arXiv：cs.AI· Ramzi Idoughi, Radouan Boukharfane, Peter Wonka·· 6 小时前
DFK-Former：用集合 Transformer 从稀疏样本学习无散度 3D 流场
Learning Incompressible 3D Flows from Sparse Samples: A Set Transformer Approach to Divergence-Free Kernel Fitting
AI 导读
研究者提出 DFK-Former，一种集合到场的 Transformer，可从稀疏、含噪且不规则采样的 3D 观测中重建无散度速度场。该模型直接在前向传播中预测无散度核参数，无需逐场迭代优化，并引入自适应各向异性无散度核以捕捉更复杂流场结构。在稀疏、噪声和遮挡实验中，DFK-Former 将端点误差最多降低 4.6 倍，单场重建耗时 7.5 ms 而非 9.8 s。
正文
Abstract:Recovering divergence-free velocity fields from sparse and corrupted observations requires reconciling physical consistency, expressive representations, and robustness to missing data. We introduce DFK-Former, a set-to-field transformer that reconstructs divergence-free velocity fields from sparse, noisy, and irregularly sampled 3D observations. The model is inherently invariant to the number and ordering of input samples and predicts the parameters of a divergence-free kernel representation, producing continuous, mesh-free velocity fields that satisfy incompressibility exactly by construction rather than approximately through soft constraints or projection operators. Unlike existing divergence-free kernel approaches, which estimate kernel coefficients via a separate optimization for each field, DFK-Former predicts these coefficients directly in a single forward pass. This eliminates costly iterative fitting while providing a learned prior that enables accurate reconstruction even when observations are insufficient to uniquely determine the field. To enhance representational power, we introduce adaptive and anisotropic divergence-free kernels that preserve exact incompressibility guarantees while capturing more complex flow structures. Extensive experiments across varying levels of sparsity, noise, and occlusion, including transfer to real Direct Numerical Simulation (DNS) turbulence data, demonstrate substantial gains in both accuracy and efficiency. DFK-Former reduces endpoint error by up to 4.6x compared to per-field divergence-free-kernel optimization, while reconstructing each field in 7.5 ms instead of 9.8 s, yielding a speedup of more than three orders of magnitude. These results establish direct, feed-forward prediction of divergence-free kernel parameters as a practical and effective paradigm for physically consistent velocity-field reconstruction from sparse observations.
| Comments: | 33 pages with supplement materials, 9 figures |
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computational Engineering, Finance, and Science (cs.CE) |
| MSC classes: | 68T07, 76M21, 65D12, 65D05 |
| ACM classes: | I.2.6; G.1.1; J.2 |
| Cite as: | arXiv:2610.16645 [cs.LG] |
| (or arXiv:2610.16645v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.16645 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Ramzi Idoughi [view email]
[v1]
Mon, 12 Oct 2026 12:06:44 UTC (3,329 KB)
来源：arXiv：cs.AI · arxiv.org