arXiv：cs.LG· Ramansh Sharma, Matthew Lowery, Houman Owhadi, Varun Shankar·· 6 小时前AI 评分39
面向不可压缩流的保性质算子学习：一种保持物理性质的核方法
Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows
AI 导读
研究者提出一种保性质的核算子学习方法，将输入函数映射到保性质核基上的输出函数展开系数，可解析且同时保持不可压缩性、周期性与湍流等物理性质。该方法在2D和3D、层流与湍流不可压缩流问题上，泛化时相对ℓ2误差最多降低六个数量级，训练速度比神经算子快最多五个数量级，并避免神经算子出现的大散度误差。
正文
Abstract:We present a novel property-preserving kernel-based operator learning method for incompressible flows governed by the incompressible Navier--Stokes equations. Traditional numerical solvers incur significant computational costs to respect incompressibility. Operator learning offers efficient surrogate models, but current neural operators fail to exactly enforce physical properties such as incompressibility, periodicity, and turbulence. Our kernel method maps input functions to expansion coefficients of output functions in a property-preserving kernel basis, ensuring that predicted velocity fields \emph{analytically} and \emph{simultaneously} preserve the aforementioned physical properties. We present universal approximation results and worst-case a priori convergence rates for our framework; empirically, the observed convergence rates exceed the pessimistic predictions across most benchmarks, motivating a formulation of more optimistic convergence rates. Another central contribution of this work is a novel computational framework revolving around streaming construction of kernel Gramians and recursive Schur-complement algorithms to solve the large block linear systems arising from property-preserving kernel methods. This kernel-based framework makes operator learning practical at scales up to $10{,}000$ training functions sampled at up to $10{,}000$ spatial locations. We evaluate the method on challenging 2D and 3D, laminar and turbulent, incompressible flow problems. Our method achieves up to six orders of magnitude lower relative $\ell_2$ errors upon generalization and trains up to five orders of magnitude faster compared to neural operators. Moreover, while our method enforces incompressibility analytically, neural operators exhibit large divergence errors. Our results show that our method provides an accurate and efficient surrogate for incompressible flows.
| Subjects: | Fluid Dynamics (physics.flu-dyn); Machine Learning (cs.LG) |
| Cite as: | arXiv:2602.15472 [physics.flu-dyn] |
| (or arXiv:2602.15472v5 [physics.flu-dyn] for this version) | |
| https://doi.org/10.48550/arXiv.2602.15472 arXiv-issued DOI via DataCite |
Submission history
From: Ramansh Sharma [view email]
[v1]
Tue, 17 Feb 2026 10:20:46 UTC (9,237 KB)
[v2]
Fri, 27 Feb 2026 03:00:26 UTC (9,238 KB)
[v3]
Tue, 17 Mar 2026 17:44:47 UTC (9,256 KB)
[v4]
Tue, 14 Apr 2026 21:41:34 UTC (9,234 KB)
[v5]
Wed, 7 Oct 2026 01:59:59 UTC (10,485 KB)
来源：arXiv：cs.LG · arxiv.org