MMGP: a Mesh Morphing Gaussian Process-based machine learning method for regression of physical problems under non-parameterized geometrical variability

When learning simulations for modeling physical phenomena in industrial designs, geometrical variabilities are of prime interest. For parameterized geometries, classical regression techniques can be successfully employed. However, in practice, the shape parametrization is generally not available in the inference stage and we only have access to a mesh discretization. Learning mesh-based simulations is challenging and most of the recent advances have been relying on deep graph neural networks in order to overcome the limitations of standard machine learning approaches. While graph neural networks have shown promising performances, they still suffer from a few shortcomings, such as the need of large datasets or their inability to provide predictive uncertainties out of the shelf. In this work, we propose a machine learning method that do not rely on graph neural networks. Complex geometrical shapes and variations with fixed topology are dealt with using well-known mesh morphing onto a common support, combined with classical dimensionality reduction techniques and Gaussian processes. The proposed methodology can easily deal with large meshes, without knowing any parametrization describing the shape, and provide predictive uncertainties, which are of primary importance for decision-making. In the considered numerical experiments, the proposed method is competitive with respect to our implementation of graph neural networks, regarding either efficiency of the training and accuracy of the predictions.