Abstract: We propose a new framework, called Poisson learning, for graph based semi-supervised learning at very low label rates. Poisson learning is motivated by the need to address the degeneracy of Laplacian semi-supervised learning at very low label rates. The method replaces the assignment of label values at training points with the placement of sources and sinks, and solves the resulting Poisson equation on the graph. The outcomes are provably more stable and informative than those of Laplacian learning. Poisson learning is fast and efficient to implement, and we present numerical experiments showing the method is superior to other recent approaches to semi-supervised learning at low label rates on the MNIST, FashionMNIST, and the WebKb datasets. We also propose a graph-cut version of Poisson learning, called Poisson MBO, that gives higher accuracy and can incorporate prior knowledge of relative class sizes.