φ-FEM-FNO: a new approach to train a Neural Operator as a fast PDE solver for variable geometries

In this paper, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and a new finite element method called φ-FEM. For that, we use the Fourier Neural Operator (FNO), a learning mapping operator. The purpose of this paper is to provide numerical evidence to show the effectiveness of this technique. We will focus here on the resolution of the Poisson equation with non-homogeneous Dirichlet boundary conditions. The key idea of our method is to address the challenging scenario of varying domains, where each problem is solved on a different geometry. The considered domains are defined by level-set functions due to the use of the φ-FEM approach. We will first recall the idea of φ-FEM and of the Fourier Neural Operator. Then, we will explain how to combine these two methods. We will finally illustrate the efficiency of this combination with some numerical results on two test cases, showing in particular that our method is faster than learning-based and finite element solvers for a fixed accuracy.

Mots clés

Partial differential equations Fourier neural operator Finite element method Varying geometry Unfitted method

Domaines

Machine Learning [stat.ML]

Fichier principal

main.pdf (24.33 Mo)

best_prediction_0.000157_1.pdf (36.98 Ko)

best_prediction_0.000177_2.pdf (33.44 Ko)

boxplots_compare_losses_1.pdf (19.7 Ko)

boxplots_compare_losses_L2_phi_FEM_1.pdf (33.36 Ko)

boxplots_new_data_L2_1.pdf (52.62 Ko)

boxplots_new_data_L2_500_2.pdf (22.8 Ko)

boxplots_new_data_compare_method_2.pdf (16.85 Ko)

boxplots_new_data_compare_method_geo_fno_1.pdf (17.09 Ko)

boxplots_new_data_compare_method_std_fem_1.pdf (16.45 Ko)

boxplots_validation_L2_Linf_1.pdf (24.96 Ko)

boxplots_validation_L2_Linf_2.pdf (28.31 Ko)

boxplots_validation_epochs_L2_error_1.pdf (20.17 Ko)

boxplots_validation_epochs_L2_error_2.pdf (22.13 Ko)

compare_computation_times_1.pdf (16.46 Ko)

compare_computation_times_ratio_1.pdf (15.31 Ko)

domain.pdf (8.5 Ko)

error_size_3_methods_2.pdf (17.24 Ko)

error_size_4_methods_1.pdf (16.95 Ko)

error_size_5_methods_1.pdf (18 Ko)

error_time_3_methods_2.pdf (20.63 Ko)

error_time_4_methods_1.pdf (20.25 Ko)

error_time_5_methods_1.pdf (21.66 Ko)

example_output_FEMs_FNO_1.pdf (23.1 Mo)

histograms_new_data_L2_1.pdf (37.74 Ko)

histograms_new_data_L2_500_2.pdf (36.84 Ko)

histograms_validation_L2_1.pdf (34.56 Ko)

histograms_validation_L2_2.pdf (36.35 Ko)

histograms_validation_Linf_1.pdf (35.82 Ko)

histograms_validation_Linf_2.pdf (37.59 Ko)

level_set_domains_216_2.pdf (33.02 Ko)

level_set_domains_241_2.pdf (33.04 Ko)

level_set_domains_39_2.pdf (34.29 Ko)

level_set_domains_965_2.pdf (33.38 Ko)

logo_smai.jpg (3.07 Ko)

losses_1.pdf (51.09 Ko)

losses_2.pdf (51.39 Ko)

masks.pdf (25.87 Ko)

mesh_phi_fem.pdf (122.78 Ko)

min_mean_max_L2_error_epochs_1.pdf (22.4 Ko)

min_mean_max_L2_error_epochs_2.pdf (22.07 Ko)

min_mean_max_L2_error_epochs_new_data_1.pdf (22.42 Ko)

min_mean_max_L2_error_epochs_new_data_2.pdf (22.12 Ko)

min_mean_max_Linf_error_epochs_new_data_1.pdf (22.64 Ko)

min_mean_max_Linf_error_epochs_new_data_2.pdf (22.35 Ko)

multiplot.pdf (107.25 Ko)

tikztest_multiplot_2-figure0.pdf (107.55 Ko)

Origine	Fichiers produits par l'(les) auteur(s)

Michel Duprez : Connectez-vous pour contacter le contributeur

https://hal.science/hal-04473794

Soumis le : jeudi 22 février 2024-17:40:33

Dernière modification le : mercredi 20 novembre 2024-15:26:06

Dates et versions

hal-04473794 , version 1 (22-02-2024)

Identifiants

HAL Id : hal-04473794 , version 1

Citer

Michel Duprez, Vanessa Lleras, Alexei Lozinski, Vincent Vigon, Killian Vuillemot. φ-FEM-FNO: a new approach to train a Neural Operator as a fast PDE solver for variable geometries. 2024. ⟨hal-04473794⟩

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