Colloques du Collège de France - Collège de France

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Clémentine Prieur : Diffeomorphism-Based Feature Learning Using Poincaré Inequalities

Clémentine Prieur

Professeure, université Grenoble Alpes, LJK, équipe/projet Inria AIRSEA

Résumé

Joint work with Romain Verdière (Inria Grenoble) and Olivier Zahm (Inria Grenoble).

During this talk, I will present a gradient-enhanced algorithm for high-dimensional function approximation which achieves outperforming accuracy on small data sets.

This algorithm, introduced in [1], proceeds in two steps: first, we reduce the input dimension by learning the relevant input features from gradient evaluations and, second, we regress the function output against the pre-learnt features. Specifically, we learn the feature map by minimizing an error bound obtained using Poincaré inequality applied either in input space or in feature space.

This results in two different strategies which we compare both theoretically and numerically, and which we position in relation to existing methods from the literature. In particular, we prove that if we seek the nonlinear feature map as the first components of a C1-diffeomorphism, then our strategy is theoretically guaranteed. Our strategy to learn the C1-diffeomorphism is based on coupling flows, a particular class of invertible neural networks defined as the composition of block-triangular maps.

Finally I will present several numerical experiments to demonstrate that the algorithm we propose outperforms the state-of-the-art competitors in terms of accuracy with little data sets.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Benjamin Peherstorfer : Dirac-Frenkel Dynamics with Momentum

Benjamin Peherstorfer

Associate professor, Courant Institute of Mathematical Sciences, New York University, USA

Résumé

Dirac-Frenkel instantaneous residual minimization evolves nonlinear parametrizations of PDE solutions in time, but ill-conditioning can render the parameter dynamics non-unique. We interpret this non-uniqueness as a gauge freedom: nullspace directions that leave the time derivative unchanged can be used to select better-conditioned parameter velocities. Building on Onsager's minimum-dissipation principle, we introduce a history variable (interpretable as momentum) and inject it only along the nullspace directions. The resulting Dirac-Frenkel-Onsager dynamics preserve instantaneous residual minimization, in contrast to standard regularization that can introduce bias, while promoting temporally smooth parameter evolutions. Examples demonstrate that the approach leads to increased robustness in singular and near-singular regimes.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Andrea Manzoni : Reduced Order Modeling and Scientific Machine Learning: Synergies and Opportunities

Andrea Manzoni

Associate Professor of Numerical Analysis, MOX - Department of Mathematics, Politecnico di Milano, Italy

Résumé

Among several recently proposed data-driven Reduced Order Models (ROMs), deep learning-based ROMs (DL-ROMs) have proved to be a successful strategy to construct non-intrusive, highly accurate surrogates for the real time solution of parametric nonlinear time-dependent PDEs. By relying on (possibly, convolutional) autoencoders, it is indeed possible to generate latent spaces where the candidate solution is then sought, as a function of parameters and time, using an additional neural network.

In this talk I will provide an overview on DL-ROMs, discussing some recent theoretical results that justify their construction, and connecting them to classical reduced basis methods. Then, I will showcase a series of possible extensions of DL-ROMs capable to (i) handle knowledge of physical laws, (ii) deal with varying geometries, (iii) identify the latent dynamics to ensure accurate out-of-training forecasts, and (iv) include uncertainty quantification.

In all these cases, we will show how the construction of a suitably expressive—and possibly explainable—latent space is essential to ensure accuracy and efficiency of reduced order models exploiting deep neural networks, drawing also some conclusions of possible interest to other contexts in scientific machine learning.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Albert Cohen : Optimal Linear and Non-Linear Dimensionality Reduction

Albert Cohen

Professeur, Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, Paris

Résumé

Understanding how to optimally approximate general compact sets by finite dimensional spaces is of central interest for designing efficient numerical methods in forward simulation or inverse problems. The concept of n-width, introduced in 1936 by Kolmogorov, is well tailored to linear approximation methods. The interest for n-width has recently been revived by the approximation of parametrized/stochastic PDEs, and the development of reduced basis methods. We briefly survey some now classical results.

We then focus on analogous concepts for nonlinear approximation which are still the object of current research, motivated in particular by the development of neural networks, and possible applications to hyperbolic parametrized PDEs for which linear methods are not effective. We discuss a general framework that allows to embrace various concepts of linear and nonlinear widths, and present some recent results and relevant open problems within this framework.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Mario Ohlberger : Reduced Order Surrogate Models for PDE-Constrained Optimization and Inverse Problems

Mario Ohlberger

Professor of applied mathematics and managing director of the Institute of Analysis and Numerics, University of Münster, Germany

Résumé

Classically, model order reduction for parameterized systems is based on a so-called offline phase, where reduced approximation spaces are constructed and the reduced parameterized system is built, followed by an online phase, where the reduced system can be cheaply evaluated in a multi-query context.

In this contribution, instead, we follow an active learning or enrichment approach where a multi-fidelity hierarchy of reduced order models is constructed on-the-fly while exploring a parameterized system. To this end we focus on learning-based reduction methods in the context of PDE constrained optimization and inverse problems and evaluate their overall efficiency. We discuss learning strategies, such as adaptive enrichment within a trust region optimization framework as well as a combination of reduced order models with machine learning approaches. Concepts of rigorous certification and convergence will be presented, as well as numerical experiments that demonstrate the efficiency of the proposed approaches.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Beatriz Moya : Avancées en modélisation hybride : vers la transition numérique des territoires

Beatriz Moya

Professeure associée et titulaire d'une chaire junior du projet ITTAI à l'École Nationale Supérieure d'Arts et Métiers (ENSAM), Paris

Résumé

Dans cette présentation, nous discuterons des avancées récentes en matière de jumeaux hybrides et de leurs synergies avec une intelligence artificielle informée par des biais allant de la physique et de la géométrie à la résilience. Des exemples seront présentés dans le contexte de l'évaluation des risques et de la gestion de crise, afin de développer des solutions adaptées fondées sur ces technologies pour renforcer la réactivité des villes et des territoires.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Jörg Fehr : From Latent Space Representations to Practical Surrogate Models for Structural Dynamical Systems

Jörg Fehr

Professor, Institute of Engineering and Computational Mechanics, University of Stuttgart, Germany

Résumé

The simulation and optimization of complex technical systems often require models that are both sufficiently accurate and computationally efficient. In engineering practice, this balance is difficult to achieve: detailed numerical models provide valuable insight, but they are frequently too costly for repeated evaluations, design optimization, uncertainty studies, or real-time applications.

In this contribution, I will discuss how mathematical methods from model order reduction, system identification, and machine learning can be transferred into practical engineering workflows for structural dynamical systems. The focus is not on replacing physics-based models, but on using data-driven latent space representations to construct surrogate models that remain connected to the underlying mechanical problem.

Several strategies are considered, ranging from black-box latent models to structure-aware identification approaches, including port-Hamiltonian formulations. Particular attention is given to practical issues that arise in technical applications: high-dimensional simulation data, limited or noisy training sets, black-box industrial solvers, multi-physics effects, and the need for reliable predictions beyond isolated benchmark examples.

The methods are illustrated using application-oriented examples such as crash simulations, multiphysics disc-brake models, and further structural and fluid-dynamical systems. The aim is to show how recent mathematical developments can support engineers in analyzing, accelerating, and optimizing complex technical systems while maintaining interpretability and physical plausibility.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Sebastian Ares de Parga Regalado : Robust Nonlinear Projection-Based Reduced-Order Models: Comparative Assessment of Closure and Manifold Strategies

Sebastian Ares de Parga Regalado

Postdoctoral researcher at the Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Barcelona, Spain

Résumé

Recent advances in nonlinear model reduction indicate that overcoming linear Kolmogorov limitations requires principled combinations of projection-based approximation and data-driven modeling [3]. In intrusive PROM settings, PROM–ANN introduced latent-space closure reconstruction of truncated modal coordinates [1], while PROM–RBF and PROM–GPR generalized this mechanism through alternative regression operators for the same closure channel [2]. In parallel, projection-compatible nonlinear latent-manifold formulations based on POD-autoencoders (POD-AE), in the spirit of POD-DL-ROM [4], provide an additional pathway to nonlinear approximation while preserving Galerkin/LSPG online dynamics.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque - Olga Mula : Gradient Flows on Neural Network Manifolds

Olga Mula

Professor of Mathematics, University of Vienna, Austria

Résumé

This talk addresses numerical methods for gradient flows in Hilbert spaces based on neural network approximations. The central idea is to represent the solution on a neural network manifold and evolve its parameters in time. At first glance, this approach appears general, elegant, and easy to implement, and it has achieved notable empirical success in machine learning and scientific computing for PDEs. A closer look, however, reveals significant challenges. Developing a proper functional framework that ensures existence of solutions and rigorously connects to practical algorithms raises subtle issues. In this talk, I will present a framework to address these challenges, and show why they are not merely technical obstacles, but rather reflect fundamental aspects of neural approximation.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexity - Angelo Iollo : Collocated Reduced-Order Models for High-Order Schemes: Analysis and Application

Angelo Iollo

Institut de Mathématiques de Bordeaux, université de Bordeaux & équipe-projet commune MONHADE Inria - Onera - université de Bordeaux

Résumé

In this talk, I will present Collocated Model Order Reduction (cMOR), a novel hyper-reduction framework specifically designed for high-order discretizations. The method evaluates the governing operator exclusively on a small subset of collocation points, identified via Non-Negative Least Squares (NNLS) sparse quadrature. By combining a restricted POD basis with carefully constructed prolongation and interpolation operators, cMOR enables sparse yet accurate residual evaluation while preserving key structural properties of the underlying scheme.