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Complex computational models are used nowadays in all fields of applied sciences to predict the behaviour of natural, economic and engineering systems. High-fidelity simulators are able to capture more and more realistic features by including multi-scale or multi-physics aspects in their governing equations, which can result in high complexity. Although computer power has attained unprecedented levels, it is still not possible to use brute force approaches such as Monte Carlo simulation for solving uncertainty propagation, sensitivity or optimization problems with those models. This is the reason why surrogate models such as polynomial chaos expansions (PCE) or Gaussian processes (GP), among others, have gained a lot of attention in the past two decades. In parallel, machine learning and in particular deep neural networks have shown tremendous performance in solving dedicated problems such as image classification or natural language processing. This has raised a lot of interest in the uncertainty quantification community, too. In this lecture, we advocate a sound cross-fertilization of the two worlds. The links between surrogate modelling and supervised learning are first underlined. We show that sparse polynomial chaos expansions can be used as a supervised learning technique in a pure data-driven sense. Through various examples, we show that their predicting capabilities are comparable to, or in some cases better, than neural networks and support vector regression, especially in the context of small data sets. In a second part, we address problems with high-dimensional input vectors for which standard PCE or GP techniques cannot be straightforwardly applied. In this context, we devise an optimal coupling strategy between dimensionality reduction techniques borrowed from machine learning and standard surrogate modelling. This allows us to develop surrogates for computational models where inputs are large time series or maps. We illustrate the performance of this approach on examples in heat conduction and wind turbine dynamics. |
