# Representing Flow Fields with Divergence-Free Kernels for Reconstruction

arXiv: https://arxiv.org/abs/2504.01913 · HTML: https://mlsanity.com/paper/2504.01913/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 78 references that are in the map.
- First posted on arXiv 2 Apr 2025.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Physics Informed Neural Fields for Smoke Reconstruction with Sparse Data](https://mlsanity.com/paper/2206.06577/) (2022)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [NeuralVDB: High-resolution Sparse Volume Representation using Hierarchical Neural Networks](https://mlsanity.com/paper/2208.04448/) (2022)
- [Global Transport for Fluid Reconstruction with Learned Self-Supervision](https://mlsanity.com/paper/2104.06031/) (2021)
- [Inferring Hybrid Neural Fluid Fields from Videos](https://mlsanity.com/paper/2312.06561/) (2023)
- [Learning Vortex Dynamics for Fluid Inference and Prediction](https://mlsanity.com/paper/2301.11494/) (2023)
- [Physics-Informed Neural Network Super Resolution for Advection-Diffusion Models](https://mlsanity.com/paper/2011.02519/) (2020)
- [A Multi-Pass GAN for Fluid Flow Super-Resolution](https://mlsanity.com/paper/1906.01689/) (2019)
- [Fluid Simulation on Neural Flow Maps](https://mlsanity.com/paper/2312.14635/) (2023)
- [Physics-Informed Learning of Characteristic Trajectories for Smoke Reconstruction](https://mlsanity.com/paper/2407.09679/) (2024)
- [Image and video compression of fluid flow data](https://mlsanity.com/paper/2301.00078/) (2022)
- [Gaussian Fluids: A Grid-Free Fluid Solver based on Gaussian Spatial Representation](https://mlsanity.com/paper/2405.18133/) (2024)
- [Fluid Implicit Particles on Coadjoint Orbits](https://mlsanity.com/paper/2406.01936/) (2024)
- [High-Order Elasticity Interpolants for Microstructure Simulation](https://mlsanity.com/paper/2303.03120/) (2023)

## Built on by (most cited first, 5 of 5)

- [LagrangianSplats: Divergence-Free Transport of Gaussian Primitives for Fluid Reconstruction](https://mlsanity.com/paper/2605.09299/) (2026)
- [Diverse Sampling in Diffusion Models with Marginal Preserving Particle Guidance](https://mlsanity.com/paper/2605.06553/) (2026)
- [A Variational Optimal Transport Operator on Incompressible Flow](https://mlsanity.com/paper/2609.13729/) (2026)
- [SplashSplat: Reconstructing Splashing Liquids from Real-World Multi-View Videos](https://mlsanity.com/paper/2609.20818/) (2026)
- [Learning Incompressible 3D Flows from Sparse Samples: A Set Transformer Approach to Divergence-Free Kernel Fitting](https://mlsanity.com/paper/2610.16645/) (2026)

## Abstract (arXiv)

Accurately reconstructing continuous flow fields from sparse or indirect measurements remains an open challenge, as existing techniques often suffer from oversmoothing artifacts, reliance on heterogeneous architectures, and the computational burden of enforcing physics-informed losses in implicit neural representations (INRs). In this paper, we introduce a novel flow field reconstruction framework based on divergence-free kernels (DFKs), which inherently enforce incompressibility while capturing fine structures without relying on hierarchical or heterogeneous representations. Through qualitative analysis and quantitative ablation studies, we identify the matrix-valued radial basis functions derived from Wendland's C4 polynomial (DFKs-Wen4) as the optimal form of analytically divergence-free approximation for velocity fields, owing to their favorable numerical properties, including compact support, positive definiteness, and second-order differentiablility. Experiments across various reconstruction tasks, spanning data compression, inpainting, super-resolution, and time-continuous flow inference, has demonstrated that DFKs-Wen4 outperform INRs and other divergence-free representations in both reconstruction accuracy and computational efficiency while requiring the fewest trainable parameters.