Communication Dans Un Congrès Année : 2025

Convolution neural operator preconditioning for the solution of some heterogeneous PDES

Résumé

This work exclusively focuses on using convolution neural operator learning for accelerating the solution of some heterogenous PDEs (including Poisson equations, Darcy flow, Diffusion-Advection equations) using the flexible GMRES [1] method. We use operator learning with U-Net [2] neural network architecture. For the sake of learning general information, the neural operator is trained with randomly generated datasets using an unsupervised approach. The trained neural operator exhibits significant generalization features with respect to different aspects. That includes the ability to address varying source terms, diffusivity terms, velocity field for advection, and varying boundary conditions for these heterogeneous equations. Furthermore, it also shows promising results for addressing wider range of the advection dominant situations, which is the challenging case for the advection-diffusion equations. Overall, this work demonstrates the efficiency of applying the neural operator learning as felxible preconditioner for subspace iterative linear solvers.

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hal-04961884 , version 1 (22-02-2025)

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  • HAL Id : hal-04961884 , version 1

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Yanfei Xiang, Luc Giraud. Convolution neural operator preconditioning for the solution of some heterogeneous PDES. DTE - AICOMAS 2025 - Digital Twins in Engineering & Artificial Intelligence and Computational Methods in Applied Science, Kenjiro Terada, John Dolbow, Ferdinando Auricchio, Harald van Brummelen, Dominique Baillargeat, Charbel Farhat, Georges Karniadakis, Stefan Kollmannsberger, Trond Kvamsdal, Vissarion Papadopoulos, Ruben Sevilla, Feb 2025, Paris, France. ⟨hal-04961884⟩
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