Learning Manifold and Itô Dynamics with Branched Neural Rough Differential Equations

• URL: http://arxiv.org/abs/2606.05272v1
• Date: Wed, 03 Jun 2026 17:12:52 GMT
• Title: Learning Manifold and Itô Dynamics with Branched Neural Rough Differential Equations
• Abstract summary: BNRDEs recast the NRDE log-ODE step as numerical integration on the state-space manifold.<n>BNRDEs offer a unified effective approach beyond Euclidean-Stratonovich setting.
• Score: 41.540565282229785
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 Itô dynamics require, nor the ordered covariant derivatives that govern Itô 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 Itô 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 Itô-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.

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