Neural SDEs as infinite-dimensional GANs

Journal article

Stochastic differential equations (SDEs) are a staple of mathematical modelling of temporal dynamics. However, a fundamental limitation has been that such models have typically been relatively inflexible, which recent work introducing Neural SDEs has sought to solve. Here, we show that the current classical approach to fitting SDEs may be approached as a special case of (Wasserstein) GANs, and in doing so the neural and classical regimes may be brought together. The input noise is Brownian motion, the output samples are time-evolving paths produced by a numerical solver, and by parameterising a discriminator as a Neural Controlled Differential Equation (CDE), we obtain Neural SDEs as (in modern machine learning parlance) continuous-time generative time series models. Unlike previous work on this problem, this is a direct extension of the classical approach without reference to either prespecified statistics or density functions. Arbitrary drift and diffusions are admissible, so as the Wasserstein loss has a unique global minima, in the infinite data limit \textit{any} SDE may be learnt.

Authors: Kidger, P; Foster, J; Li, X; Oberhauser, H; Lyons, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Journal of Machine Learning Research
• Pages: 5453-5463
• Series: Proceedings of Machine Learning Research
• Series number: 139
• Publication date: 2021-07-01
• Acceptance date: 2021-05-08
• Event title: Thirty-eighth International Conference on Machine Learning (ICML 2021)
• Event location: Virtual event
• Event website: https://icml.cc/Conferences/2021
• Event start date: 2021-07-18
• Event end date: 2021-07-24
• EISSN: 2640-3498
• Language: English
• Keywords: FFR