Statistical closure modeling for reduced-order models of stationary systems by the ROMES method
Statistical closure modeling for reduced-order models of stationary systems by the ROMES method
复制标题
采用 ROMES 方法对平稳系统降阶模型进行统计闭包建模
DOI：
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发表时间：
2019
影响因子：
1.7
通讯作者：
K. Carlberg
中科院分区：
文献类型：
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作者：
S. Pagani;A. Manzoni;K. Carlberg
这项工作提出了一种技术，用于构建一个统计闭包模型的降阶模型（ROMs）应用于作为代数方程的参数化系统建模的平稳系统。该技术将降阶模型误差代理（ROMES）方法扩展到闭包建模。原始的ROMES方法应用高斯过程回归构建了一个统计模型，该模型将可廉价计算的误差指标（例如，残差范数，双加权残差）映射到一个随机变量，用于(1)状态误差的范数或(2)感兴趣的标量量中的误差。这项工作不是针对这两种类型的误差，而是提出为状态误差本身构建一个统计模型；它通过构建广义坐标的统计模型来实现这一点，该统计模型既描述了平面内误差（即试验子空间中的误差），也描述了平面外误差的低维近似。前者可以被认为是统计闭包模型，因为它量化了ROM广义坐标中的误差。由于任何数量的兴趣都可以作为状态的函数来计算，因此所提出的方法使得任何数量的兴趣误差都可以在后验统计量化，因为状态误差模型可以通过相关的数量的兴趣函数来传播。在线性和非线性平稳系统上进行的数值实验说明了该技术的能力：(1)将（预期的）ROM预测精度提高一个数量级，(2)统计量化任意感兴趣量的误差，以及(3)在非线性系统的情况下实现比仅使用ROM方法更经济有效的方法来减少误差。
This work proposes a technique for constructing a statistical closure model for reduced-order models (ROMs) applied to stationary systems modeled as parameterized systems of algebraic equations. The proposed technique extends the reduced-order-model error surrogates (ROMES) method to closure modeling. The original ROMES method applied Gaussian-process regression to construct a statistical model that maps cheaply computable error indicators (e.g., residual norm, dual-weighted residuals) to a random variable for either (1) the norm of the state error or (2) the error in a scalar-valued quantity of interest. Rather than target these two types of errors, this work proposes to construct a statistical model for the state error itself; it achieves this by constructing statistical models for the generalized coordinates characterizing both the in-plane error (i.e., the error in the trial subspace) and a low-dimensional approximation of the out-of-plane error. The former can be considered a statistical closure model, as it quantifies the error in the ROM generalized coordinates. Because any quantity of interest can be computed as a functional of the state, the proposed approach enables any quantity-of-interest error to be statistically quantified a posteriori, as the state-error model can be propagated through the associated quantity-of-interest functional. Numerical experiments performed on both linear and nonlinear stationary systems illustrate the ability of the technique (1) to improve (expected) ROM prediction accuracy by an order of magnitude, (2) to statistically quantify the error in arbitrary quantities of interest, and (3) to realize a more cost-effective methodology for reducing the error than a ROM-only approach in the case of nonlinear systems.