SimDFKs: Divergence-Free Kernels as Spatial Representations for Flow Simulations

Fluid dynamics has posed enduring challenges in computational physics and computer graphics, where resolving rich, multiscale flow structures demands spatial representations that are both expressive and computationally efficient. Recent studies have shown that kernel-based spatial representations, which approximate flow fields as superpositions of localized basis functions, can outperform implicit neural representations (INRs) in both flow reconstruction and simulation while retaining a compact continuous parameterization. In this work, we extend divergence-free kernels (DFKs) from flow reconstruction to forward fluid simulation, introducing a key reformulation that makes evolving the continuous representation under PDEs practical and stable. Building on this representation, our optimization-based framework evolves the kernel parameters under advection, external forces, and complex boundary conditions while preserving incompressibility by construction. Our reformulation retains the compactness and expressiveness of the original DFK formulation while substantially improving the efficiency and numerical accuracy of derivative-sensitive PDE optimization. Without requiring training on precomputed data, our method produces high-quality forward simulations of a variety of 2D and 3D fluid phenomena from given initial conditions. The same framework also enables simulation recovery, i.e., reconstructing a fluid-dynamics-consistent flow field sequence from sparse vorticity observations of results produced by an existing solver. This capability is not directly available to conventional forward numerical solvers.