Representing Flow Fields with Divergence-Free Kernels for Reconstruction

TL;DR

This work tackles the challenge of reconstructing incompressible flow fields from sparse, indirect, or partially observed data by embedding incompressibility directly into a kernel-based representation. It introduces divergence-free kernels (DFKs), specifically DFKs-Wen4 derived from Wendland’s $\mathcal{C}^4$ polynomial, to produce analytically divergence-free velocity fields via $\bm{\psi}_i(\mathbf{x})=(-\mathbf{I}\nabla^2+\nabla\nabla^T)\phi_i(\mathbf{x})$ with $\phi_i$ built from $R_{\mathrm{Wen4}}$. The paper demonstrates through extensive experiments across fitting, inpainting, super-resolution, and time-continuous inference that DFKs-Wen4 outperform INRs and other divergence-free representations in accuracy and efficiency while using far fewer trainable parameters. Key advantages include dipolar flow representation, compact support, positive definiteness, and second-order differentiability, enabling accurate, localized, and boundary-aware reconstructions suitable for both inverse problems and potential forward simulations. By providing a robust, scalable alternative to neural representations for flow-field reconstruction, this approach offers immediate practical impact for CFD, graphics, and data-driven flow analysis, with clear paths toward anisotropic and manifold extensions in future work.

Abstract

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.

Figures (15)

• Figure 1: Fitting experiments of the analytic vortices, with the resulting vorticity fields rendered. Physics-embedded methods clearly demonstrate superior fitting capabilities over physics-informed ones. Additionally, kernel-based approaches excel at capturing local details compared to neural networks-base representrations. Among all tested approaches, DFK-Wen4 achieves the lowest fitting error, as shown in Tab. \ref{['tab:losses']}.
• Figure 2: Fitting experiments of the simple plume, with the resulting vorticity fields rendered. SIREN, Curl SIREN, and Regular RBF produce overly smooth results, failing to capture finer details. In contrast, Curl Kernel and DFK-Wen4 closely match the ground truth (see Tab. \ref{['tab:losses']}), delivering accurate and high-fidelity representations, with only $4.3\%$ of DoFs used compared to the raw data.
• Figure 3: Illustration of different divergence-free kernels in 2D. The left four images display streamline plots, where brighter background colors indicate higher velocity. For Curl Kernel (\ref{['fig:illu_curl']}), the vector field is constructed as $(\partial/\partial y,-\partial/\partial x)\,R_\mathrm{Wen4}$, which generalizes to $\bm{\nabla}\times R_\mathrm{Wen4}\bm{\omega}$ in 3D. For DFKs (\ref{['fig:illu_gauss']}--\ref{['fig:illu_flowpek']}), the fields are formulated as $(-\bm{I}\bm{\nabla}^2+\bm{\nabla}\bm{\nabla}^\top)\,\phi\,\bm{\omega}$, where $\phi$ takes the forms $R_\mathrm{Gau}=\exp{(-9r^2/2)}$, $R_\mathrm{Poly6}=(1-r^2)^3_+$, and $R_\mathrm{Wen4}$, respectively. For comparison, $\bm{\omega}$ is set to $(1,0)$. Note that these plots also serve as 2D cross-sections of their 3D counterparts. The rightmost image (\ref{['fig:illu_vort']}) shows the corresponding vorticity field of DFK-Wen4, with cool and warm colors indicating opposite rotation directions.
• Figure 4: Projection experiments of the Taylor vortex. The input is illustrated using HSV color encoding, with its divergence plotted. The vorticity fields of ground truth and experimental results are presented with jet color mapping, where our method provides the most accurate projection. The PSNR/SSIM values are as follows: Curl SIREN: 33.68/0.965; Curl Kernel: 29.80/0.970; DFK-Wen4:39.30/0.994.
• Figure 5: Projection experiments of the vortex ring collision. The vortice fields of input and results are rendered. As demonstrated in Tab. \ref{['tab:losses']}, DFK-Wen4 has the lowest loss in this case, though Curl Kernel also provides comparable visualization. Note that the vorticity field output by the INR is very diffuse.