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WAN3DNS: Weak Adversarial Networks for Solving 3D Incompressible Navier-Stokes Equations

Wenran Li, Xavier Cadet, Miloud Bessafi, Cédric Damour, Yu Li, Alain Miranville, Peter Chin, Rong Yang, Xinguang Yang, Frederic Cadet

TL;DR

WAN3DNS introduces a weak-form minimax neural solver for the 3D incompressible Navier–Stokes equations, enabling learning directly from weak solutions and avoiding stringent regularity requirements of standard PINNs. The method provides a Galerkin-type error bound that ties the $L^{2}$ training error to residual norms, giving theoretical grounding for empirical performance. Empirically, WAN3DNS delivers superior accuracy on 2D Kovasznay and 3D Beltrami benchmarks and shows robust behavior for the 3D lid-driven cavity, outperforming several baselines including DeepXDE, NSFnets, WAN-Biharmonic, and PINNs in key metrics. The work bridges weak solution theory and deep learning for complex, underresolved fluid flows and releases code at the provided repository for reproducibility and further development.

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. Despite extensive research, the full mathematical understanding of the 3D incompressible Navier-Stokes equations continues to elude scientists, highlighting the depth and difficulty of the problem. 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, allowing 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 evaluated WAN3DNS's accuracy and effectiveness in three benchmark cases: the 2D Kovasznay, 3D 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. This implies that for neural networks with small loss, the corresponding $L^{2}$ error will also be small. 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. Code: https://github.com/Wenran-Li/WAN3DNS

WAN3DNS: Weak Adversarial Networks for Solving 3D Incompressible Navier-Stokes Equations

TL;DR

WAN3DNS introduces a weak-form minimax neural solver for the 3D incompressible Navier–Stokes equations, enabling learning directly from weak solutions and avoiding stringent regularity requirements of standard PINNs. The method provides a Galerkin-type error bound that ties the training error to residual norms, giving theoretical grounding for empirical performance. Empirically, WAN3DNS delivers superior accuracy on 2D Kovasznay and 3D Beltrami benchmarks and shows robust behavior for the 3D lid-driven cavity, outperforming several baselines including DeepXDE, NSFnets, WAN-Biharmonic, and PINNs in key metrics. The work bridges weak solution theory and deep learning for complex, underresolved fluid flows and releases code at the provided repository for reproducibility and further development.

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. Despite extensive research, the full mathematical understanding of the 3D incompressible Navier-Stokes equations continues to elude scientists, highlighting the depth and difficulty of the problem. 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, allowing 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 evaluated WAN3DNS's accuracy and effectiveness in three benchmark cases: the 2D Kovasznay, 3D Beltrami, and 3D lid-driven cavity flows. Furthermore, using Galerkin's theory, we conduct a rigorous error analysis and show that the training error is controllably bounded by the architectural parameters of the network and the norm of residues. This implies that for neural networks with small loss, the corresponding error will also be small. 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. Code: https://github.com/Wenran-Li/WAN3DNS

Paper Structure

This paper contains 32 sections, 1 theorem, 30 equations, 9 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. 3D Navier-Stokes equations
  4. Strong Formulation vs Weak Formulation
  5. Weak Adversarial Networks (WAN)
  6. WAN3DNS
  7. Weak formulation
  8. Loss function and Network Architecture
  9. Error analysis
  10. Numerical Validation
  11. Baselines
  12. What is the strength/weakness?
  13. What were they tested on before?
  14. Why use these 3 baselines for the first example, 2 baseline models for the second example, and just 1 baseline model for the last one?
  15. Experiments
  16. ...and 17 more sections

Key Result

Theorem 1

Let $\mathbf{u} \in H^1$, $\mathbf{u} \in H^2(0, A; H^1_0(\Omega)) \cap L^\infty(0, A; H^2(\Omega))$ be a weak solution of the Navier-Stokes equations. Let $\mathbf{u}_\theta$, $\mathbf{v}_\theta$, $q_\theta$ be functions constructed by the set of network parameters $\theta, \phi$, which are the out where:

Figures (9)

  • Figure 1: Workflow of WAN3DNS. In the figure, the input $x,y,z,t$ are independent variables, the primary network is a fully connected network for generating the velocity $u,v,w$ and pressure $p$. The adversarial network generates a test function $S$ for divergence-free condition and $l,m,n$, the elements of $\mathbf{v}$, the test function for governing equations. The loss function set according to the equations and conditions finally outputs the solution.
  • Figure 2: Results of Kovasznay flow on x-axis direction $u$. First row (from left to right): exact solution and WAN3dNS solution; Second row: Solution of $u$ from NSFnets, WAN-Biharmonic, DeepXDE; Last row: Absolute pointwise error of $u$ from WAN3DNS, NSFnets, WAN-Biharmonic and DeepXDE.
  • Figure 3: Results of the Beltrami flow experiment on $w$, the heatmap when $t=1, z=0$. First row: the exact solution of $w$; Second row: the approximate solution of WAN3DNS, NSFnets and DeepXDE; Third row: the absolute error of WAN3DNS, NSFnets and DeepXDE.
  • Figure 4: Results of Kovasznay flow on y-axis direction $v$. First row (from left to right): exact solution and WAN3dNS solution; Second row: Solution of $v$ from NSFnets, WAN-Biharmonic, DeepXDE; Last row: Absolute pointwise error of $v$ from WAN3DNS, NSFnets, WAN-Biharmonic and DeepXDE.
  • Figure 5: Results of Kovasznay flow on pressure $p$. DeepXDE performs better than WAN3DNS, but the NSFnets totally loss it's effectiveness.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Theorem 1