Understanding fluid behavior is an essential and challenging problem in all scientific and engineering disciplines. Motion estimation from visual observations and numerical simulation are two approaches to understand fluid behavior.

For fluid motion estimation, techniques such as Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV) are primarily optimized for controlled laboratory environments and rely heavily on high-quality imaging. These dependencies limit their use with low-quality images with sparse tracers, such as scenarios of environmental flow analysis or satellite imagery interpretation. Moreover, the methods struggle with efficiency in processing high-resolution video streams, restricting their real-time application. We propose a variational optical flow inspired unsupervised learning method for fluid motion estimation. Further, we incorporate numerical fluid simulation, proposing a prediction-correction approach to enhance motion estimation. We demonstrate the proposed methods are effective both in accuracy and efficiency on synthetic and real-world data.

For numerical fluid simulations, solving the Navier-Stokes equations numerically requires substantial computational resources and often encounters stability and accuracy challenges, especially in complex scenarios needing high-fidelity simulations. However, many systems are multi-resolution where certain regions are dynamically critical while others remain quasi-stationary. While a uniform, high-resolution mesh can be utilized to capture the dynamics, this often wastes finite computational resources. Mesh adaptation offers a strategy to accelerate the solving of PDEs, but traditional methods such as the Monge-Amp\'ere technique are computationally demanding. Excessive computational costs can outweigh the benefits of frequent mesh adaptation. We propose the first learning-based mesh movement networks to accelerate the mesh adaptation. Building on this, we further propose universal mesh movement networks utilizing graph transformer and graph attention networks. The proposed model, once trained on PDE-independent datasets, can be applied in a non-intrusive, zero-shot manner to move meshes with different size distributions and structures, for solvers applicable to different PDE types and boundary geometries.