基于ROMES方法的稳态系统降阶模型的统计闭合建模
Statistical closure modeling for reduced-order models of stationary systems by the ROMES method
AI总结：
本文将ROMES方法扩展至稳态系统降阶模型的统计闭合建模，通过为状态误差的广义坐标构建统计模型，实现了任意物理量误差的后验统计量化，并显著提升了预测精度与计算性价比。
摘要：
本文提出了一种技术，用于为应用于建模为参数化代数方程组的稳态系统的降阶模型（ROMs）构建统计闭合模型。所提出的技术将降阶模型误差代理（ROMES）方法扩展到闭合建模。原始的ROMES方法应用高斯过程回归来构建一个统计模型，该模型将可低成本计算的误差指标（例如，残差范数，对偶加权残差）映射到随机变量，用于表示（1）状态误差的范数或（2）标量值物理量中的误差。本文不针对这两类误差，而是提出为状态误差本身构建统计模型；它通过为表征面内误差（即试探子空间中的误差）和面外误差的低维近似的广义坐标构建统计模型来实现这一点。前者可以被视为统计闭合模型，因为它量化了ROM广义坐标中的误差。由于任何物理量都可以计算为状态的泛函，所提出的方法能够对任意物理量误差进行后验的统计量化，因为状态误差模型可以通过相关的物理量泛函进行传播。在线性和非线性稳态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）统计量化任意物理量中的误差，以及（3）在非线性系统的情况下，实现一种比仅使用ROM的方法更具成本效益的降低误差的方法。
英文摘要：
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.
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