UWEC CERCA 2026

Deep learning models are increasingly used to analyze complex scientific data, yet the internal structure of these models remains poorly understood. Central to every such model is a latent space (LS): a compressed representation of the input data that encodes what the model has learned. We develop a framework for characterizing the shape and structure of LSs using Topological Data Analysis (TDA) and sub-Riemannian geometry (sRG). Specifically, persistent homology is used to quantifies global features such as clusters and holes of the Ls, and sRG is used to discover curvature and distance in high-dimensional, constrained spaces. Together, these tools provide a principled, interpretable window into how scientific deep learning models organize learned representations.  We apply this framework to two scientific domains:  1. Carlton applies the framework to single-cell tracking, learning a neural stochastic differential equation model for cell trajectories, and examining whether LS structure predicts position-estimation error over time. By comparing results across antibody types, this work aims to identify structural signatures in the LS that are characteristic of tracking performance. 2. Scott trains a neural operator model to reconstruct three-dimensional dark matter density fields from two-dimensional gravitational lensing images. The role and impact of the learned LS structure on dark matter density field reconstructions is addressed. Together, these projects demonstrate that TDA and sub-Riemannian geometry offer actionable insight into how scientific deep learning models represent and process complex physical data.