Nonparametric probabilistic approach of model uncertainties introduced by a projection-based nonlinear reduced-order model

Abstract

The paper is devoted to model uncertainties (or model form uncertainties) induced by modeling errors in computational sciences and engineering (such as in computational structural dynamics, fluid-structure interaction, and vibroacoustics, etc.) for which a parametric high-fidelity computational model (HFM) is used for addressing optimization problems (such as a robust design optimization problem), which are solved by introducing a parametric reducedorder model (ROM) constructed using an adapted reduced-order basis (ROB) derived from the parametric HFM. Two main methodologies are available to take into account such modeling errors. The first one is the usual output-predictive error method that has been introduced for many years. This approach can induce some difficulties because the parametric HFM and ROM do not learn from data. The second one is the nonparametric probabilistic approach of model uncertainties introduced in the framework of structural dynamics fifteen years ago. This approach is adapted, but is mainly limited to linear operators of the parametric HFM. The present paper deals with this challenging problem and proposes a novel nonparametric probabilistic approach of the modeling errors for any parametric nonlinear HFM for which a parametric nonlinear ROM can be constructed from the HFM. The methodology proposed consists in substituting the deterministic ROB with a stochastic ROB for which the probability measure in constructed on a subset of a compact Stiefel manifold. The stochastic model depends on a small number of hyperparameters for which the identification is performed by solving a statistical inverse problem. An application is presented in nonlinear computational structural dynamics.