WAN3DNS: WEAK ADVERSARIAL NETWORKS FOR SOLVING 3D INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Wenran LI, Xavier Cadet, BESSAFI MILOUD, Cédric Damour, Yu LI, Alain Miranville, Peter Chin, Rong Yang, Xinguang Yang, Frederic CADET

Keywords: Generative models; Computational Fluid Dynamics

Abstract: The 3D incompressible Navier-Stokes equations model essential fluid phenomena, including turbulence and aerodynamics, but are challenging to solve due to nonlinearity and limited solution regularity. Classical solvers are costly, and neural network-based methods typically assume strong solutions, limiting their use in underresolved regimes. We introduce WAN3DNS, a weak-form neural solver that recasts the equations as a minimax optimization problem, enabling learning directly from weak solutions. Using the weak formulation, WAN3DNS circumvents the stringent differentiability requirements of classical physics-informed neural networks (PINNs) and accommodates scenarios where weak solutions exist, but strong solutions may not. We evaluate WAN3DNS's accuracy and effectiveness on three benchmark cases: the Kovasznay, Beltrami, and 3D lid-driven cavity flows. Furthermore, using Galerkin's theory, we conduct a rigorous error analysis and show that the $L^{2}$-training error is controllably bounded by the architectural parameters of the network and the norm of residues. It implies that for neural networks with a small loss, the corresponding $L^{2}$-error will be small as well. This work bridges the gap between weak solution theory and deep learning, offering a robust alternative for complex fluid flow simulations with reduced regularity constraints.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 11699