Most systems in the physical world are spatio-temporal in nature. The clouds move over our heads, vehicles travel on the roads and electricity is transmitted through vast spatial networks. Machine learning offers many opportunities to understand and forecast the evolution of these systems by making use of large amounts of collected data. However, building useful models of such systems requires taking both spatial and temporal correlations into account. We can not accurately forecast the weather in Linköping without knowing if there is hot air blowing in from the south. Similarly, we can not predict if a vehicle is about to make a left turn without knowing its position and velocity relative to other vehicles on the road. This thesis proposes a set of methods for accurately capturing such spatio-temporal dependencies in machine learning models. 

At the core of the thesis is the idea of using graphs as a way to represent the spatial relationships in spatio-temporal systems. Graphs offer a highly flexible framework for this purpose, in particular for situations where observation locations do not lie on a regular spatial grid. Throughout the thesis, spatial graphs are constructed by letting nodes correspond to spatial locations and edges the relationships between them. These graphs are then used to construct different machine learning models, including graph neural networks and probabilistic graphical models. Combining such graph-based components with machine learning methods for time series modeling then allows for capturing the full spatio-temporal structure of the data. 

The main contribution of the thesis lies in exploring a number of methods using graph-based modeling for spatio-temporal data. This includes extending temporal graph neural networks to handle data observed irregularly over time. Temporal graph neural networks are also used to develop a model for vehicle trajectory forecasting, where the edges of the graph correspond to interactions between traffic agents. The thesis additionally includes work on Bayesian modeling, where a connection between Gaussian Markov random fields and graph neural networks allows for building scalable probabilistic models for data defined using graphs. 

A motivation for the methods developed in this thesis is the increasing use of machine learning in earth science. Capturing relevant spatio-temporal relationships is central for building useful models of the earth system. The thesis includes numerous experiments making use of weather and climate data, as well as application-driven work specifically targeting weather forecasting. Recent years have seen rapid progress in using machine learning models for weather forecasting, and the thesis makes multiple contributions in this direction. A probabilistic weather forecasting model is developed by combining graph-based methods with a latent variable formulation. Lastly, machine learning limited area models are also explored, where graph neural networks are used for regional weather forecasting.