发表机构
University of California San Diego（加州大学圣迭戈分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出mollified-sharp分解，将激波位置视为随机变量进行概率正则化，实现含移动激波流场的参数化降阶建模，在跨音速翼型数据上显著降低压力误差。
AI 中文摘要
本文介绍了mollified-sharp分解，一种对参数化降阶模型中移动激波进行概率正则化的方法。每个检测到的激波位置被视为具有指定概率密度的随机变量。对该人工分布进行平均，将局部压力变化替换为平滑过渡，其空间范围由密度宽度决定，而非通过对压力场直接滤波。每个快照被精确分解为一个正则化的mollified场和一个恢复激波的局部sharp修正。这种概率构造和精确的加性分解定义了通用方法；检测器、核函数、多激波处理、对齐坐标和回归均为实现选择。在当前实现中，校准指示器检测激波，紧支撑的Wendland核对其进行mollify处理，由每个激波位置概率密度导出的峰值归一化且在必要时分区的权重定义了其局部对齐域的质心、主轴和尺度。分离的POD-GPR模型分别表示mollified场和对齐修正，并附加激波存在性和对齐的回归。该方法在Catalani等人（2023）的跨音速翼型压力数据上进行了演示，并与基于相同数据和POD能量准则构建的标准POD-GPR模型进行了比较。mollified-sharp模型将测试集平均相对$L^2$压力误差降低了31.2%，将测试集平均表面压力系数误差降低了33.2%，同时改善了预测的激波位置和压力变化；代价是在线评估时间中位数延长至2.24倍。该构造原则上适用于其他具有移动尖锐特征（如多相流中的移动材料界面）的参数依赖场。
英文摘要
This paper introduces the mollified-sharp decomposition, a probabilistic regularization of moving shocks in parametric reduced-order models. Each detected shock location is treated as a random variable with a prescribed probability density. Averaging over this artificial distribution replaces the localized pressure change by a smooth transition whose spatial extent is set by the density width, rather than by direct filtering of the pressure field. Each snapshot is decomposed exactly into a regularized mollified field and a local sharp correction that restores the shock. This probabilistic construction and exact additive split define the general method; the detector, kernel, treatment of multiple shocks, alignment coordinates, and regression are implementation choices. In the present realization, a calibrated indicator detects shocks, a compactly supported Wendland kernel mollifies them, and a peak-normalized and, where necessary, partitioned weight derived from each shock-location probability density defines the centroid, principal axes, and scales of its local alignment domain. Separate POD-GPR models represent the mollified field and aligned corrections, with additional regressions for shock presence and alignment. The method is demonstrated on the transonic airfoil pressure data of Catalani et al. (2023) and compared with a standard POD-GPR model constructed from the same data and POD-energy criterion. The mollified-sharp model reduces the mean test-set relative $L^2$ pressure error by 31.2% and the mean test-set surface-pressure-coefficient error by 33.2%, while also improving the predicted shock locations and pressure changes; the trade-off is a median online evaluation time 2.24 times as long. The construction is applicable in principle to other parameter-dependent fields with moving sharp features, such as moving material interfaces in multiphase flows.
