Learning Representation of Turbulent Vector Fields via Efficient Moving Least Squares Based on Monte Carlo Method

In this study, we develop a numerical method to represent turbulent flow in various 2D vector fields using the Monte Carlo method-based MLS (Moving Least Squares) from a density field and express it as a learning representation through a neural network. Conventional MLS performs high-order interpolation solely based on vector-based constraints, making it difficult to effectively reflect the characteristics of a density field, which limits its applicability in various fields. Additionally, equations of higher degree require significant computational effort, making them costly in terms of computation time and resources when applied to simulation tasks that require per-frame calculations. To address these issues, this study integrates the Monte Carlo method-based weighting into MLS to efficiently consider the characteristics of the density field in the input data and design an algorithm that represents it as various forms of vector fields. Furthermore, we extend the solver to express this approach as a learning representation through a neural network. To validate the applicability of our method, we conducted experiments extracting turbulent vector fields from various density fields. Since conventional MLS does not guarantee temporal continuity, directly applying it to simulations results in noise. To resolve this, our method generates turbulent flow by analyzing the angular variation between the generated velocity and the underlying fluid velocity, ensuring stable advection of the density. As a result, our method efficiently and accurately extracts turbulent flow from a density field and integrates it into an underlying fluid solver, enabling the practical use of high-order interpolation in physics-based simulations. Experimental results across various scenarios demonstrate that our method improves both computation time and quality compared to previous methods, yielding enhanced turbulent flow fields.