Colloques du Collège de France - Collège de France

Maria Melchior
Santé publique
Collège de France
Année 2025-2026

Colloque - Mental Health and Addiction - Lisa Berkman: Social Determinants of Health in Adults: a life course perspective on the role of social networks and work policies

Session 4: Social Determinants of Mental Health over the Lifecourse

Lisa Berkman
Professor Harvard University, School of Public Health

Résumé

Social determinants of health play a major role in shaping population health and healthy aging. In this presentation we will focus on conditions in mid life particularly around working conditions and social isolation that lead to healthy population aging. Using a life course approach we focus on those conditions in mid life that predict successful aging in terms of physical and mental health.

Yvon Maday
Chaire Informatique et sciences numériques
Collège de France
Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Karen E. Willcox : Multifidelity Proper Orthogonal Decomposition

Karen E. Willcox
Professor, Director of Oden Institute, University of Texas at Austin, USA

Résumé

The proper orthogonal decomposition (POD) is widely used to compute a low-dimensional basis that underpins a subsequent dimension reduction or reduced-order modeling step. POD is data-driven in the sense that it requires a training data set of high-fidelity solutions, typically referred to as snapshots. For many complex scientific applications, the computational cost of generating these snapshots is prohibitive, especially when their generation requires sampling over a high-dimensional parameter space. This talk presents a multifidelity POD (mfPOD) formulation that leverages cheaper, lower-fidelity snapshots to reduce the computational cost of computing the POD basis. MFPOD then weights high- and low-fidelity snapshot data via a control-variate formulation to guarantee an unbiased estimate of the expected high-fidelity least-squares projection error. For restrictive computational budgets, the MFPOD cost function has (under some assumptions) lower variance than the POD cost function, which makes the MFPOD subspace more robust against variations in the training data and thus less prone to overfitting. Numerical results show that mfPOD achieves an order of magnitude in computational speedup, translating into useful gains in large-scale problems. Joint work with Nicole Aretz.

Yvon Maday
Chaire Informatique et sciences numériques
Collège de France
Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Tommaso Taddei : Registration in Bounded Domains for Model Reduction of Parametric Conservation Laws

Tommaso Taddei
Associate Professor of Numerical Analysis, Department of Mathematics Guido Castelnuovo, Sapienza University of Rome, Italy

Résumé

In this talk, I review recent efforts on the development of registration methods for parametric model order reduction (MOR), with emphasis on advection-dominated flows. In computer vision and pattern recognition, registration refers to the process of finding a parametric transformation that aligns two datasets; in model order reduction, registration methods seek a parametric bijection that tracks coherent structures (e.g., shocks, shear layers) of the solution field. The ultimate goal is to enhance performance of traditional linear compression methods (e.g., POD) and mesh adaptation techniques for the mapped solution field.

We discuss the application of registration techniques to model reduction. First, we illustrate the combination of registration with projection-based reduced-order models and parametric mesh adaptation. Second, we discuss the application of registration to nonlinear interpolation. We present numerical results for two- and three-dimensional parametric compressible flows, to show the potential of the method.

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.

Yvon Maday
Chaire Informatique et sciences numériques
Collège de France
Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Élise Grosjean : A Doubly Reduced Approximation for the Solution to PDEs Based on a Domain Truncation and a Reduced Basis Method: Application to Navier-Stokes Equations

Élise Grosjean
Enseignante-chercheuse Inria, Équipe IDEFIX de l'Unité de Mathématiques Appliquées, ENSTA, Institut Polytechnique de Paris

Résumé

During this talk, I will present the NIRB two-grid method, together with recent extensions applied to the Navier–Stokes equations, aimed at further reducing the computational cost of the algorithm. The NIRB two-grid method, introduced in [1], is based on two stages. First, during an offline phase, a reduced basis is constructed from high-fidelity solutions computed on a fine mesh, involving a large number of degrees of freedom, using a standard discretisation technique. Then, during the online phase, the parametric problem is solved on a coarser mesh, and the resulting solution is projected onto the reduced space, thereby substantially decreasing the computational cost.

We extend this framework by further reducing the complexity of the online stage. As a representative application, we consider a classical benchmark problem in fluid mechanics: the two-dimensional Backward-Facing Step (BFS). In particular, we simplify the online computation by (i) using a coarse uniform mesh, rather than refining it near the re-entrant corner, and (ii) significantly truncating the outflow section of the channel. Both choices would typically be regarded as detrimental to the accuracy of a high-fidelity flow representation. To overcome this difficulty, we construct two reduced bases and introduce a deterministic linear mapping that enables the transfer from one basis to the other. Additional numerical simulations, including three-dimensional and time-dependent configurations, demonstrate the efficiency of the proposed approach.