# Feature Engineering with Regularity Structures

arXiv: https://arxiv.org/abs/2108.05879 · HTML: https://mlsanity.com/paper/2108.05879/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 54 references that are in the map.
- First posted on arXiv 12 Aug 2021.

## Builds on (most cited first, 16 of 16)

- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [MeshingNet: A New Mesh Generation Method based on Deep Learning](https://mlsanity.com/paper/2004.07016/) (2020)
- [Learning Spatial-Semantic Context with Fully Convolutional Recurrent Network for Online Handwritten Chinese Text Recognition](https://mlsanity.com/paper/1610.02616/) (2016)
- [Sparse arrays of signatures for online character recognition](https://mlsanity.com/paper/1308.0371/) (2013)
- [Persistence paths and signature features in topological data analysis](https://mlsanity.com/paper/1806.00381/) (2018)
- [Neural Networks Trained to Solve Differential Equations Learn General Representations](https://mlsanity.com/paper/1807.00042/) (2018)
- [A signature-based machine learning model for bipolar disorder and borderline personality disorder](https://mlsanity.com/paper/1707.07124/) (2017)
- [An augmented Lagrangian deep learning method for variational problems with essential boundary conditions](https://mlsanity.com/paper/2106.14348/) (2021)
- [Learning To Solve Differential Equations Across Initial Conditions](https://mlsanity.com/paper/2003.12159/) (2020)

## Built on by (most cited first, 7 of 7)

- [Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics](https://mlsanity.com/paper/2110.10249/) (2021)
- [Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs](https://mlsanity.com/paper/2204.06255/) (2022)
- [The Volterra signature](https://mlsanity.com/paper/2603.04525/) (2026)
- [2-d signature of images and texture classification](https://mlsanity.com/paper/2205.11236/) (2022)
- [SPDEBench: An Extensive Benchmark for Learning Stochastic PDEs](https://mlsanity.com/paper/2505.18511/) (2025)
- [Autoregressive models for biomedical signal processing](https://mlsanity.com/paper/2304.11070/) (2023)
- [Path Signatures for Seizure Forecasting](https://mlsanity.com/paper/2308.09312/) (2023)

## Abstract (arXiv)

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.