Uncertainty Tube Visualization of Particle Trajectories

This figure compares (a) a spaghetti plot of ensemble members, (b) a circular tube, and (c) our uncertainty tube for visualizing model uncertainty. Previous methods face challenges such as visual clutter (a) or the assumption of symmetric uncertainty (a, b), but our uncertainty tube (c), constructed using superellipses, provides a more accurate visualization of asymmetric uncertainty. Its superelliptical shape distinctly improves the visualization of the uncertainty orientation and its evolution along trajectories, as highlighted in the boxes. The visualization is further enhanced with a color palette that uses gray for low uncertainty, blue for large asymmetric uncertainty, and yellow for large symmetric uncertainty. Predicting particle trajectories with neural networks (NNs) has substantially enhanced many scientific and engineering domains. However, effectively quantifying and visualizing the inherent uncertainty in predictions remains challenging. Without an understanding of the uncertainty, the reliability of NN models in applications where trustworthiness is paramount is significantly compromised. This paper introduces the uncertainty tube, a novel, computationally efficient visualization method designed to represent this uncertainty in NN-derived particle paths. By integrating well-established uncertainty quantification techniques, such as Deep Ensembles, Monte Carlo Dropout (MC Dropout), and Stochastic Weight Averaging-Gaussian (SW AG), we demonstrate the practical utility of the uncertainty tube, showcasing its application on both synthetic and simulation datasets. Understanding and analyzing flow field data is fundamental for numerous scientific and engineering disciplines, including fluid dynamics, atmospheric science, and material processing. Traditional computational fluid dynamics (CFD) simulations are often computationally intensive, a limitation that has led researchers to explore more efficient paradigms. This exploration has given rise to neural networks (NNs) as a transformative tool in this domain, driven by their capacity to overcome these computational bottlenecks. Notably, recent work, such as Han et al. [26, 27], leverages NNs to learn Lagrangian-based flow maps, enabling efficient and robust particle tracing in time-varying fields. These data-driven models demonstrate remarkable accuracy and speed, making them increasingly indispensable for accelerating discovery and design cycles in fluid dynamics. Despite these advancements, a significant challenge remains in providing a comprehensive understanding of the confidence associated with NN predictions in flow fields.