法之下的多级数据代表点。