Surrogate Models for Oscillatory Systems Using Sparse Polynomial Chaos Expansions and Stochastic Time Warping

Abstract

Polynomial chaos expansions (PCEs) have proven efficiency in a number of fields for propagating parametric uncertainties through computational models of complex systems, namely structural and fluid mechanics, chemical reactions and electromagnetism, etc. For problems involving oscillatory, time-dependent output quantities of interest, it is well known that achieving reasonable accuracy in PCE-based approaches is difficult in the long term. To address this issue, we propose a fully nonintrusive approach based on stochastic time warping: each realization (trajectory) of the model response is first rescaled to its own time scale so as to put all sampled trajectories in-phase in a common virtual time line. Principal component analysis is introduced to compress the information contained in these transformed trajectories, and then sparse PCE representations using least angle regression are used to approximate the components. The approach shows a remarkably small prediction error not only for particular trajectories but also for second-order statistics of the latter. The approach is illustrated on different benchmark problems that are well-known in the time-dependent PCE literature, such as rigid body dynamics, chemical reactions, and forced oscillations of a nonlinear system.