Efficient forward and inverse uncertainty quantification for dynamical systems based on dimension reduction and Kriging surrogate modeling in functional space

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.