MATH-AmSud NeuralFEM (2026-2027)

Overview

Title

Optimal and certified NEURAL network predictions thanks to tools coming from Finite Element Methods

International coordinator

Michel Duprez

Funding

MATH-AmSud regional programme

Budget

€48.5k

Abstract

Neural networks offer speed and flexibility for approximating the solutions of parametrised partial differential equations: once trained, they predict in near real time, as some intra-operative guidance applications require. But nothing guarantees the accuracy of these predictions, which can occasionally be completely wrong — a major obstacle wherever the stakes are critical, as in biomedical engineering.

NeuralFEM brings to these networks the a priori and a posteriori analysis tools developed for finite elements, so as to obtain schemes that are both optimised and certified. The project targets non-smooth and nonlinear problems — variational inequalities arising from contact and plasticity, fractional PDEs modelling anomalous diffusion — for which classical methods reach their limits, particularly in the presence of steep gradients, singularities or discontinuities.

The project’s second objective is to strengthen the ties between the scientific communities of Uruguay, Colombia, Chile and France: collaboration on specific problems, co-supervision of students, and organisation of meetings. A workshop bringing together all members will be held in France in the first year, and a second in Montevideo in the second.

Work packages

  • T1 — Methods: design of networks for nonlinear and non-smooth problems (fractional operators, contact), including hybrid approaches combining finite elements and networks.
  • T2 — A priori analysis: mathematical study of the schemes from T1, function spaces suited to approximation by networks, convergence of gradient descent algorithms.
  • T3 — A posteriori analysis: error estimators inspired by finite elements to drive adaptive strategies, with proofs of reliability and efficiency.
  • T4 — Reduced bases: implementation and evaluation of ROM approaches on the same problems, for comparison with neural methods.
  • T5 — Applications: biomechanics and biomedicine — tissue compression, stent–artery interaction, anomalous diffusion in tissue.

The implementations will build on FEniCSx and PyTorch, and will be released as open access.

Partners

The consortium brings together 16 researchers from five countries.

Uruguay — Universidad de la República, Montevideo
Franz Chouly (national coordinator), Juan Pablo Borthagaray

Colombia — Universidad Nacional de Colombia and Universidad EAFIT, Medellín
Manuela Bastidas (national coordinator), Diego Alejandro Muñoz, Nicolás Guarín-Zapata

Chile — Universidad de Concepción, Universidad de Santiago de Chile, Universidad del Bío-Bío
Rodolfo Araya (national coordinator), Patrick Vega, Jorge Aguayo

France — Inria, ENPC-CERMICS, École Polytechnique
Michel Duprez (international coordinator, MIMESIS team), Pablo Alvarez, Stéphane Cotin, Raphaël Bulle, Martin Genet, Alexandre Ern, Virginie Ehrlacher (national coordinator)