x ∈ U in an n-dimensional manifold (with U ⊆ M) can be mapped to n-dimensional Euclidean space through a continuous, bijective function. These neighborhoods and associated mappings are also known as charts. For example, the surface of the Earth is a 2D manifold and we can map the curved surface of the Earth to a flat Euclidean plane (i.e., a 2D Euclidean space) using a chart in order to measure properties such as distances or areas. The general nature of manifolds allows them to represent a broad range of structures, shapes, and complex geometric objects in molecular simulations (e.g., surface of a protein or a nanoparticle). Manifolds can also have encoded data on them (e.g., Earth surface temperature), which is captured using a continuous function f : M → R. In Figure , we present a manifold representation for a 2D simulation of water. Here, the simulation domain (e.g., a 2D plane) is a 2D manifold and we define a continuous function that captures the time-averaged density of water at each location in the domain.