Réseaux de neurones informés par la physique pour la résolution d’équations de mécanique des fluides

Abstract

This thesis explores the application of Physics-Informed Neural Networks (PINNs) to moving interface problems using the level set method, as well as to free surface flows governed by the Shallow Water Equations (SWE). Specifically, we highlight the performance of the PirateNet architecture, which integrates improvements such as sequence-to-sequence training, random weight factorization, gradient-norm-based loss term balancing, causal training, and the incorporation of random Fourier features. For moving interface problems, the enhanced PINN demonstrated superior capabilities in solving Zalesak’s disk and a complex case involving significant interface deformation, the time-reversed vortex flow, achieving an error of L2 = 0.81% in the latter. Unlike classical numerical methods, PINNs can capture interface evolution without the need for upwind numerical stabilization or geometric reinitialization. Furthermore, although the addition of an Eikonal-based regularization term may improve results under certain conditions, it must be carefully weighted to avoid adverse effects. Regarding the Shallow Water Equations, the enhanced PINN outperformed the standard PINN by accurately predicting free surface evolution, including in the presence of variable bathymetry and interfering waves. An error of L2 ℎ = 0.30% was achieved for the 1D wave propagation over a hump test case, while maintaining very low mass loss and accurately capturing the wave speed. However, despite reliable height prediction, PINNs struggle to model velocity fields accurately in regions with strong gradients. Numerical viscosity can help reduce this error, but it significantly increases computational cost, especially in 2D. A key finding of this study is that, despite their promising performance, PINNs remain substantially more computationally expensive than traditional numerical methods. While inference is fast, the initial model training requires significant time, limiting their practicality for direct PDE problems. In summary, this thesis demonstrates that the enhanced PINN equipped with the discussed improvements represents a meaningful advancement in applying PINNs to moving interface problems and free surface flows. While further development is needed to fully match the precision of classical numerical methods, PINNs offer an appealing alternative in contexts where real data can be integrated to refine predictions and improve physical modeling.

Date	22 Apr 2025
Original language	French
Awarding Institution	École de technologie supérieure
Supervisor	Azzeddine Soulaïmani (Supervisor)