Decoupled Divergence-Free Neural Networks Basis Method for Incompressible Fluid Problems

TL;DR

This paper introduces Decoupled-DFNN, a neural network approach that exactly enforces divergence-free velocity fields for incompressible flows, enabling efficient and accurate solutions to fluid dynamics problems.

Contribution

It presents a novel decoupled formulation that separates velocity and pressure solutions, improving computational efficiency while maintaining incompressibility.

Findings

Reduces computational cost compared to existing methods.

Strictly preserves the divergence-free condition.

Employs a sequential solution strategy for velocity and pressure.

Abstract

We propose a decoupled divergence-free neural networks basis (Decoupled-DFNN) method for solving incompressible flow problems, including the Stokes and Navier-Stokes equations. To ensure the divergence free property exactly, the velocity field is represented as the curl of a stream function in two dimensions and as the curl of a vector potential in three dimensions. Beyond classical stream-function or velocity-vorticity formulations, we further utilize the properties of the curl operator to derive two specific decoupled subproblems for the velocity (through the stream function or vector potential) and the pressure, respectively. The proposed formulations enable a sequential solution strategy, in which the velocity and pressure are solved independently. To resolve the inherent nonlinearity of the Navier-Stokes equations, we employ a Gauss-Newton linearization strategy, transforming the nonlinear velocity subproblem into a sequence of linear subproblems. These decoupled subproblems for velocity and pressure are subsequently solved using the TransNet framework. Compared with existing methods, the proposed approach reduces computational cost while strictly preserving the incompressibility constraint.