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 - 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.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Kathrin Smetana : Certified Randomized Model Order Reduction Methods for High-Dimensional Approximation

Kathrin Smetana

Tenure-track Assistant professor in the Department of Mathematical Sciences at the Stevens Institute of Technology, Hoboken, New Jersey, USA

Résumé

In this talk, we present randomized methods that provide high-probability guarantees for the accuracy of reduced order approximations of parametric partial differential equations (PDEs) with high-dimensional parameter sets. The underlying philosophy is to combine classical reduced basis and greedy approximation ideas with concentration phenomena and data-dependent sampling to obtain certified approximations in high dimensions.

We first present non-asymptotic error bounds for the Proper Orthogonal Decomposition (POD) under the sole assumption that the parameter-to-solution map is uniformly bounded for almost all parameter values. In contrast to existing results, the leading term in our bounds is governed by the sum of the neglected eigenvalues and scales inversely with the number of samples, thereby allowing one to exploit rapid eigenvalue decay. The resulting estimates are independent of the dimension of the parameter space. Consequently, even a modest number of samples can be sufficient for the empirical POD approximation to perform comparably to the ideal POD constructed from the full parameter distribution, including in infinite-dimensional parameter settings.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Geneviève Dusson : Metric-Based Nonlinear Model Order Reduction with Applications to Quantum Chemistry

Geneviève Dusson

Chargée de recherche, CNRS, Laboratoire de mathématiques de Besançon, université Franche-Comté

Résumé

A broad class of problems in science and engineering involves the repeated solution of partial differential equations (PDEs) for different parameter values. Linear reduced order models are a powerful tool to decrease the computational cost of these simulations by approximating the solutions in a low-dimensional space. They have proven highly effective in many settings; however, they often perform poorly for transport-dominated PDEs, where key solution features such as translations cannot be accurately represented in a linear subspace.

To overcome these limitations, several nonlinear reduced order models have recently been proposed, including approaches based on quadratic or polynomial mappings and neural networks. In this talk, I will present an alternative metric-based approach to nonlinear model order reduction. The central idea is to replace linear combinations in low-dimensional spaces with barycenters taken with respect to a suitably chosen metric, computed from a small number of representative solutions. In particular, I will provide constructions based on the Wasserstein distance from optimal transport, which is well adapted to capturing translations. I will also show how the choice of metric can be adapted to incorporate physical constraints, such as sparsity or prescribed marginals. The proposed methodology will be illustrated through numerical examples involving the approximation of electronic densities and pair densities arising in quantum chemistry.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Anthony Nouy : Stable Nonlinear Manifold Approximation Using Compositional Networks

Anthony Nouy

Professeur au département de mathématiques, Centrale Nantes – Nantes Université

Résumé

We consider the problem of approximating a subset M of a Hilbert space X by a low-dimensional manifold Mn. A large class of nonlinear methods can be described by a decoder D: IRn à X whose range is the nonlinear manifold Mn, and an encoder E : E à IRn which extracts n pieces of information E(u) from an element u in M.

Here, we introduce a nonlinear method where E is linear and D is a stable decoder which is obtained by a tree-structured composition of polynomial maps, estimated sequentially from samples in M. Rigorous error and stability analyses are provided, as well as an adaptive strategy for constructing a decoder which guarantees an approximation of the set M with controlled mean-squared or worst-case errors, and a controlled stability (Lipschitz continuity) of the encoder and decoder pair.

Also, we discuss on the definition of optimal encoders and provide concrete strategies for their estimation.

Joint work with A. Bensalah, J. Soffo, A. Somacal.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Ludovic Chamoin : Integrated Structural Health Monitoring with Real-time Data Assimilation and Hybrid Twins

Ludovic Chamoin

Professeur, LMPS, ENS Paris-Saclay

Résumé

The design of smart autonomous mechanical structures able to perform online control of their integrity, and take anticipated actions during service before downtime or failure occur, has become an active research area. It is a critical need in various industrial sectors (transport, energy, etc.) for more reliability but also more performance and durability of equipment (aircrafts, wind turbines, bridges, etc.). Implementing such an advanced technology would permit optimized maintenance and capability to operate in degraded mode, managing the decrease of loading capabilities by adapting the operating plan.

However, the real-time monitoring of damage in engineering systems, by dynamically coupling predictive simulation tools (in terms of digital twins) and sensor observations, is made very difficult in practice due to several issues. In particular, the complex nonlinear multiscale phenomena which are involved may be associated with computationally intensive simulations (hardly compatible with real-time), which requires reduced order modeling and effective strategies for data assimilation and control. In addition, the problem is plagued with model bias, uncertain environment, and measurement noise, which need to be taken into account for accurate diagnosis and prognosis, and safe decision-making. In this context, an appealing trend is to refer to hybrid twins, in which an a priori physics-guided model is updated and enriched on-the-fly with data-based information, thus making benefit of all knowledge available.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Evie Nielen : Efficient Greedy Sampling for Model Order Reduction

Evie Nielen

Doctoral Candidate, Mathematics and Computer Science, Computational Science, University of Technology Eindhoven, Netherlands

Résumé

This talk presents the Polytope Division Method (PDM), a greedy algorithm for solving high-dimensional configuration optimization problems—such as those arising in model reduction and optimal experimental design—where one seeks an optimal sampling of parameter spaces. Classical approaches like standard greedy sampling rely on fixed training sets and quickly suffer from the curse of dimensionality. PDM replaces global sampling with an adaptive, geometry-driven strategy based on recursive polytope subdivision. At each step, the method evaluates the objective only at samples in dynamically refined regions. This yields a sampling complexity that scales linearly with dimension, avoiding exponential growth. The approach requires no a priori choice of training set size and focuses computational effort where it matters most. Applications to reduced basis methods and empirical interpolation demonstrate strong performance gains. Numerical results show that PDM achieves comparable accuracy to classical methods at significantly lower offline reduced cost.

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Helin Gong : AI-Driven Complexity Reduction and Multi-Physics Digital Twins: From Theory to Engineering Implementation in Nuclear Reactors

Helin Gong

Associate Professor, Paris Elite Institute of Technology, Shanghai Jiao Tong University, Shanghai, China.

Résumé

To meet the rigorous demands of Best-Estimate Plus Uncertainty (BEPU) in modern nuclear engineering, it is essential to characterize safety margins and system dynamics with both high fidelity and high efficiency. Building upon foundational complexity reduction methods—such as the Generalized Empirical Interpolation Method (GEIM) and Reduced Basis methods—this talk presents the recent advancements in applying these mathematical tools to real-world nuclear engineering practices.

By integrating Model Order Reduction (ROM) with Artificial Intelligence (AI) and Data Assimilation, we have developed a data-enabled, physics-informed digital twin framework. This approach effectively resolves high-dimensional multi-physics coupling problems and allows for ultra-real-time state estimation and parameter identification. Furthermore, the presentation will highlight the engineering implementation of these methodologies, demonstrating how theoretical reduced-order models are deployed into industrial software and platform architectures (e.g., AI-Enhanced Digital Twin Engineering Platform) for the online monitoring and predictive simulation of commercial nuclear reactor cores.