Surrogate modeling for uncertainty quantification in nonlinear dynamics

TL;DR

This paper reviews surrogate modeling techniques for uncertainty quantification in nonlinear dynamical systems, emphasizing methods to efficiently approximate time-dependent responses under uncertainty.

Contribution

It classifies time-dependent UQ problems and discusses various surrogate approaches, including PCA with polynomial chaos, time warping, and NARX models, with illustrative examples.

Findings

Surrogate models significantly reduce computational costs in UQ for nonlinear dynamics.

Different surrogate approaches are suitable for various types of time-dependent problems.

Practical examples demonstrate the effectiveness of each surrogate method.

Abstract

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.