Self-Supervised Machine Learning of ROM-nets for the Mechanics of Materials

Abstract

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 grounded in Céa’s Lemma. The proposed methodology, called ROM-net [2], involves 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, in which 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 discussed at the end of the lecture.

References

[1] Learning projection-based reduced-order models
D. Ryckelynck, F. Casenave, N. Akkari
Manifold Learning: Model Reduction in Engineering, 9–37, (2024).

[2] Model order reduction assisted by deep neural networks (ROM-net)
T. Daniel, F. Casenave, N. Akkari, D. Ryckelynck
*Advanced Modeling and Simulation in Engineering Sciences* 7 (1), 1–27, (2020).

David Ryckelynck

David Ryckelynck is a full professor at MINES Paris – PSL and a researcher at CEMEF, the Center for Material Forming.
His work focuses on computational mechanics, materials modeling, and the development of advanced numerical methods for complex engineering systems. He is particularly known for his contributions to model order reduction and hyper-reduction techniques, which aim to speed up large-scale nonlinear simulations while preserving their predictive accuracy.
More recently, his research has also explored the links between reduced-order modeling, manifold learning, deep learning, and scientific machine learning, with applications to digital twins and real-time simulation. Beyond his research, he has played an important role in doctoral training in computational mechanics and materials, and contributes to the development of data-driven approaches for scientific engineering.