# Learning stochastic differential equations using RNN with log signature features

arXiv: https://arxiv.org/abs/1908.08286 · HTML: https://mlsanity.com/paper/1908.08286/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Aug 2019.

## Built on by (most cited first, 13 of 13)

- [Sig-Wasserstein GANs for Time Series Generation](https://mlsanity.com/paper/2111.01207/) (2021)
- [Framing RNN as a kernel method: A neural ODE approach](https://mlsanity.com/paper/2106.01202/) (2021)
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- [Variational Inference for SDEs Driven by Fractional Noise](https://mlsanity.com/paper/2310.12975/) (2023)
- [Deep Signature FBSDE Algorithm](https://mlsanity.com/paper/2108.10504/) (2021)
- [AFE-CNN: 3D Skeleton-based Action Recognition with Action Feature Enhancement](https://mlsanity.com/paper/2208.03444/) (2022)
- [Sig-DEG for Distillation: Making Diffusion Models Faster and Lighter](https://mlsanity.com/paper/2508.16939/) (2025)
- [A Neural RDE-based model for solving path-dependent PDEs](https://mlsanity.com/paper/2306.01123/) (2023)
- [Deep learning-based estimation of time-dependent parameters in Markov models with application to nonlinear regression and SDEs](https://mlsanity.com/paper/2312.08493/) (2023)
- [The path to a goal: Understanding soccer possessions via path signatures](https://mlsanity.com/paper/2508.12930/) (2025)
- [Semi-Supervised Bayesian GANs with Log-Signatures for Uncertainty-Aware Credit Card Fraud Detection](https://mlsanity.com/paper/2509.00931/) (2025)
- [Signature and Log-signature for the Study of Empirical Distributions Generated with GANs](https://mlsanity.com/paper/2203.03226/) (2022)
- [Generating drawdown-realistic financial price paths using path signatures](https://mlsanity.com/paper/2309.04507/) (2023)

## Abstract (arXiv)

This paper contributes to the challenge of learning a function on streamed multimodal data through evaluation. The core of the result of our paper is the combination of two quite different approaches to this problem. One comes from the mathematically principled technology of signatures and log-signatures as representations for streamed data, while the other draws on the techniques of recurrent neural networks (RNN). The ability of the former to manage high sample rate streams and the latter to manage large scale nonlinear interactions allows hybrid algorithms that are easy to code, quicker to train, and of lower complexity for a given accuracy. We illustrate the approach by approximating the unknown functional as a controlled differential equation. Linear functionals on solutions of controlled differential equations are the natural universal class of functions on data streams. Following this approach, we propose a hybrid Logsig-RNN algorithm that learns functionals on streamed data. By testing on various datasets, i.e. synthetic data, NTU RGB+D 120 skeletal action data, and Chalearn2013 gesture data, our algorithm achieves the outstanding accuracy with superior efficiency and robustness.