Wenzhuo Liu, doctorante dans le projet HSA de l’IRT SystemX, a soutenu sa thèse de l’Université Paris-Saclay le 19 avril 2023 sur le sujet suivant : « Deep Graph Neural Networks for Numerical Simulation of PDEs ».

Cette Thèse est co-encadrée par Mouadh Yagoubi (IRT SystemX) et Marc Schoenauer, directeur de thèse (Inria).

Résumé de la thèse :

Partial differential equations (PDEs) are an essential modeling tool for the numerical simulation of complex systems. However, their accurate numerical resolution usually requires a high computational cost. In recent years, deep Learning algorithms have demonstrated impressive successes in learning from examples, and their direct application to databases of existing solutions of a PDE could be a way to tackle the excessive computational cost of classical numerical approaches: Once a neural model has been learned, the computational cost of inference of the solution on new example is very low. However, many issues remain that this Ph.D. thesis investigates, focusing on three major hurdles: handling unstructured meshes, which can hardly be done accurately by simply porting the neural successes on image processing tasks; generalization issues, in particular for Outof-Distribution examples; and the too high computational costs for generating the training data. We propose three contributions, based on Graph Neural Networks, to tackle these problems: A hierarchical model inspired by the multi-grid techniques of Numerical Analysis; The use of Meta-Learning to improve the performance of Out-of-Distribution data; and Transfer Learning between multi-fidelity datasets to reduce the computational cost of data generation. The proposed approaches are experimentally validated on different physical systems.

Keywords : Deep learning, Graph Neural Networks, PDEs

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