# LagrangianSplats: Divergence-Free Transport of Gaussian Primitives for Fluid Reconstruction

arXiv: https://arxiv.org/abs/2605.09299 · HTML: https://mlsanity.com/paper/2605.09299/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 51 references that are in the map.
- First posted on arXiv 10 May 2026.

## Builds on (most cited first, 24 of 24)

- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [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)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [4D Gaussian Splatting for Real-Time Dynamic Scene Rendering](https://mlsanity.com/paper/2310.08528/) (2023)
- [PIFu: Pixel-Aligned Implicit Function for High-Resolution Clothed Human Digitization](https://mlsanity.com/paper/1905.05172/) (2019)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Deformable 3D Gaussians for High-Fidelity Monocular Dynamic Scene Reconstruction](https://mlsanity.com/paper/2309.13101/) (2023)
- [Dynamic 3D Gaussians: Tracking by Persistent Dynamic View Synthesis](https://mlsanity.com/paper/2308.09713/) (2023)
- [4D-Rotor Gaussian Splatting: Towards Efficient Novel View Synthesis for Dynamic Scenes](https://mlsanity.com/paper/2402.03307/) (2024)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [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)
- [FluidNexus: 3D Fluid Reconstruction and Prediction from a Single Video](https://mlsanity.com/paper/2503.04720/) (2025)
- [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)
- [ScalarFlow: A Large-Scale Volumetric Data Set of Real-world Scalar Transport Flows for Computer Animation and Machine Learning](https://mlsanity.com/paper/2011.10284/) (2020)
- [Physics-Informed Learning of Characteristic Trajectories for Smoke Reconstruction](https://mlsanity.com/paper/2407.09679/) (2024)
- [Physics-Informed Deformable Gaussian Splatting: Towards Unified Constitutive Laws for Time-Evolving Material Field](https://mlsanity.com/paper/2511.06299/) (2025)
- [Representing Flow Fields with Divergence-Free Kernels for Reconstruction](https://mlsanity.com/paper/2504.01913/) (2025)
- [Fluid Implicit Particles on Coadjoint Orbits](https://mlsanity.com/paper/2406.01936/) (2024)

## Built on by (most cited first, 2 of 2)

- [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)

Reconstructing 3D fluid velocity fields from sparse 2D video observations is a highly ill-posed inverse problem, demanding both transport consistency with observed motion and physical validity under fluid laws. Existing methods typically impose these constraints through soft penalties, often leading to compromised accuracy and convergence issues. We introduce a reconstruction framework that structurally enforces both constraints. Specifically, we parameterize the reconstructed velocity using a continuous Divergence-Free Kernel representation, driving the advection of a Lagrangian 3D Gaussian Splatting representation. This formulation intrinsically guarantees both flow incompressibility and long-range transport coherence by construction. To enable the efficient optimization of such a constrained system, we introduce a novel Sliding Window scheme that propagates gradients over meaningful temporal horizons while maintaining tractable training costs. Experiments on synthetic and real-world datasets demonstrate that our method outperforms state-of-the-art baselines in both transport consistency and physical accuracy, enabling applications such as high-quality re-simulation and flow analysis.