Learning Manifold and Itô Dynamics with Branched Neural Rough Differential Equations

2026-06-03

AI Summary

This work addresses the challenge of unifying irregularly sampled data, Itô stochastic dynamics, and covariant evolution under manifold constraints within a single neural differential equation framework. The authors propose Branched Neural Rough Differential Equations (B-NRDEs), which leverage a Hopf algebraic structure to align the algebraic properties of the driving signal with the calculus type of the target dynamics, enabling geometric numerical integration that preserves manifold structure over coarse time steps. This approach provides the first unified model for Itô processes in Euclidean space, covariant dynamics on manifolds equipped with connections, and Stratonovich interpretations, while explicitly capturing quadratic variation through branched signature kernels. Empirical results demonstrate significant improvements over existing methods in tasks including rough Bergomi volatility modeling, SO(3) attitude simulation, and SPD covariance evolution, effectively supporting non-Euclidean and Itô stochastic dynamics.

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.

Problem

• Neural Rough Differential Equations
• Itô dynamics
• manifold-valued processes
• quadratic variation
• stochastic calculus

Innovation

• Branched Neural Rough Differential Equations
• Itô dynamics
• manifold-valued learning
• Hopf algebra
• signature kernel