Recent years have enjoyed a significant interest in exploiting tensor networks in the context of machine learning, both as a tool for the formulation of new learning algorithms and for enhancing the mathematical understanding of existing methods. In this talk, we will explore two readings of such a connection. On the one hand, we will consider the task of identifying the underlying non-linear governing equations, required both for obtaining an understanding and making future predictions. We will see that this problem can be addressed in a scalable way making use of tensor network based parameterizations for the governing equations. On the other hand, we will investigate the expressive power of tensor networks in probabilistic modelling. Inspired by the connection of tensor networks and machine learning, and the natural correspondence between tensor networks and probabilistic graphical models, we will provide a rigorous analysis of the expressive power of various tensor-network factorizations of discrete multivariate probability distributions. Joint work with A. Goeßmann, M. Götte, I. Roth, R. Sweke, G. Kutyniok, I. Glasser, N. Pancotti, J. I. Cirac.