The Local Deep Galerkin Method Applied to the (2+1)-Dimensional Navier-Stokes and Cahn-Hilliard Equations

Résumé fourni par la source

Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative approach to numerically solving Partial Differential Equations. This study examines the Local Deep Galerkin Method (LDGM) applied to two benchmark systems: the (2+1)-dimensional Cahn-Hilliard (2D-CH) equation, modeling phase separation phenomena such as biofilm dynamics, and the (2+1)-dimensional incompressible Navier-Stokes equations (2D-NS). The LDGM is trained on a loss function that minimizes the sum of the squares of the residuals of the system of equations. The LDGM is compared to a numerical simulation based on the Finite Element Method for the 2D-CH and the analytical solution of the Taylor-Green Vortex for the 2D-NS. Results indicate that LDGM struggles to capture high-frequency initial conditions in the 2D-CH system, yielding errors exceeding 80%. However, when applied to the Taylor-Green vortex problem, LDGM demonstrates promising performance in predicting velocity fields. These findings highlight both the potential and limitations of Deep Galerkin Methods for complex fluid systems, with implications for future work in turbulence modeling and multi-phase flow in aerospace contexts. Additionally, the mesh-free nature of Deep Galerkin Methods offers promise for simulations involving complex geometries where traditional CFD methods struggle.

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