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Learning Manifold and Ito Dynamics with Branched Neural Rough Differential Equations

arXiv 2606.05272 ↗First posted 3 Jun 2026cs.LGcited by 0

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Abstract (arXiv)

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 Ito dynamics require, nor the ordered covariant derivatives that govern Ito 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 Ito 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 Ito-consistent law matching by making quadratic-variation terms visible during training. On rough Bergomi volatility, sim-to-real 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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 29 of its 74 references that are in the map. First posted on arXiv 3 Jun 2026.

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29 of 74 references.

References · 29

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17 more
  1. 2021Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Controlcited by 38
  2. 2023Non-adversarial training of Neural SDEs with signature kernel scorescited by 37
  3. 2024Log Neural Controlled Differential Equations: The Lie Brackets Make a Differencecited by 35
  4. 2021KNODE-MPC: A Knowledge-based Data-driven Predictive Control Framework for Aerial Robotscited by 26
  5. 2023Generalizing Graph ODE for Learning Complex System Dynamics across Environmentscited by 23
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  8. 2019The Oxford Multimotion Dataset: Multiple SE(3) Motions with Ground Truthcited by 12
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  15. 2025Continuous-Time SO(3) Forecasting with Savitzky--Golay Neural Controlled Differential Equationscited by 2
  16. 2025Scalable Machine Learning Algorithms using Path Signaturescited by 1
  17. 2025Explicit and Effectively Symmetric Schemes for Neural SDEs on Lie Groupscited by 1

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