The paper presents a novel methodology to build surrogate models of complicated functions by an active learning-based sequential decomposition of the input random space and construction of localized polynomial chaos expansions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE). The approach utilizes sequential decomposition of the input random space into smaller sub-domains approximated by low-order polynomial expansions. This allows approximation of functions with strong nonlinearties, discontinuities, and/or singularities. Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon for these types of problems and confines error to a very small vicinity near the non-linearity. The global behavior of the surrogate model is therefore significantly better than existing methods as shown in numerical examples. The whole process is driven by an active learning routine that uses the recently proposed $\Theta$ criterion to assess local variance contributions. The proposed approach balances both \emph{exploitation} of the surrogate model and \emph{exploration} of the input random space and thus leads to efficient and accurate approximation of the original mathematical model. The numerical results show the superiority of the DAL-PCE in comparison to (i) a single global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method developed as an accurate surrogate model and which is based on a similar domain decomposition process. This method represents general framework upon which further extensions and refinements can be based, and which can be combined with any technique for non-intrusive polynomial chaos expansion construction.

翻译：本文提出了一种新颖的方法，通过基于主动学习的输入随机空间顺序分解和局部多项式混沌展开的构建，来建立复杂函数的代理模型，称为域自适应局部多项式混沌展开（DAL-PCE）。该方法利用输入随机空间的顺序分解，将其划分为由低阶多项式展开逼近的较小子域。这使得能够逼近具有强非线性、不连续性和/或奇异性的函数。输入随机空间的分解和局部逼近减轻了这类问题的吉布斯现象，并将误差限制在非线性附近的极小区域内。如数值算例所示，代理模型的全局行为因此显著优于现有方法。整个过程由主动学习流程驱动，该流程使用最近提出的$\Theta$准则来评估局部方差贡献。所提方法平衡了代理模型的\emph{利用}与输入随机空间的\emph{探索}，从而实现了对原始数学模型的高效且精确的逼近。数值结果表明，与（i）单一全局多项式混沌展开和（ii）最近提出的基于类似域分解过程构建精确代理模型的随机谱嵌入（SSE）方法相比，DAL-PCE具有优越性。该方法提供了一个通用框架，可在此基础上进行进一步扩展和优化，并可与非侵入式多项式混沌展开构建的任何技术相结合。