Learning Manifold and It\^o Dynamics with Branched Neural Rough Differential Equations

Researchers introduced Branched Neural Rough Differential Equations (B-NRDEs), a Hopf-algebraic framework that extends neural rough differential equations to handle Itô dynamics and manifold-valued data. The method uses algebraic structures based on rooted trees to preserve quadratic-variation terms and manifold constraints, enabling accurate simulation of stochastic processes and geometric flows. B-NRDEs demonstrated improved performance on rough Bergomi volatility, SO(3) dynamics forecasting, and SPD covariance dynamics, offering a unified approach for problems beyond the Euclidean-Stratonovich setting.

arXiv:2606.05272v1 Announce Type: new Abstract: Neural rough differential equations NRDEs stay accurate under irregular sampling while taking far fewer integration steps than standard neural differential equations, summarising a finely sampled driver by its log-signature and advancing the hidden state over coarse intervals using the log-ODE method. This efficiency rests on the shuffle algebra, the algebraic counterpart of Stratonovich calculus. This reliance means NRDEs cannot expose the quadratic-variation terms It\^o dynamics require, nor the ordered covariant derivatives that govern It\^o flows on connection-equipped manifolds. Ameliorating this, we introduce Branched Neural Rough Differential Equations B-NRDEs , a Hopf-algebraic framework that recasts the NRDE 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 It\^o dynamics, Munthe-Kaas--Wright planar rooted trees for ordered covariant derivatives on manifolds, and the shuffle algebra in the classical Stratonovich case. This yields intrinsic coarse-step dynamics that exactly preserve manifold constraints. Finally, we introduce a branched signature-kernel objective to enable It\^o-consistent law matching by making quadratic-variation terms visible during training. On rough Bergomi volatility, sim-to-real $\mathrm{SO} 3 $ dynamics forecasting, and SPD covariance dynamics, B-NRDEs offer a unified, effective approach to stochastic and manifold-valued dynamics beyond the Euclidean--Stratonovich setting.