Spatial heterogeneity has been regarded as an important issue in space–time prediction. Although some statistical methods of space–time predictions have been proposed to address spatial heterogeneity, the linear assumption makes it difficult for these methods to predict geographical processes accurately because geographical processes always involve complicated nonlinear characteristics. An extreme learning machine (ELM) has the advantage of approximating nonlinear relationships with a rapid learning speed and excellent generalization performance. However, determining how to incorporate spatial heterogeneity into an ELM to predict space–time data is an urgent problem. For this purpose, a new method called geographically weighted ELM (GWELM) is proposed to address spatial heterogeneity based on an ELM in this article. GWELM is essentially a locally varying ELM in which the parameters are regarded as functions of spatial locations, and geographically weighted least squares is applied to estimate the parameters in a local model. The proposed method is used to analyze two groups of different data sets, and the results demonstrate that the GWELM method is superior to the comparative method, which is also developed to address spatial heterogeneity.