LEARNING VORTEX DYNAMICS FOR FLUID INFERENCE AND PREDICTION

Abstract

We propose a novel differentiable vortex particle (DVP) method to infer and predict fluid dynamics from a single video. Lying at its core is a particle-based latent space to encapsulate the hidden, Lagrangian vortical evolution underpinning the observable, Eulerian flow phenomena. Our differentiable vortex particles are coupled with a learnable, vortex-to-velocity dynamics mapping to effectively capture the complex flow features in a physically-constrained, low-dimensional space. This representation facilitates the learning of a fluid simulator tailored to the input video that can deliver robust, long-term future predictions. The value of our method is twofold: first, our learned simulator enables the inference of hidden physics quantities (e.g., velocity field) purely from visual observation; secondly, it also supports future prediction, constructing the input video's sequel along with its future dynamics evolution. We compare our method with a range of existing methods on both synthetic and real-world videos, demonstrating improved reconstruction quality, visual plausibility, and physical integrity. 1 

1. INTRODUCTION

As small as thin soap films, and as large as atmospheric eddies observable from outer space, fluid systems can exhibit intricate dynamic features on different mediums and scales. However, despite recent progress, it remains an open problem for scientific machine learning to effectively represent these flow features, identify the underlying dynamics system, and predict the future evolution, due to the noisy data, imperfect modeling, and unavailable, hidden physics quantities. Here, we identify three fundamental challenges that currently hinder the success of such endeavors. First, flow features are difficult to represent. Traditional methods learn the fluid dynamics by storing velocity fields either using regularly-spaced grids or smooth neural networks. These approaches have demonstrated promising results for fluid phenomena that are relatively damped and laminar (e.g., Chu et al., 2022) , but for fluid systems that can exhibit turbulent features on varying scales, these methods fall short due to the problem's curse of dimensionality (high-resolution space and time), local non-smoothness, and hidden constraints. As a result, more compact and structured representation spaces and data structures are called for. Secondly, hidden flow dynamics is hard to learn. Fluid systems as prescribed by the Navier-Stokes equations tightly couple multiple physical quantities (i.e., velocity, pressure, and density), and yet, only the density information can be accessibly measured. Due to the system's complexity, ambiguity, and non-linearity, directly learning the underlying dynamics from the observable density space is infeasible; and successful learning usually relies on velocity or pressure supervision, a requirement that distances these methods from deployment in real-world scenarios. Exciting recent progress has been made in hidden dynamics inference by PDE-based frameworks such as Raissi et al. (2020) , which uncover the underlying physics variables solely from density observations. However, this type of methods encounter the third fundamental challenge, which is that performing future prediction is difficult. As we will demonstrate, although strong results



Our video results, code, and data can be found at our project website: https://yitongdeng. github.io/vortex_learning_webpage.

