Graph-based learning is concerned with applying machine learning (e.g., classification, clustering, or regression) to graph-structured data. A graph structure encodes interdependencies among constituents, such as social media users, images, videos, or physical or biological agents, and provides a convenient representation of high dimensional data that has proven to be highly effective in machine learning. In this talk, we will show how a variety of machine learning problems on graphs can be interpreted as numerical schemes for solving partial differential equations (PDEs). This connection between PDEs and graph-based learning allows us to utilize theoretical and computational tools from the field of PDEs to study machine learning problems and develop new algorithms. This talk will overview 3 problems at the intersection of graph-based learning and PDEs. We will first present a new approach to graph-based semi-supervised learning called Poisson learning that is well-posed at arbitrarily low label rates, and discuss its application to active learning on graphs. Second, we will present a new method for detecting the boundary of a point cloud that has theoretical guarantees for solving PDE boundary-value problems. And finally, we will discuss provably robust approximations of graph distances obtained through solving graph Hamilton-Jacobi equations, and present applications to data depth.