Gaussian process modeling of transient mechanical random fields: a complete study from simulation to identification – GAME

Although the diverse uncertainty quantification tools they provide, there are still settings where Gaussian processes would be relevant but are not necessarily explored. This project focuses on simulation and identification problems where random fields are inputs in mechanical systems. Our contributions are mainly fivefold. First, we seek to encode expert knowledge into kernels modeling mechanical fields. For instance, we consider applications describing spatial flows in porous media and wear topographies in automotive brake pads, both applications referring to major current issues in engineering related to the reduction of costs and nuisances. There, since datasets are often not rich enough to allow capturing the dynamics of simulations, the choice of the kernel is key to avoid model misspecification and to reduce predictions errors. We propose to construct new kernels relying on dynamical principles (e.g. by Darcy's law) and/or properties from the kernel design theory (e.g. composite kernels). The resulting models can then be used to generate a large number of virtual experiments to be processed through machine learning techniques aiming to characterize other random events (e.g. the injection accidents in component manufacturing). Second, we focus on mechanical systems where inputs are fields, and outputs are functions, more precisely, time-series. For instance, we consider outputs describing acoustic pressures when modeling friction-induced vibrations. To the best of our knowledge, there are no previous works on this topic. We also address the case where inputs are probability distribution functions. This latter case is inspired by a mechanical application where different experimental tests with the same input parametrization can lead to different output profiles. There, we propose to associate an empirical probability distribution to the set of experimental realizations with the same input parametrization. The resulting probability distribution is then passed as the input of the mechanical system. This implies further investigating proper distances over probability distributions and Gaussian process models with distribution inputs and functional outputs. Third, we provide theoretical guarantees of estimators for Gaussian processes with (multivariate or distribution) functional inputs, guarantees that can be used for the certification of the proposed Gaussian process frameworks. Fourth, we tackle the inverse problem of identifying parameters associated to mechanical fields from measurable quantities. This identification task allows characterizing pessimistic scenarios of mechanical problems in manufacturing and comfort. In this step, inverse models are fed by virtual simulations generated in the previous steps. Finally, we provide R/Python codes which will be disseminated to the scientific community as part of an open-source toolbox. The successful achievement of our contributions allows modeling random fields in our two mechanical applications within a unified framework where both simulation and identification problems are addressed.

The ANR JCJC GAME project proposes both innovative theoretical and numerical contributions that are of interest in research fields such as applied mathematics, machine learning and mechanics (but not limited to them). Our theoretical contributions are based on Gaussian process models with (multivariate or distribution) functional inputs and functional outputs. Since they may be independent of the mechanical applications, the resulting models can then be used in other research fields (e.g. coastal engineering). The extensive numerical tests proposed throughout this project will lead to further recommendations that can be useful in mechanics and other engineering domains.