Neural Networks-based Random Vortex Methods for Modelling Incompressible Flows

TL;DR

This work tackles 2D incompressible Navier–Stokes modeling by learning the vorticity field $\omega^{\theta}$ with a neural network and recovering velocity through a Poisson solve $-\Delta_x u^{\theta} = \nabla_x \wedge \omega^{\theta}$, thereby enforcing incompressibility and no-slip boundaries without requiring the Biot–Savart kernel. By grounding the training loss in the stochastic Random Vortex Dynamics representation, the method links NN optimization to the physical dynamics while deferring the velocity computation to a traditional elliptic solver. The authors demonstrate that this vorticity-first approach can achieve accurate velocity and vorticity fields, handle complex domains, and readily incorporate measurement data via data assimilation. Overall, the approach provides a physically consistent, kernel-free, and data-augmentable framework for NN-based CFD in two dimensions with potentially broad applicability to domains where the Biot–Savart kernel is unknown.

Abstract

In this paper we introduce a novel Neural Networks-based approach for approximating solutions to the (2D) incompressible Navier--Stokes equations, which is an extension of so called Deep Random Vortex Methods (DRVM), that does not require the knowledge of the Biot--Savart kernel associated to the computational domain. Our algorithm uses a Neural Network (NN), that approximates the vorticity based on a loss function that uses a computationally efficient formulation of the Random Vortex Dynamics. The neural vorticity estimator is then combined with traditional numerical PDE-solvers, which can be considered as a final implicit linear layer of the network, for the Poisson equation to compute the velocity field. The main advantage of our method compared to the standard DRVM and other NN-based numerical algorithms is that it strictly enforces physical properties, such as incompressibility or (no slip) boundary conditions, which might be hard to guarantee otherwise. The approximation abilities of our algorithm, and its capability for incorporating measurement data, are validated by several numerical experiments.

Figures (7)

• Figure 1.1: Training cycle of our NN-based algorithm. To simulate the Navier--Stokes equations, these training cycles are repeated several times until $\omega^{\theta}$ can be expected to sufficiently approximate $\omega$.
• Figure 7.1: The loss function at different times $t$.
• Figure 7.2: The fully-bounded flow \ref{['velocity_NE1']}: the first row represents the approximation at different times $t$; the second row contains the true solution at the times; the third row represents the absolute l2-error of the approximation.
• Figure 7.3: The fully-bounded flow \ref{['velocity_NE1']}: the first row contains the vorticity learned at different times $t$; the second represents the true vorticity at the times.
• Figure 7.4: The relative l2 error with and without data assimilation as function of time $t$.

Theorems & Definitions (6)

• Remark 1
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• Remark 4
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• Remark 6