PDE-AI

Numerical analysis, optimal control and optimal transport for AI : new architectures for machine learning

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The aim is to develop mathematical analysis methods (numerical analysis, control, optimal transport) for the study and the improvement of artificial intelligence systems, and to develop rigorous, reliable AI for numerical analysis and simulation.

Antonin Chambolle, Research director at CNRS

The main aim of the FOUNDRY project is to support the creation of a group of applied mathematicians specializing in machine learning issues, and to stimulate the cross-over between mathematical analysis, optimal control and optimal transport, leading to new architectures for machine learning models. Three main themes are being investigated: a first theme focuses on the analysis of learning methods, a second on new deep architectures (where specific architectures for numerical simulation will be studied in particular), and a final research angle focusing on generative methods and diffusion from an analytical point of view.

Keywords : Nonlinear analysis, partial differential equations (PDE), numerical analysis, control, optimization, machine learning, neural networks

Project web site : https://pde-ai.math.cnrs.fr/

Missions

Our researches


Study and better understand neural network training dynamics and top-down gradient optimizations chemes

Study the various networks (e.g. residuals, transformers) seen as approximations to Partial Differential Equations (PDEs) that emerge as mean-field equations for neural networks with a large number of neurons. Work towards a better understanding of learning dynamics, based on gradient flows


Design new deep architectures, based on the analysis of PDEs, including the development/analysis of AI techniques for simulation and numerical analysis

Analyze the approximation and stability properties of networks with techniques drawn from the analysis of PDE discretization schemes, while developing the use of AI for numerical simulation


Study generative methods and diffusion from an analytical point of view, exploiting the analogy with dynamical systems and partial differential equations

Understand the ability of generative networks to model disconnected or multimodal distributions, in order to define statistical bounds on the optimal transport distance to the target distribution. In addition to theoretical guarantees, new sampling strategies can be developed based on gradient stream theory for particle ensembles, in optimal transport metrics

Consortium

Université Paris Sciences et Lettres, Université Paris-Dauphine, CNRS, INRIA (Nice), Université Paris-Cité, Sorbonne Université, Université Paris-Saclay, Institut National Polytechnique de Toulouse, Université de Bordeaux, CREST-GENEST (ENSAE), Université de Strasbourg, Université Paris 1 Panthéon-Sorbonne, Ecole Nationale de l’aviation civile, Université Toulouse Capitole, Université Toulouse 3 Paul Sabatier, Université de Lyon 1, INSA Lyon, Université de Côte d’Azur

Consortium location

Publication


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