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.

翻译：我们研究了正则性结构理论中的模型作为机器学习任务特征的应用。模型是时空信号的多项式函数，旨在良好逼近偏微分方程的解，即使在低正则性条件下也是如此。模型可被视为路径签名的自然多维推广；因此，我们的工作旨在将近期数据科学中签名方法的用途从时间有序数据扩展到更广泛的范畴。我们给出了与时空信号关联的模型特征向量的灵活定义，并提出了两种算法来展示这些特征与线性回归结合的途径。我们将这些算法应用于多个数值实验，以学习给定强迫项和边界条件下偏微分方程的解。实验包括含强迫项的半线性抛物型方程和波动方程，以及无强迫项的Burgers方程。相比于若干替代方法，我们的算法表现出明显优势。此外，在Burgers方程实验中，我们发现当观测数据中加入噪声时，模型仍具有非平凡的预测能力。