  We present a grid-free fluid solver featuring a novel Gaussian
representation. Drawing inspiration from the expressive capabilities of 3D
Gaussian Splatting in multi-view image reconstruction, we model the continuous
flow velocity as a weighted sum of multiple Gaussian functions. Leveraging this
representation, we derive differential operators for the field and implement a
time-dependent PDE solver using the traditional operator splitting method.
Compared to implicit neural representations as another continuous spatial
representation with increasing attention, our method with flexible 3D Gaussians
presents enhanced accuracy on vorticity preservation. Moreover, we apply
physics-driven strategies to accelerate the optimization-based time integration
of Gaussian functions. This temporal evolution surpasses previous work based on
implicit neural representation with reduced computational time and memory.
Although not surpassing the quality of state-of-the-art Eulerian methods in
fluid simulation, experiments and ablation studies indicate the potential of
our memory-efficient representation. With enriched spatial information, our
method exhibits a distinctive perspective combining the advantages of Eulerian
and Lagrangian approaches.
