Project and Generate: Divergence-Free Neural Operators for Incompressible Flows
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arXiv
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| Autores principales: | , , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915891103399936 |
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| author | Li, Xigui Zhang, Hongwei Jiang, Ruoxi Chen, Deshu Lin, Chensen Han, Limei Qi, Yuan Guo, Xin Cheng, Yuan |
| author_facet | Li, Xigui Zhang, Hongwei Jiang, Ruoxi Chen, Deshu Lin, Chensen Han, Limei Qi, Yuan Guo, Xin Cheng, Yuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24500 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Project and Generate: Divergence-Free Neural Operators for Incompressible Flows Li, Xigui Zhang, Hongwei Jiang, Ruoxi Chen, Deshu Lin, Chensen Han, Limei Qi, Yuan Guo, Xin Cheng, Yuan Machine Learning Fluid Dynamics 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. |
| title | Project and Generate: Divergence-Free Neural Operators for Incompressible Flows |
| topic | Machine Learning Fluid Dynamics |
| url | https://arxiv.org/abs/2603.24500 |