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

A splash lives for a fraction of a second: sheets tear into ligaments and droplets, appearance is view-dependent and nearly textureless, and little persists long enough to track. Reconstruction research has consequently focused on smoke, synthetic liquids, or gently deforming surfaces. To our knowledge, no synchronized multi-view dataset of splashing liquids exists. We therefore introduce a benchmark of 20 real scenes, from coherent streams to violent splashes, captured by seven synchronized, calibrated 4K cameras at 60 fps, with manually refined per-view liquid and container masks and fixed evaluation splits. We further present SplashSplat, built on a single principle: impose physical structure only where the observations can constrain it. Per-frame liquid SDFs fused from the masks provide the geometry, level-set transport between consecutive SDFs yields a coarse velocity field, and Lagrangian carriers advected along this flow, corrected against each new observation and reseeded where coverage is lost, decode local Gaussians for differentiable rendering. SplashSplat outperforms state-of-the-art dynamic Gaussian splatting methods on our real captures and on a synthetic benchmark, with physically more plausible motion and a lower training cost. The same representation supports temporal interpolation and style transfer without re-optimization.

Reconstructing continuous flow fields from sparse surface-mounted sensors is central to aerodynamic design, flow control, and digital-twin instrumentation. Existing neural methods for this task typically encode sensor readings into implicit latent codes with little spatial interpretability and limited formal guidance on how representational capacity should scale with observation count. Inspired by 3D Gaussian Splatting, we introduce FLUIDSPLAT, a sensor-conditioned model that predicts K anisotropic Gaussian primitives forming a partition-of-unity scaffold, a spatially explicit and interpretable intermediate representation of the flow. For an idealized Gaussian primitive estimator, we prove an $O(K^{-s/d})$ approximation rate for fields with Sobolev smoothness $s$; incorporating $N$ noisy observations yields a squared-risk decomposition with bias $O(K^{-2s/d})$ and variance $O(σ^{2}K/N)$.Balancing the two yields $K^{*}\!\sim\!(N/σ^{2})^{d/(2s+d)}$: primitive count cannot grow freely under sparse sensing, revealing a variance bottleneck that motivates complementing the scaffold with a state-conditioned residual decoder. Across four benchmarks spanning 2D and 3D, FLUIDSPLAT achieves 11-28% error reduction over several strong baselines on cylinder flow, AirfRANS, FlowBench LDC-3D, and PhySense-Car 3D benchmarks.

Feed-forward 3D Gaussian Splatting models offer fast single-pass reconstruction,but scaling them to match per-scene optimization quality is fundamentally hindered by the scarcity of large-scale 3D annotations. A practical compromise is predict-then-refine,where post-prediction optimization compensates for the limited capacity of the feed-forward network. However,standard feed-forward 3DGS is trained solely for zero-step rendering error,ignoring whether its output constitutes a good initialization for the downstream optimizer. We present ForeSplat,an optimization-aware training framework that equips feed-forward 3DGS models to produce initializations explicitly designed for rapid,effective refinement. By offloading part of the scene-modeling burden to the optimizer,ForeSplat substantially reduces the capacity pressure on the feed-forward model,making high-quality reconstruction feasible even with compact networks. At its core is MetaGrad,a lightweight multi-anchor meta-gradient training rule that bypasses costly higher-order differentiation through the 3DGS optimizer. MetaGrad unrolls a short inner-loop refinement trajectory,samples anchor states,and back-propagates aggregated first-order gradients to the prediction head as a surrogate optimization-aware signal. This fine-tuning adds no inference cost and enables high-quality reconstruction within seconds after a few refinement steps. We instantiate ForeSplat on diverse backbones,including AnySplat,Pi3X,and a distilled variant tailored for edge deployment. Across all tested architectures,a ForeSplat-trained initialization converges in fewer refinement steps and reaches a higher peak reconstruction quality than its vanilla counterpart,even fully converged. The framework consistently bridges the gap between amortized prediction and per-scene optimization,establishing a practical path toward lightweight,high-fidelity 3D reconstruction.