Surrogate models are extensively employed for forward and inverse uncertainty quantification in complex, computation-intensive engineering problems. Nonetheless, constructing high-accuracy surrogate models for complex dynamical systems with limited training samples continues to be a challenge, as capturing the variability in high-dimensional dynamical system responses with a small training set is inherently difficult. This study introduces an efficient Kriging modeling framework based on functional dimension reduction (KFDR) for conducting forward and inverse uncertainty quantification in dynamical systems. By treating the responses of dynamical systems as functions of time, the proposed KFDR method first projects these responses onto a functional space spanned by a set of predefined basis functions, which can deal with noisy data by adding a roughness regularization term. A few key latent functions are then identified by solving the functional eigenequation, mapping the time-variant responses into a low-dimensional latent functional space. Subsequently, Kriging surrogate models with noise terms are constructed in the latent space. With an inverse mapping established from the latent space to the original output space, the proposed approach enables accurate and efficient predictions for dynamical systems. Finally, the surrogate model derived from KFDR is directly utilized for efficient forward and inverse uncertainty quantification of the dynamical system. Through three numerical examples, the proposed method demonstrates its ability to construct highly accurate surrogate models and perform uncertainty quantification for dynamical systems accurately and efficiently.

中文翻译：

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基于降维和函数空间 Kriging 代理建模的动力学系统正向和逆不确定性量化

代理模型广泛用于复杂的计算密集型工程问题中的正向和反向不确定性量化。尽管如此，用有限的训练样本为复杂的动力学系统构建高精度的代理模型仍然是一个挑战，因为用较小的训练集捕捉高维动力学系统响应的变化本身就很困难。本研究介绍了一种基于函数降维 （KFDR） 的高效 Kriging 建模框架，用于在动力学系统中进行正向和反向不确定性量化。通过将动力系统的响应视为时间的函数，所提出的 KFDR 方法首先将这些响应投影到由一组预定义的基函数跨越的函数空间上，该函数可以通过添加粗糙度正则化项来处理噪声数据。然后通过求解函数特征方程来确定一些关键的潜在函数，将时变响应映射到低维潜在函数空间。随后，在潜在空间中构建具有噪声项的 Kriging 代理模型。通过建立从潜在空间到原始输出空间的逆映射，所提出的方法能够对动力学系统进行准确和有效的预测。最后，从 KFDR 得出的代理模型直接用于动力学系统的高效正向和逆向不确定性量化。通过三个数值示例，所提出的方法证明了其构建高精度代理模型和准确有效地对动力学系统进行不确定性量化的能力。