Surrogate modeling for uncertainty quantification in nonlinear dynamics

Predicting the behavior of complex systems in engineering often involves significant uncertainty about operating conditions, such as external loads, environmental effects, and manufacturing variability. As a result, uncertainty quantification (UQ) has become a critical tool in modeling-based engineering, providing methods to identify, characterize, and propagate uncertainty through computational models. However, the stochastic nature of UQ typically requires numerous evaluations of these models, which can be computationally expensive and limit the scope of feasible analyses. To address this, surrogate models, i.e., efficient functional approximations trained on a limited set of simulations, have become central in modern UQ practice. This book chapter presents a concise review of surrogate modeling techniques for UQ, with a focus on the particularly challenging task of capturing the full time-dependent response of dynamical systems. It introduces a classification of time-dependent problems based on the complexity of input excitation and discusses corresponding surrogate approaches, including combinations of principal component analysis with polynomial chaos expansions, time warping techniques, and nonlinear autoregressive models with exogenous inputs (NARX models). Each method is illustrated with simple application examples to clarify the underlying ideas and practical use.