# Uncertainty Tube Visualization of Particle Trajectories

arXiv: https://arxiv.org/abs/2508.13505 · HTML: https://mlsanity.com/paper/2508.13505/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 68 references that are in the map.
- First posted on arXiv 19 Aug 2025.

## Builds on (most cited first, 17 of 17)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Ensemble deep learning: A review](https://mlsanity.com/paper/2104.02395/) (2021)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Uncertainty Quantification and Deep Ensembles](https://mlsanity.com/paper/2007.08792/) (2020)
- [Deep Learning for Reduced Order Modelling and Efficient Temporal Evolution of Fluid Simulations](https://mlsanity.com/paper/2107.04556/) (2021)
- [Is MC Dropout Bayesian?](https://mlsanity.com/paper/2110.04286/) (2021)
- [Exploratory Lagrangian-Based Particle Tracing Using Deep Learning](https://mlsanity.com/paper/2110.08338/) (2021)
- [Void-and-Cluster Sampling of Large Scattered Data and Trajectories](https://mlsanity.com/paper/1907.05073/) (2019)
- [Uncertainty-Aware Deep Neural Representations for Visual Analysis of Vector Field Data](https://mlsanity.com/paper/2407.16119/) (2024)
- [Glyph-Based Uncertainty Visualization and Analysis of Time-Varying Vector Fields](https://mlsanity.com/paper/2409.00042/) (2024)
- [Coherent structure coloring: identification of coherent structures from sparse data using graph theory](https://mlsanity.com/paper/1610.00197/) (2016)
- [Interactive Visualization of Time-Varying Flow Fields Using Particle Tracing Neural Networks](https://mlsanity.com/paper/2312.14973/) (2023)

## Abstract (arXiv)

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. Our key innovation is the design and implementation of a superelliptical tube that accurately captures and intuitively conveys nonsymmetric uncertainty. By integrating well-established uncertainty quantification techniques, such as Deep Ensembles, Monte Carlo Dropout (MC Dropout), and Stochastic Weight Averaging-Gaussian (SWAG), we demonstrate the practical utility of the uncertainty tube, showcasing its application on both synthetic and simulation datasets.