In this talk we show how to learn the underlying geometry of data using multiscale data diffusion, and then combine this with deep learning for prediction and inference in several different settings. First we look at capturing graphs using multiscale diffusion based geometric scattering within neural frameworks. We show how to make such networks end-to-end differentiable in order to learn rich representations spaces from which to classify and generate graphs. We then show how to extend this type of analysis to manifolds, where point-clouds of data can be similarly featurized using cascades of wavelets on data graphs to create a manifold scattering transform. Next we show how to derive Wasserstein distances between pointclouds of such data using multiscale diffusion distances. FInally we move from static to dynamic optimal transport using neural ODEs in order to learn dynamic trajectories from static snapshot data—a key problem in inference from single cell data. Throughout the talk, we present examples of such techniques being applied to massively high throughput and high dimensional datasets from biology and medicine.