arXiv subjects

Jingrui Xing

Publications and source records attributed to Jingrui Xing.

3 recordsLinked to original sources

arXiv2026

DDFKs: Fluid Simulation with Dynamic Divergence-Free Kernels

Fluid simulations based on memory-efficient spatial representations like implicit neural spatial representations (INSRs) and Gaussian spatial representation (GSR), where the velocity fields are parameterized by neural networks or weighted Gaussian functions, has been an emerging research area. Though advantages over traditional discretizations like spatial adaptivity and continuous differentiability of these spatial representations are leveraged by fluid solvers, solving the time-dependent PDEs that governs the fluid dynamics remain challenging, especially in incompressible fluids where the divergence-free constraint is enforced. In this paper, we propose a grid-free solver Dynamic Divergence-Free Kernels (DDFKs) for incompressible flows based on divergence-free kernels (DFKs). Each DFK is incorporated with a matrix-valued radial basis function and a vector-valued weight, yielding a divergence-free vector field. We model the continuous flow velocity as the sum of multiple DFKs, thus enforcing incompressibility while being able to preserve different level of details. Quantitative and qualitative results show that our method achieves comparable accuracy, robustness, ability to preserve vortices, time and memory efficiency and generality across diverse phenomena to state-of-the-art methods using memory-efficient spatial representations, while excels at maintaining incompressibility. Though our first-order solver are slower than fluid solvers with traditional discretizations, our approach exhibits significantly lower numerical dissipation due to reduced discretization error. We demonstrate our method on diverse incompressible flow examples with rich vortices and various solid boundary conditions.

physics.flu-dyn↗

arXiv2025

Representing Flow Fields with Divergence-Free Kernels for Reconstruction

Accurately reconstructing continuous flow fields from sparse or indirect measurements remains an open challenge, as existing techniques often suffer from oversmoothing artifacts, reliance on heterogeneous architectures, and the computational burden of enforcing physics-informed losses in implicit neural representations (INRs). In this paper, we introduce a novel flow field reconstruction framework based on divergence-free kernels (DFKs), which inherently enforce incompressibility while capturing fine structures without relying on hierarchical or heterogeneous representations. Through qualitative analysis and quantitative ablation studies, we identify the matrix-valued radial basis functions derived from Wendland's $\mathcal{C}^4$ polynomial (DFKs-Wen4) as the optimal form of analytically divergence-free approximation for velocity fields, owing to their favorable numerical properties, including compact support, positive definiteness, and second-order differentiablility. Experiments across various reconstruction tasks, spanning data compression, inpainting, super-resolution, and time-continuous flow inference, has demonstrated that DFKs-Wen4 outperform INRs and other divergence-free representations in both reconstruction accuracy and computational efficiency while requiring the fewest trainable parameters.

cs.GR↗

arXiv2024

Gaussian Fluids: A Grid-Free Fluid Solver based on Gaussian Spatial Representation

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. This representation is continuously differentiable, which enables us to derive spatial differentials directly and solve the time-dependent PDE via a custom first-order optimization tailored to fluid dynamics. Compared to traditional discretizations, which typically adopt Eulerian, Lagrangian, or hybrid perspectives, our approach is inherently memory-efficient and spatially adaptive, enabling it to preserve fine-scale structures and vortices with high fidelity. While these advantages are also sought by implicit neural representations, GSR offers enhanced robustness, accuracy, and generality across diverse fluid phenomena, with improved computational efficiency during temporal evolution. Though our first-order solver does not yet match the speed of fluid solvers using explicit representations, its continuous nature substantially reduces spatial discretization error and opens a new avenue for high-fidelity simulation. We evaluate the proposed solver across a broad range of 2D and 3D fluid phenomena, demonstrating its ability to preserve intricate vortex dynamics, accurately capture boundary-induced effects such as K\'arm\'an vortex streets, and remain robust across long time horizons - all without additional parameter tuning. Our results suggest that GSR offers a compelling direction for future research in fluid simulation.

cs.GR↗