基于主动学习的域自适应局部多项式混沌展开
Active Learning-based Domain Adaptive Localized Polynomial Chaos Expansion
AI总结：
本文提出域自适应局部多项式混沌展开（DAL-PCE），通过主动学习序贯分解输入随机空间并构建局部低阶展开，以高效准确近似强非线性、不连续或奇异函数，数值实验表明其优于全局多项式混沌展开和随机谱嵌入方法。
摘要：
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序贯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，该方法被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序贯分解，将其划分为更小的子域，并用低阶多项式展开进行近似。这使得能够近似具有强非线性、不连续性和/或奇异性的函数。输入随机空间的分解和局部近似缓解了这类问题中的吉布斯现象，并将误差限制在非线性附近非常小的邻域内。因此，如数值示例所示，代理模型的全局行为显著优于现有方法。整个过程由主动学习例程驱动，该例程使用最近提出的 $Θ$ 准则来评估局部方差贡献。所提出的方法平衡了代理模型的\emph{利用}和输入随机空间的\emph{探索}，从而实现对原始数学模型的高效且准确的近似。数值结果表明，DAL-PCE 相较于（i）单一的全局多项式混沌展开和（ii）最近提出的随机谱嵌入（SSE）方法具有优越性，SSE 是作为一种精确代理模型而开发的，并且基于类似的域分解过程。该方法代表了一个通用框架，可以在此基础上进行进一步的扩展和改进，并且可以与任何非侵入式多项式混沌展开构建技术相结合。
英文摘要：
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 $Θ$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
