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

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.

• The paper introduces Decoupled-DFNN, which represents velocity through stream or vector potentials and removes pressure with the curl operator, enforcing incompressibility to machine precision.
• The method solves sequential least-squares subproblems with randomized extreme-learning-machine features, reducing runtime by more than 60% in 2D and roughly half in 3D compared with coupled neural solvers.
• The experiments show up to four orders of magnitude lower errors than PINNs for Stokes flow and strong low-viscosity performance, while requiring higher regularity and good initialization for nonlinear Navier–Stokes problems.

Overview and motivation

This paper by Cheng, Huang, Wang, and Zhou (2603.17906) introduces the Decoupled-DFNN method, a randomized neural network basis approach for the steady Stokes and Navier–Stokes equations in two and three dimensions. The method addresses two persistent weaknesses of neural PDE solvers for incompressible flow: the approximate (penalty-based) enforcement of the divergence-free constraint, and the computational burden of solving fully coupled velocity–pressure systems. The central idea is to represent the velocity as the curl of a potential — a stream function in 2D, a vector potential in 3D — so that $\nabla \cdot \mathbf{u} = 0$ holds identically, and then to apply the curl operator to the momentum equation to eliminate pressure, yielding single-field subproblems that can be solved sequentially.

The solver infrastructure builds on TransNet, a single-hidden-layer extreme learning machine (ELM) with hidden weights fixed via a uniform-distribution initialization strategy, so training reduces to a linear least-squares problem over output-layer coefficients. Nonlinearity in Navier–Stokes is handled by Gauss–Newton linearization at the continuous level.

Mathematical formulation

The derivation rests on classical Helmholtz-type decompositions: for simply-connected Lipschitz domains, a divergence-free field with zero boundary flux is exactly the curl of a stream function (2D) or a vector potential with vanishing curl on the boundary (3D). Applying curl to the momentum equation removes the pressure gradient. Two new identities for the nonlinear term are established:

• In 2D, with $\mathbf{u} = \nabla^\perp \phi$, the curl of the advection term satisfies $$\nabla \times ((\mathbf{u}\cdot\nabla)\mathbf{u}) = -(\nabla^\perp\phi \cdot \nabla)\Delta\phi$$.
• In 3D, with $\mathbf{u} = \nabla\times\boldsymbol{\phi}$ and $\nabla\cdot\boldsymbol{\phi}=0$, it satisfies $$\nabla \times ((\mathbf{u}\cdot\nabla)\mathbf{u}) = (\Delta\boldsymbol{\phi}\cdot\nabla)\boldsymbol{\phi} - (\nabla\times\boldsymbol{\phi}\cdot\nabla)\Delta\boldsymbol{\phi}$$.

These yield decoupled velocity subproblems: a biharmonic equation in $\phi$ for Stokes, and its fourth-order nonlinear counterpart for Navier–Stokes, closed by no-slip conditions translated into clamped-type conditions on the potential ($\phi = \partial_n\phi = 0$ in 2D; $\nabla\cdot\boldsymbol{\phi} = 0$, $\nabla\times\boldsymbol{\phi} = \mathbf{0}$, $\mathbf{u} = \nabla^\perp \phi$0 in 3D). Once the velocity is known, pressure is recovered from a separate gradient problem with a pointwise gauge $\mathbf{u} = \nabla^\perp \phi$1. The authors note this recovers a classical solution, unlike weak formulations common in prior stream-function work, and that the 3D formulation extends the "pure" stream-function approach beyond what earlier vorticity–vector-potential methods (which couple potential and vorticity) achieve.

Two assumptions deserve emphasis: the derivations require $\mathbf{u} = \nabla^\perp \phi$2 and potentials in $\mathbf{u} = \nabla^\perp \phi$3, i.e., higher regularity than weak formulations demand; and the framework is stated for steady problems on simply-connected domains, though the authors assert straightforward extension to unsteady cases without demonstrating it.

Discretization and complexity

Each subproblem is collocated at interior and boundary points and solved as an over-determined linear least-squares system using NumPy routines; all derivative matrices are assembled via automatic differentiation of the fixed random features. For Navier–Stokes, the Fréchet derivative of the advection operator produces a Gauss–Newton iteration whose Jacobian blocks are expressed compactly with Hadamard products; three Picard-style linearization schemes from prior work serve as initializers or alternatives.

Because velocity and pressure are solved sequentially rather than jointly, the least-squares matrices are smaller than those of coupled TransNet. Per-iteration cost scales as $\mathbf{u} = \nabla^\perp \phi$4 plus $\mathbf{u} = \nabla^\perp \phi$5 in 2D versus $\mathbf{u} = \nabla^\perp \phi$6 for coupled TransNet, with analogous savings in 3D. This is the structural source of the roughly twofold runtime advantage observed empirically.

Numerical results

Experiments run on a laptop-class machine compare Decoupled-DFNN against coupled TransNet and PINN baselines across manufactured solutions in 2D and 3D.

Stokes problems. With $\mathbf{u} = \nabla^\perp \phi$7, divergence errors reach at least $\mathbf{u} = \nabla^\perp \phi$8 in 2D and $\mathbf{u} = \nabla^\perp \phi$9 in 3D — effectively machine precision — versus penalty-based enforcement in TransNet/PINN. Against PINN, Decoupled-DFNN achieves velocity and pressure errors four orders of magnitude smaller while PINN's divergence error stalls near $$\nabla \times ((\mathbf{u}\cdot\nabla)\mathbf{u}) = -(\nabla^\perp\phi \cdot \nabla)\Delta\phi$$0; wall-clock time is about 2 seconds versus roughly 180 seconds for PINN. In low-viscosity regimes ($$\nabla \times ((\mathbf{u}\cdot\nabla)\mathbf{u}) = -(\nabla^\perp\phi \cdot \nabla)\Delta\phi$$1), velocity accuracy exceeds TransNet by nearly two orders of magnitude, indicating robustness at high Reynolds number. A notable concession appears in 3D: TransNet attains better pressure accuracy by about one order of magnitude for large basis sizes, so the accuracy advantage is not uniform across all fields.

Navier–Stokes problems. Since both TransNet and Decoupled-DFNN use the same PINN-derived initial guess and Newton-type iterations, velocity and pressure accuracies are comparable; the decisive gains are computational and structural: more than 60% reduction in runtime per solve in 2D, roughly half the cost of TransNet in 3D, and strict preservation of incompressibility throughout the iteration. Accuracy improves further as viscosity decreases below $$\nabla \times ((\mathbf{u}\cdot\nabla)\mathbf{u}) = -(\nabla^\perp\phi \cdot \nabla)\Delta\phi$$2 in 3D. The authors attribute the generally weaker Navier–Stokes performance relative to Stokes to initial-guess quality in the Gauss–Newton loop — an acknowledged limitation, since local convergence makes the method dependent on a good starting iterate supplied here by a PINN pre-run or Scheme I iterations.

Limitations and open questions

Three caveats bear directly on interpreting the results. First, exact divergence-freeness is purchased with fourth-order regularity requirements on the potential; whether the method degrades on solutions with reduced regularity is not examined. Second, the theory and experiments cover simply-connected domains; multiply-connected geometries would require modified potential representations (harmonic components), which are not addressed. Third, the Gauss–Newton strategy is only locally convergent, and the paper itself identifies improved initialization as an open problem. Error analysis for the discrete scheme is also absent; the evidence is entirely empirical.

Conclusion

Decoupled-DFNN combines a structure-preserving potential representation with curl-based decoupling and ELM-style least-squares solving to enforce incompressibility to machine precision while halving the cost of coupled randomized-network solvers. Its strongest empirical claims — divergence errors of $$\nabla \times ((\mathbf{u}\cdot\nabla)\mathbf{u}) = -(\nabla^\perp\phi \cdot \nabla)\Delta\phi$$3, order-of-magnitude accuracy gains over PINN at low viscosity, and ~50–60% runtime reductions — are well supported within the tested settings, provided one accepts the higher-regularity assumption and the reliance on external initial guesses for the nonlinear case.