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

Abstract

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 efficient optimization of such a constrained system, we introduce a Sliding Window scheme that propagates gradients over meaningful temporal horizons while maintaining tractable training costs.

Overview

Our method couples a Lagrangian 3D Gaussian Splatting representation with a continuous Divergence-Free Kernel velocity field, enforcing incompressibility by construction while maintaining transport coherence over long time horizons. A sliding-window optimization scheme makes the constrained reconstruction tractable, allowing gradients to propagate through meaningful temporal intervals without requiring full-sequence backpropagation.

Dataset

We release the LagrangianSplats dataset for fluid reconstruction research. The dataset is available on Hugging Face.

Citation

@article{tao2026lagrangiansplats,
  title={LagrangianSplats: Divergence-Free Transport of Gaussian Primitives for Fluid Reconstruction},
  author={Tao, Ningxiao and Chen, Baoquan and Chu, Mengyu},
  journal={arXiv preprint arXiv:2605.09299},
  year={2026}
}