Low-rank statistical finite elements for scalable model-data synthesis
Low-rank statistical finite elements for scalable model-data synthesis
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DOI：
10.1016/j.jcp.2022.111261
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发表时间：
2021-09
期刊：
影响因子：
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通讯作者：
Connor Duffin;E. Cripps;T. Stemler;M. Girolami
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文献类型：
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作者：
Connor Duffin;E. Cripps;T. Stemler;M. Girolami
统计学习添加到物理导出的数学模型在文献中越来越受欢迎。最近的一种方法是用数据驱动的贝叶斯统计方法来增强控制方程的基础物理。该方法通过在控制方程中嵌入随机强迫来消除模型的错误设定。在收到额外的数据，离散有限元解的后验分布更新使用经典的贝叶斯滤波技术。由此产生的后验联合量化与模型误指定和数据旨在代表感兴趣的真实过程的普遍存在的问题相关联的不确定性。尽管有这种吸引力，计算的可扩展性是一个挑战statFEM的应用程序通常在物理和工业环境中经历的高维问题。本文通过嵌入底层密集协方差矩阵的低秩近似来克服这一障碍，该近似是从满秩替代方案的首阶模式中获得的。证明了一系列的反应扩散问题的维数增加，使用实验和模拟数据，该方法重建稀疏观测的数据生成过程，以最小的信息损失，在后验均值和方差，铺平了道路，进一步整合的物理和概率的方法来复杂的系统。
Statistical learning additions to physically derived mathematical models are gaining traction in the literature. A recent approach has been to augment the underlying physics of the governing equations with data driven Bayesian statistical methodology. Coined statFEM, the method acknowledgesa priorimodel misspecification, by embedding stochastic forcing within the governing equations. Upon receipt of additional data, the posterior distribution of the discretised finite element solution is updated using classical Bayesian filtering techniques. The resultant posterior jointly quantifies uncertainty associated with the ubiquitous problem of model misspecification and the data intended to represent the true process of interest. Despite this appeal, computational scalability is a challenge to statFEM's application to high-dimensional problems typically experienced in physical and industrial contexts. This article overcomes this hurdle by embedding a low-rank approximation of the underlying dense covariance matrix, obtained from the leading order modes of the full-rank alternative. Demonstrated on a series of reaction-diffusion problems of increasing dimension, using experimental and simulated data, the method reconstructs the sparsely observed data-generating processes with minimal loss of information, in both the posterior mean and variance, paving the way for further integration of physical and probabilistic approaches to complex systems.