Luke Thompson

Abstract

Neural rough differential equations (NRDEs) stay accurate under irregular sampling while taking far fewer integration steps than standard neural differential equations. They summarise a finely sampled driver by its log-signature and advance the hidden state over coarse intervals using the log-ODE method. But this efficiency rests on the shuffle algebra, the algebraic counterpart of Stratonovich calculus, so NRDEs cannot expose the quadratic-variation terms Itô dynamics require, nor the ordered covariant derivatives governing Itô flows on connection-equipped manifolds. We introduce Branched Neural Rough Differential Equations (B-NRDEs), which recast the log-ODE step as geometric numerical integration on the state-space manifold, matching the driving algebra to the governing calculus: Grossman-Larson rooted trees for Euclidean Itô dynamics, Munthe-Kaas-Wright planar trees for ordered covariant derivatives on manifolds, and the shuffle algebra in the Stratonovich case. A branched signature-kernel objective then makes quadratic variation visible during training, enabling Itô-consistent law matching. We demonstrate B-NRDEs on rough Bergomi volatility, sim-to-real SO(3) forecasting, and SPD covariance dynamics.

Our speaker

Luke Thompson is a PhD student at the University of Sydney working on machine learning applications of rough paths in non-Euclidean settings such as Lie groups and homogeneous spaces. His recent work develops branched neural rough differential equations for learning manifold and Itô dynamics, along with structure-preserving schemes for neural SDEs on Lie groups. More broadly, he is interested in geometric approaches to deep learning, with work appearing at venues including ICML and ICLR.

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