Project and Generate: Divergence-Free Neural Operators for Incompressible Flows

TL;DR

This paper introduces divergence-free neural operators that enforce incompressibility in fluid simulations, ensuring physically consistent and stable results by integrating spectral projections and divergence-free probability measures.

Contribution

It presents a unified framework combining spectral Leray projection and divergence-free Gaussian measures to enforce incompressibility in neural fluid models, improving stability and physical accuracy.

Findings

Exact incompressibility achieved up to discretization error

Enhanced stability and physical consistency in simulations

Effective projection methods for deterministic and generative models

Abstract

Learning-based models for fluid dynamics often operate in unconstrained function spaces, leading to physically inadmissible, unstable simulations. While penalty-based methods offer soft regularization, they provide no structural guarantees, resulting in spurious divergence and long-term collapse. In this work, we introduce a unified framework that enforces the incompressible continuity equation as a hard, intrinsic constraint for both deterministic and generative modeling. First, to project deterministic models onto the divergence-free subspace, we integrate a differentiable spectral Leray projection grounded in the Helmholtz-Hodge decomposition, which restricts the regression hypothesis space to physically admissible velocity fields. Second, to generate physically consistent distributions, we show that simply projecting model outputs is insufficient when the prior is incompatible. To address this, we construct a divergence-free Gaussian reference measure via a curl-based pushforward, ensuring the entire probability flow remains subspace-consistent by construction. Experiments on 2D Navier-Stokes equations demonstrate exact incompressibility up to discretization error and substantially improved stability and physical consistency.