FluidGaussian: Gaussian Fluid Representations

• FluidGaussian is a framework that represents fluid states using localized Gaussian primitives instead of conventional Eulerian grids.
• It couples Gaussian splatting with fluid simulations to analytically compute derivatives like gradients, vorticity, and uncertainty.
• The method enhances computational efficiency and compression, supporting grid-free solvers and active sensing in fluid visualization.

Searching arXiv for papers on “FluidGaussian” and closely related Gaussian-based fluid representations. “FluidGaussian” denotes a family of fluid and flow representations in which Gaussian or Gaussian-like primitives serve as the primary state variables for geometry, velocity, density, or uncertainty, rather than a conventional Eulerian grid. In the most direct usage, FluidGaussian is a plug-and-play reconstruction method that couples 3D Gaussian Splatting with fluid-structure simulations to assess and reduce physics-relevant reconstruction defects (Liu et al., 22 Mar 2026). Closely related work uses Gaussian primitives to encode turbulent velocity fields continuously (Shenoy et al., 7 Apr 2026), to build grid-free Navier–Stokes solvers based on Gaussian spatial representations (Xing et al., 2024), and to advect Gaussian splats under a divergence-free velocity field for fluid reconstruction from sparse video (Tao et al., 10 May 2026). Across these formulations, the common premise is that Gaussian parameterizations can be compact, continuous, interpretable, and differentiable, while exposing quantities such as gradients, vorticity, enstrophy, optical depth, or simulation-induced uncertainty in forms that are difficult to obtain as efficiently from dense voxel grids.

1. Origins and scope of the term

The term “FluidGaussian” appears explicitly as the title of “FluidGaussian: Propagating Simulation-Based Uncertainty Toward Functionally-Intelligent 3D Reconstruction” (Liu et al., 22 Mar 2026), where it denotes a method for active 3D reconstruction that uses fluid simulation as a diagnostic of geometry quality. A closely related usage appears in “Gaussian Field Representations for Turbulent Flow: Compression, Scale Separation, and Physical Fidelity” (Shenoy et al., 7 Apr 2026), which describes a continuous parametric encoding of a three-dimensional turbulent velocity field as a superposition of localized Gaussian primitives. That work states that the formulation is “almost exactly about what you’re calling a ‘FluidGaussian’ representation,” defining the velocity field as a sum of kernels with learnable positions, amplitudes, and scales.

Other arXiv works extend the same design pattern in adjacent directions. “A Grid-Free Fluid Solver based on Gaussian Spatial Representation” (Xing et al., 2024) models the continuous flow velocity as a weighted sum of multiple Gaussian functions and advances the state through an operator-splitting solver. “LagrangianSplats: Divergence-Free Transport of Gaussian Primitives for Fluid Reconstruction” (Tao et al., 10 May 2026) parameterizes velocity with a continuous Divergence-Free Kernel representation and density with Lagrangian 3D Gaussian Splatting primitives. “3D Gaussian Modeling and Ray Marching of OpenVDB datasets for Scientific Visualization” (Sharma et al., 14 Sep 2025) uses volumetric Gaussians derived from sparse OpenVDB data for optical-depth accumulation in volumes such as clouds, fire, and fluids.

Taken together, these works suggest an umbrella interpretation of FluidGaussian as a Gaussian-primitive approach to fluid states, fluid-derived uncertainty, or fluid visualization. This suggests a unifying editorial shorthand—FluidGaussian framework (Editor’s term)—for methods that encode fluid-relevant quantities by localized Gaussian primitives and exploit the resulting continuity, compactness, and differentiability. Where the literature uses the term more narrowly, the narrow meaning is preserved below.

2. Gaussian primitives as fluid state variables

In the turbulent-flow compression formulation, the velocity field $\mathbf{u}(\mathbf{x})$ is represented as a continuous function

$$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$

where $N$ is the number of kernels, $\mathbf{a}_i \in \mathbb{R}^3$ is the vector amplitude, $\boldsymbol{\mu}_i \in \mathbb{R}^3$ is the center, and $w_i(\mathbf{x})$ is a normalized spatial weight (Shenoy et al., 7 Apr 2026). The unnormalized response is Gaussian,

$$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$

with $$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$, and

$$w_i(\mathbf{x}) = \frac{G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i)} {\sum_{j=1}^{N} G(\mathbf{x}; \boldsymbol{\mu}_j, \boldsymbol{\sigma}_j) + \varepsilon}.$$

The learnable parameters are therefore $$\{\mathbf{a}_i,\boldsymbol{\mu}_i,\boldsymbol{\sigma}_i\}_{i=1}^{N}$$, optimized from data by minimizing a normalized $$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$0 velocity reconstruction loss.

The grid-free solver of (Xing et al., 2024) adopts a similar but not identical parameterization. Its Gaussian Spatial Representation defines

$$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$1

where each Gaussian has mean $$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$2, covariance $$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$3, and a vector coefficient $$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$4. The covariance is reparameterized by rotation and scales, and Gaussians are clamped outside a small support so that evaluation is local. This representation is used directly as the velocity field in an incompressible-fluid solver.

In fluid reconstruction from video, (Tao et al., 10 May 2026) splits the state across two Gaussian-based structures. The density field is represented by 3D Gaussian splats with mean position $$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$5, covariance $$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$6, opacity, and spherical-harmonics appearance parameters, while the velocity field is represented by divergence-free kernels,

$$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$7

Here the Gaussians are Lagrangian carriers of density, and the divergence-free kernel field supplies their transport.

For scientific visualization of fluid-like volumes, (Sharma et al., 14 Sep 2025) treats a 3D Gaussian as a volumetric basis function,

$$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$8

or in anisotropic form as

$$\hat{\mathbf{u}}(\mathbf{x}) = \sum_{i=1}^{N} w_i(\mathbf{x})\,\mathbf{a}_i,$$9

In that setting, the Gaussian stores density or opacity, derived from grouped OpenVDB voxels, and supports analytic line integration for optical-depth accumulation.

These formulations differ in what the Gaussian carries—velocity amplitude, density, uncertainty score, or optical density—but converge on a common structure: a fluid-relevant field is expressed by a finite set of localized primitives in $N$0, with continuous evaluation at arbitrary points.

3. Differential structure, transport, and incompressibility

A principal reason Gaussian parameterizations are attractive in fluid problems is that derivative-based quantities are available analytically or semi-analytically. In the turbulent-flow representation, the reconstructed field is an explicit analytic function, so the gradient satisfies

$N$1

and for a Gaussian kernel,

$N$2

This permits analytic evaluation of vorticity $N$3, enstrophy density $N$4, global enstrophy, $N$5-criterion isosurfaces, and energy spectra from the compressed representation itself (Shenoy et al., 7 Apr 2026).

The grid-free solver makes the differential structure explicit in its formulation. For

$N$6

the gradient is

$N$7

with divergence given by the trace and curl or vorticity assembled from the antisymmetric part of the gradient tensor (Xing et al., 2024). This analytic differential calculus enables operator splitting for incompressible flow: a Lagrangian advection step updates Gaussian centers, covariances, and weights via the flow map and its Jacobian, and a projection step optimizes Gaussian weights and scales so that divergence is reduced while advected vorticity is preserved.

The Lagrangian reconstruction method of (Tao et al., 10 May 2026) hard-wires incompressibility at the representation level rather than by penalty. It defines

$N$8

where $N$9 is a compactly supported scalar radial basis function. By construction,

$\mathbf{a}_i \in \mathbb{R}^3$0

for any coefficients $\mathbf{a}_i \in \mathbb{R}^3$1. The Gaussian splat centers then obey

$\mathbf{a}_i \in \mathbb{R}^3$2

so transport coherence is enforced structurally, not by framewise matching alone. A vorticity-transport penalty

$\mathbf{a}_i \in \mathbb{R}^3$3

adds a soft momentum-consistency term.

This body of work shows two dominant strategies. One uses unconstrained Gaussian fields and derives differential quantities directly, accepting that incompressibility is approximate unless enforced by optimization (Shenoy et al., 7 Apr 2026, Xing et al., 2024). The other restricts the velocity basis itself to the divergence-free subspace, so incompressibility is exact by construction (Tao et al., 10 May 2026). A plausible implication is that the main methodological divide within FluidGaussian research is not “Gaussian versus non-Gaussian,” but “general Gaussian basis versus physically constrained Gaussian basis.”

4. Compression, scale separation, and reconstruction quality

The turbulent-flow paper quantifies the compression–accuracy trade-off most directly. A $\mathbf{a}_i \in \mathbb{R}^3$4 vector field contains $\mathbf{a}_i \in \mathbb{R}^3$5 scalar values, whereas a diagonal Gaussian model with $\mathbf{a}_i \in \mathbb{R}^3$6 kernels stores $\mathbf{a}_i \in \mathbb{R}^3$7 scalars. At $\mathbf{a}_i \in \mathbb{R}^3$8, this yields $\mathbf{a}_i \in \mathbb{R}^3$9 parameters and a compression ratio of approximately $\boldsymbol{\mu}_i \in \mathbb{R}^3$0; at $\boldsymbol{\mu}_i \in \mathbb{R}^3$1, $\boldsymbol{\mu}_i \in \mathbb{R}^3$2 parameters and a compression ratio of approximately $\boldsymbol{\mu}_i \in \mathbb{R}^3$3 (Shenoy et al., 7 Apr 2026). In the early smooth regime at $\boldsymbol{\mu}_i \in \mathbb{R}^3$4, $\boldsymbol{\mu}_i \in \mathbb{R}^3$5 gives $\boldsymbol{\mu}_i \in \mathbb{R}^3$6 error $\boldsymbol{\mu}_i \in \mathbb{R}^3$7 and enstrophy error $\boldsymbol{\mu}_i \in \mathbb{R}^3$8, while $\boldsymbol{\mu}_i \in \mathbb{R}^3$9 gives $w_i(\mathbf{x})$0 error $w_i(\mathbf{x})$1 and enstrophy error $w_i(\mathbf{x})$2. In the turbulent regime at $w_i(\mathbf{x})$3, however, $w_i(\mathbf{x})$4 gives $w_i(\mathbf{x})$5 error $w_i(\mathbf{x})$6 but enstrophy error $w_i(\mathbf{x})$7, and $w_i(\mathbf{x})$8 gives $w_i(\mathbf{x})$9 error $$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$0 but enstrophy error $$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$1. The main conclusion stated there is that the principal limitation of baseline Gaussian representations is geometric expressiveness rather than parameter count.

The same study compares several structure-aware extensions at $$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$2. Relative enstrophy errors at $$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$3 are 82.00% for the baseline Gaussian, 78.51% for adaptive placement, 81.01% for multi-resolution Gaussian, 71.42% for anisotropic Gaussian, and 62.24% for a compact-support Beta basis (Shenoy et al., 7 Apr 2026). The anisotropic Gaussian is the best among the Gaussian families; the Beta basis lowers enstrophy further but introduces visible spatial artifacts.

The grid-free solver does not present the topic as compression, but its memory efficiency follows from representing the velocity field with a finite number of Gaussians rather than a dense Eulerian grid. Reported Gaussian counts include 5000→10643 particles for the Taylor vortex, 5000→9652 for leapfrog vortices, and 68208→73334 for 3D vortex rings, with per-timestep costs of 3.5 minutes, 9.3 minutes, and 3.2 minutes for the first 17 frames respectively (Xing et al., 2024). The paper states that its temporal evolution surpasses previous work based on implicit neural representation with reduced computational time and memory, while also reporting better vorticity preservation than the compared implicit neural spatial representation.

For active reconstruction, FluidGaussian (Liu et al., 22 Mar 2026) measures success by both visual and simulation metrics. On Blender scenes, ActiveNeRF improves from PSNR 21.15 to 21.76 and FisherRF from 23.42 to 24.74 when augmented with FluidGaussian re-ranking; on DrivAerNet++, ActiveNeRF improves from 17.38 to 18.87 and FisherRF from 20.11 to 21.07. The same paper reports lower geometry-induced divergence under fluid simulations: on Blender, ActiveNeRF drops from 0.0262 to 0.0145 and FisherRF from 0.0144 to 0.0124; on Mip-NeRF360, ActiveNeRF drops from 0.0130 to 0.0049 and FisherRF from 0.0025 to 0.0022. The abstract summarizes the best observed effects as up to +8.6% visual PSNR and -62.3% velocity divergence (Liu et al., 22 Mar 2026).

For sparse-volume visualization, (Sharma et al., 14 Sep 2025) treats primitive reduction itself as a performance metric. On smoke2, a dataset with about 20.74M voxels, the reported Gaussian counts range from 2.60M at LOD-1 to 0.01M at LOD-5, corresponding to 12.5% down to 0.05% of the voxel count, with PSNR values from about 25.00 dB at dense 8×8×8 grouping with smart sparse handling down to 23.02 dB for a single Gaussian per sparse leaf. This suggests that Gaussian volumes can act as an explicit sparse surrogate for large scientific fluid datasets, not only as a simulation state.

5. Physics-aware uncertainty and active sensing

The most distinctive narrow meaning of FluidGaussian is the active-reconstruction framework of (Liu et al., 22 Mar 2026). There, a current 3D Gaussian Splatting reconstruction $$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$4 is converted into a rigid object for a Divergence-Free Smoothed Particle Hydrodynamics simulation. Short SPH rollouts are run for five initial conditions: incoming flow from $$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$5, $$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$6, and from above $$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$7. For each fluid particle $$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$8, velocity divergence is estimated by

$$G(\mathbf{x}; \boldsymbol{\mu}_i, \boldsymbol{\sigma}_i) = \exp\left( -(\mathbf{x} - \boldsymbol{\mu}_i)^T \mathbf{D}_i^{-1} (\mathbf{x} - \boldsymbol{\mu}_i) \right),$$9

and the scalar magnitude $$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$0 is aggregated onto nearby Gaussians. For Gaussian $$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$1, using fluid particles within radius $$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$2 with $$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$3,

$$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$4

This produces a per-Gaussian uncertainty score induced not by image variance but by fluid-structure inconsistency.

Candidate next-best views are first proposed by a visual method such as ActiveNeRF or FisherRF. FluidGaussian then re-ranks the top-$$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$5 candidates using a physics score

$$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$6

where $$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$7 is an area-based visibility weight derived from projected Gaussian size. The selected next view is

$$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$8

This means that the reconstruction backbone remains photometric, but camera planning becomes function-aware.

The paper’s interpretation is explicitly tied to function-critical regions such as aerodynamic or hydrodynamic surfaces. On DrivAerNet++, views that expose regions with strong fluid interaction receive higher scores, and the resulting reconstructions improve both visual metrics and divergence specifically on those function-critical regions (Liu et al., 22 Mar 2026). This suggests a broader implication for FluidGaussian as an active-sensing principle: fluid response can be used as a proxy for geometric salience when geometry is intended for downstream physical simulation.

6. Variants, neighboring methods, and limitations

Several neighboring methods help delimit what FluidGaussian is and is not. “Gaussian Splashing: Unified Particles for Versatile Motion Synthesis and Rendering” (Feng et al., 2024) uses 3D Gaussian splats as unified particles for scene geometry, Position-Based Dynamics solids, and Position-Based Fluids, with normals and physically based rendering for reflective dynamic fluids. This is a Gaussian-based fluid pipeline, but its emphasis is motion synthesis and rendering rather than continuous field representation or physics-aware reconstruction. “LagrangianSplats” (Tao et al., 10 May 2026) is closer to a physically constrained FluidGaussian in that it combines Gaussian density primitives with a divergence-free velocity basis and enforces long-range transport consistency from sparse observations. “A Grid-Free Fluid Solver based on Gaussian Spatial Representation” (Xing et al., 2024) is a FluidGaussian in the strongest solver sense, because the Gaussian basis itself is the simulation state.

The limitations identified across the literature are consistent. The turbulent-flow representation reports that baseline Gaussian models behave like built-in low-pass filters, preserving low-wavenumber energy well but strongly attenuating high-wavenumber structure and enstrophy (Shenoy et al., 7 Apr 2026). The main bottleneck is geometric expressiveness: axis-aligned or unrotated kernels cannot align efficiently with thin vortical filaments or sheets. The grid-free solver states that it does not surpass state-of-the-art Eulerian methods in overall simulation quality, even though it improves over implicit neural representations in vorticity preservation (Xing et al., 2024). The Lagrangian reconstruction method is limited to incompressible flows because the divergence-free kernel representation enforces $$\mathbf{D}_i = \mathrm{diag}(\sigma_{ix}^2, \sigma_{iy}^2, \sigma_{iz}^2)$$9 identically (Tao et al., 10 May 2026). FluidGaussian as active reconstruction incurs a notable runtime overhead because repeated SPH rollouts are used inside the next-best-view loop (Liu et al., 22 Mar 2026).

A common misconception would be to treat Gaussian primitives as merely a compression trick. The evidence does not support that narrow view. In (Shenoy et al., 7 Apr 2026), Gaussian primitives are also a means of continuous querying and analytic differentiation. In (Xing et al., 2024), they define the solver state and support operator splitting. In (Tao et al., 10 May 2026), they are Lagrangian carriers for transport-consistent density reconstruction. In (Liu et al., 22 Mar 2026), 3D Gaussians become anchors for simulation-based uncertainty propagation. Another misconception would be that Gaussianity automatically implies physical fidelity. The papers collectively show the opposite: without anisotropy, divergence-free structure, or additional physics-aware constraints, Gaussian models can remain visually accurate while failing on enstrophy, incompressibility, or downstream fluid behavior.

This suggests that FluidGaussian is best understood not as a single algorithm but as a research program. Its central question is how far localized Gaussian primitives can replace or augment grids, particles, and implicit neural fields in representing fluid states, fluid transport, and fluid-informed uncertainty. The present literature indicates that the answer is already substantive for compression, continuous querying, view planning, and sparse-observation reconstruction, while high-Reynolds-number turbulence, complex boundaries, and full dynamical fidelity remain open problems (Shenoy et al., 7 Apr 2026).