Feature Engineering with Regularity Structures

We investigate the use of models from the theory of regularity structure as features in machine learning tasks. A model is a multi-linear function of a space-time signal designed to well-approximate solutions to partial differential equations (PDEs), even in low regularity regimes. Models can be seen as natural multi-dimensional generalisations of signatures of paths; our work therefore aims to extend the recent use of signatures in data science beyond the context of time-ordered data. We provide a flexible definition of a model feature vector associated to a space-time signal, along with two algorithms which illustrate ways in which these features can be combined with linear regression. We apply these algorithms in several numerical experiments designed to learn solutions to PDEs with a given forcing and boundary data. Our experiments include semi-linear parabolic and wave equations with forcing, and Burgers' equation with no forcing. We find an advantage in favour of our algorithms when compared to several alternative methods. Additionally, in the experiment with Burgers' equation, we noticed stability in the prediction power when noise is added to the observations.