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 acknowledges a priori model 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 posterior mean and the variance, paving the way for further integration of physical and probabilistic approaches to complex systems.

中文翻译：

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用于可扩展模型数据合成的低秩统计有限元

对物理派生数学模型的统计学习补充在文献中越来越受欢迎。最近的一种方法是使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。创造了 statFEM，该方法通过在控制方程中嵌入随机强迫来确认先验模型错误指定。收到附加数据后，离散化有限元解的后验分布将使用经典贝叶斯过滤技术进行更新。由此产生的后验联合量化了与普遍存在的模型错误指定问题和旨在代表真正感兴趣的过程的数据相关的不确定性。尽管有这种吸引力，但计算可扩展性对 statFEM 来说是一个挑战 s 应用于通常在物理和工业环境中遇到的高维问题。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，该矩阵是从全秩替代的前导阶模式中获得的。该方法在一系列维度增加的反应扩散问题上进行了演示，使用实验和模拟数据，在后验均值和方差中以最小的信息损失重建稀疏观察到的数据生成过程，为进一步整合铺平了道路复杂系统的物理和概率方法。从满秩替代的前导阶模式中获得。该方法在一系列维度增加的反应扩散问题上进行了演示，使用实验和模拟数据，在后验均值和方差中以最小的信息损失重建稀疏观察到的数据生成过程，为进一步整合铺平了道路复杂系统的物理和概率方法。从满秩替代的前导阶模式中获得。该方法在一系列维度增加的反应扩散问题上进行了演示，使用实验和模拟数据，在后验均值和方差中以最小的信息损失重建稀疏观察到的数据生成过程，为进一步整合铺平了道路复杂系统的物理和概率方法。