Colloque - David Ryckelynck : Self Supervised Machine Learning of ROM-nets for Mechanics of Materials

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - David Ryckelynck : Self Supervised Machine Learning of ROM-nets for Mechanics of Materials

David Ryckelynck

Professeur à Mines Paris – PSL

Résumé

We propose a general framework for projection-based model order reduction using self-supervised machine learning [1]. For parametric elliptic equations this approach is theoretically based on Céa's Lemma. The proposed methodology, called ROM-net [2], consists in using deep learning techniques to adapt the reduced-order model to a stochastic input tensor whose nonparametrized variabilities strongly influence the quantities of interest for a given physics problem. In particular, we introduce the concept of dictionary-based ROM-nets, where deep neural networks recommend a suitable local reduced-order model from a dictionary. The dictionary of local reduced-order models is constructed from a clustering of vector subspaces in a Grassmann manifold.

It enables the identification of the local low-dimensional subspace in which the solutions evolve for different input tensors. This methodology is applied to an anisothermal elastoplastic problem in structural mechanics coupled to a stochastic thermal field. When using deep neural networks, the selection of the best reduced-order model for a given thermal loading is 60 times faster than when following the clustering procedure used in the training phase. The implementation of local hyper-reduction schemes using a dictionary-based ROM-net is straightforward. The extension to variational inequalities will be addressed at the end of the lecture.