Feature Engineering with Regularity Structures

TL;DR

This paper explores the use of regularity structure models as features for machine learning tasks involving PDE solutions, extending signature methods to space-time data and demonstrating their effectiveness through numerical experiments.

Contribution

It introduces a new framework for feature extraction using regularity structures and provides algorithms for combining these features with linear regression in PDE learning tasks.

Findings

Models outperform alternative methods in numerical experiments.

Features retain predictive power even with noisy data.

Effective for semi-linear parabolic, wave, and Burgers' equations.

Abstract

We investigate the use of models from the theory of regularity structures as features in machine learning tasks. A model is a polynomial 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 find non-trivial predictive power when noise is added to the observations.