Colloques du Collège de France - Collège de France

Odcinki

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  Colloque - Karen E. Willcox : Multifidelity Proper Orthogonal Decomposition 24.06.2026 47min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Karen E. Willcox : Multifidelity Proper Orthogonal DecompositionKaren E. WillcoxProfessor, Director of Oden Institute, University of Texas at Austin, USARésuméThe 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.
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  Colloque - Tommaso Taddei : Registration in Bounded Domains for Model Reduction of Parametric Conservation Laws 24.06.2026 37min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Tommaso Taddei : Registration in Bounded Domains for Model Reduction of Parametric Conservation LawsTommaso TaddeiAssociate Professor of Numerical Analysis, Department of Mathematics Guido Castelnuovo, Sapienza University of Rome, ItalyRésuméIn 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.
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  Colloque - David Ryckelynck : Self Supervised Machine Learning of ROM-nets for Mechanics of Materials 24.06.2026 38min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - David Ryckelynck : Self Supervised Machine Learning of ROM-nets for Mechanics of MaterialsDavid RyckelynckProfesseur à Mines Paris – PSLRésuméWe 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.
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  Colloque - 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 24.06.2026 28min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : 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 EquationsÉlise GrosjeanEnseignante-chercheuse Inria, Équipe IDEFIX de l'Unité de Mathématiques Appliquées, ENSTA, Institut Polytechnique de ParisRésuméDuring 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.
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  Colloque - Kathrin Smetana : Certified Randomized Model Order Reduction Methods for High-Dimensional Approximation 24.06.2026 38min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Kathrin Smetana : Certified Randomized Model Order Reduction Methods for High-Dimensional ApproximationKathrin SmetanaTenure-track Assistant professor in the Department of Mathematical Sciences at the Stevens Institute of Technology, Hoboken, New Jersey, USARésuméIn 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.
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  Colloque - Geneviève Dusson : Metric-Based Nonlinear Model Order Reduction with Applications to Quantum Chemistry 24.06.2026 40min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Geneviève Dusson : Metric-Based Nonlinear Model Order Reduction with Applications to Quantum ChemistryGeneviève DussonChargée de recherche, CNRS, Laboratoire de mathématiques de Besançon, université Franche-ComtéRésuméA 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.
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  Colloque - Anthony Nouy : Stable Nonlinear Manifold Approximation Using Compositional Networks 24.06.2026 44min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Anthony Nouy : Stable Nonlinear Manifold Approximation Using Compositional NetworksAnthony NouyProfesseur au département de mathématiques, Centrale Nantes – Nantes UniversitéRésuméWe 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.
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  Colloque - Ludovic Chamoin : Integrated Structural Health Monitoring with Real-time Data Assimilation and Hybrid Twins 23.06.2026 49min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : 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 ChamoinProfesseur, LMPS, ENS Paris-SaclayRésuméThe 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.
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  Colloque - Evie Nielen : Efficient Greedy Sampling for Model Order Reduction 23.06.2026 32min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Evie Nielen : Efficient Greedy Sampling for Model Order ReductionEvie NielenDoctoral Candidate, Mathematics and Computer Science, Computational Science, University of Technology Eindhoven, NetherlandsRésuméThis 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.
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  Colloque - Helin Gong : AI-Driven Complexity Reduction and Multi-Physics Digital Twins: From Theory to Engineering Implementation in Nuclear Reactors 23.06.2026 32min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : 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 ReactorsHelin GongAssociate 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.
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  Colloque - Clémentine Prieur : Diffeomorphism-Based Feature Learning Using Poincaré Inequalities 23.06.2026 39min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Clémentine Prieur : Diffeomorphism-Based Feature Learning Using Poincaré InequalitiesClémentine PrieurProfesseure, université Grenoble Alpes, LJK, équipe/projet Inria AIRSEARésuméJoint 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.
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  Colloque - Benjamin Peherstorfer : Dirac-Frenkel Dynamics with Momentum 23.06.2026 41min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Benjamin Peherstorfer : Dirac-Frenkel Dynamics with MomentumBenjamin PeherstorferAssociate professor, Courant Institute of Mathematical Sciences, New York University, USARésuméDirac-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.
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  Colloque - Andrea Manzoni : Reduced Order Modeling and Scientific Machine Learning: Synergies and Opportunities 23.06.2026 51min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Andrea Manzoni : Reduced Order Modeling and Scientific Machine Learning: Synergies and OpportunitiesAndrea ManzoniAssociate Professor of Numerical Analysis, MOX - Department of Mathematics, Politecnico di Milano, ItalyRésuméAmong 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.
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  Colloque - Albert Cohen : Optimal Linear and Non-Linear Dimensionality Reduction 23.06.2026 55min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Albert Cohen : Optimal Linear and Non-Linear Dimensionality ReductionAlbert CohenProfesseur, Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, ParisRésuméUnderstanding 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.
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  Colloque - Mario Ohlberger : Reduced Order Surrogate Models for PDE-Constrained Optimization and Inverse Problems 22.06.2026 40min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Mario Ohlberger : Reduced Order Surrogate Models for PDE-Constrained Optimization and Inverse ProblemsMario OhlbergerProfessor of applied mathematics and managing director of the Institute of Analysis and Numerics, University of Münster, GermanyRésuméClassically, 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.
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  Colloque - Beatriz Moya : Avancées en modélisation hybride : vers la transition numérique des territoires 22.06.2026 35min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Beatriz Moya : Avancées en modélisation hybride : vers la transition numérique des territoiresBeatriz MoyaProfesseure associée et titulaire d'une chaire junior du projet ITTAI à l'École Nationale Supérieure d'Arts et Métiers (ENSAM), ParisRésuméDans 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.
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  Colloque - Jörg Fehr : From Latent Space Representations to Practical Surrogate Models for Structural Dynamical Systems 22.06.2026 38min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Jörg Fehr : From Latent Space Representations to Practical Surrogate Models for Structural Dynamical SystemsJörg FehrProfessor, Institute of Engineering and Computational Mechanics, University of Stuttgart, GermanyRésuméThe 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.
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  Colloque - Sebastian Ares de Parga Regalado : Robust Nonlinear Projection-Based Reduced-Order Models: Comparative Assessment of Closure and Manifold Strategies 22.06.2026 40min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : 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 StrategiesSebastian Ares de Parga RegaladoPostdoctoral researcher at the Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Barcelona, SpainRésuméRecent 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.
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  Colloque - Olga Mula : Gradient Flows on Neural Network Manifolds 22.06.2026 37min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque - Olga Mula : Gradient Flows on Neural Network ManifoldsOlga MulaProfessor of Mathematics, University of Vienna, AustriaRésuméThis 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.
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  Colloque - Angelo Iollo : Collocated Reduced-Order Models for High-Order Schemes: Analysis and Application 22.06.2026 38min

  Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexity - Angelo Iollo : Collocated Reduced-Order Models for High-Order Schemes: Analysis and ApplicationAngelo IolloInstitut de Mathématiques de Bordeaux, université de Bordeaux & équipe-projet commune MONHADE Inria - Onera - université de BordeauxRésuméIn 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.