In this paper, an alternative to solving Bayesian inverse problems for structural health monitoring based on a variational formulation with so-called transport maps is examined. The Bayesian inverse formulation is a widely used tool in structural health monitoring applications. While Markov Chain Monte Carlo (MCMC) methods are often implemented in these settings, they come with the problem of using many model evaluations, which in turn can become quite costly. We focus here on recent developments in the field of transport theory, where the problem is formulated as finding a deterministic, invertible mapping between some easy to evaluate reference density and the posterior. The resulting variational formulation can be solved with integration and optimization methods. We develop a general formulation for the application of transport maps to vibration-based structural health monitoring. Further, we study influences of different integration approaches on the efficiency and accuracy of the transport map approach and compare it to the Transitional MCMC algorithm, a widely used method for structural identification. Both methods are applied to a lower-dimensional dynamic model with uni- and multi-modal properties, as well as to a higher-dimensional neural network surrogate system of an airplane structure. We find that transport maps have a significant increase in accuracy and efficiency, when used in the right circumstances.

中文翻译：

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MCMC 与传输图贝叶斯后验估计在结构健康监测中的效率比较

在本文中，研究了基于变分公式和所谓的传输图来解决结构健康监测的贝叶斯逆问题的替代方案。贝叶斯逆公式是结构健康监测应用中广泛使用的工具。虽然马尔可夫链蒙特卡罗 (MCMC) 方法通常在这些设置中实现，但它们存在使用许多模型评估的问题，而这反过来又会变得相当昂贵。我们在这里关注传输理论领域的最新发展，其中问题被表述为在一些易于评估的参考密度和后验之间找到确定性的、可逆的映射。由此产生的变分公式可以通过积分和优化方法来求解。我们开发了一种将传输图应用于基于振动的结构健康监测的通用公式。此外，我们研究了不同集成方法对传输图方法的效率和准确性的影响，并将其与过渡 MCMC 算法（一种广泛使用的结构识别方法）进行比较。这两种方法都应用于具有单模态和多模态特性的低维动态模型，以及飞机结构的高维神经网络代理系统。我们发现，在正确的情况下使用交通地图可以显着提高准确性和效率。