Representing Flow Fields with Divergence-Free Kernels for Reconstruction

2025-04-02

AI Summary

Reconstructing continuous fluid velocity fields from sparse or indirect measurements suffers from oversmoothing, hardware heterogeneity dependencies, and high computational overhead due to the need for complex physics-based loss terms—e.g., enforcing incompressibility (∇·u = 0)—in implicit neural representations (INRs). Method: We propose a novel INR framework built upon analytically divergence-free kernels (DFKs), introducing DFKs-Wen4: a compactly supported, positive-definite, matrix-valued, radially symmetric, twice-differentiable divergence-free basis function. Crucially, DFKs-Wen4 intrinsically satisfies ∇·u = 0 without auxiliary physics losses. Contribution/Results: Our method achieves state-of-the-art accuracy and efficiency across flow field compression, inpainting, super-resolution, and continuous-time inference. It reduces parameter count by up to 67% versus existing INRs and divergence-free models, while significantly improving fine-structure fidelity and generalization.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Web Mining and Content Analysis: Large pretrained models with web data

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.

Problem

• Reconstruct flow fields from sparse measurements accurately
• Overcome oversmoothing and computational inefficiency in current methods
• Enforce incompressibility without complex hierarchical representations

Innovation

• Divergence-free kernels enforce incompressibility
• Matrix-valued radial basis functions optimize accuracy
• Compact support and differentiability enhance efficiency