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

Andi Han; Dai Shi; Junbin Gao; Lequan Lin; Luke Thompson

arxiv: 2606.05272 · v1 · pith:VC7D66TMnew · submitted 2026-06-03 · 💻 cs.LG

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

Luke Thompson , Dai Shi , Lequan Lin , Junbin Gao , Andi Han This is my paper

Pith reviewed 2026-06-28 07:06 UTC · model grok-4.3

classification 💻 cs.LG

keywords branched neural rough differential equationsItô calculusmanifold-valued dynamicsrooted treesgeometric numerical integrationbranched signaturessignature kernelrough differential equations

0 comments

The pith

Branched neural rough differential equations use rooted trees to handle Itô dynamics and manifold-valued processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard neural rough differential equations rely on the shuffle algebra, which suits Stratonovich calculus but cannot expose quadratic-variation terms needed for Itô dynamics or ordered covariant derivatives on manifolds. The paper introduces branched neural rough differential equations that recast the log-ODE step using Grossman-Larson rooted trees for Euclidean Itô cases and Munthe-Kaas-Wright planar rooted trees for manifold connections. This matching of driving algebra to governing calculus produces intrinsic coarse-step updates that preserve manifold constraints exactly. A branched signature-kernel objective then makes quadratic variations visible during training for consistent law matching. Demonstrations cover rough Bergomi volatility, SO(3) forecasting, and SPD covariance dynamics.

Core claim

B-NRDEs provide 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. A branched signature-kernel objective enables Itô-consistent law matching by making quadratic-variation terms visible during training.

What carries the argument

Branched signatures using rooted trees (Grossman-Larson for Itô, Munthe-Kaas-Wright for manifolds) to align driving algebra with the target calculus in the log-ODE step

If this is right

NRDEs can now address Itô dynamics by surfacing quadratic-variation terms that the shuffle algebra hides.
Coarse integration steps on manifolds become intrinsic and constraint-preserving without post-processing.
A single framework unifies Stratonovich, Itô, and connection-equipped manifold flows.
The branched signature-kernel objective supports direct Itô-consistent distribution matching in training.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The tree-based construction may extend naturally to other geometric structures such as Lie groups beyond SO(3) or additional covariance manifolds.
In finance applications the visibility of quadratic variations could improve calibration of rough volatility models under Itô semantics.
Sim-to-real transfer tasks involving orientation or pose data might see gains from the exact manifold preservation property.

Load-bearing premise

Recasting the NRDE log-ODE step as geometric numerical integration via specific rooted trees on the manifold will exactly preserve constraints and expose quadratic-variation terms during training.

What would settle it

Running the SPD covariance experiments and checking whether B-NRDE trajectories stay on the manifold without extra projection steps while also showing better quadratic-variation matching than standard NRDEs on rough Bergomi paths.

Figures

Figures reproduced from arXiv: 2606.05272 by Andi Han, Dai Shi, Junbin Gao, Lequan Lin, Luke Thompson.

**Figure 1.** Figure 1: Illustration of the log-ODE method for branched neural rough differential equations (B-NRDEs). The control segment Xs,t is lifted to XH s,t ∈ H for the chosen Hopf algebra H ∈ {H, HGL, HMKW}, then transformed to a log-signature λs,t in primitive coordinates p ∈ Bprim H . The tangent vector fields W are lifted through the pseudo bialgebra map FW to matching primitiveindexed vector fields on the state mani… view at source ↗

**Figure 2.** Figure 2: Riemann–Stieltjes approximations under different evaluation conventions. Redrawn from Bellingeri et al. (2024). 2.3. Vector Fields and Pseudo Bialgebra Maps Pseudo-bialgebra maps (Kern & Lyons, 2023) identify how signature coordinates act on functions through vector fields. Let M be a smooth manifold, let Γ(TM) denote its smooth vector fields, and let Diff(M) be the algebra of linear differential operato… view at source ↗

**Figure 3.** Figure 3: rBergomi sample paths generated by B-NRDE showing good fit to the underlying law. 4.3. Sim-to-Real Dynamics Forecasting Three-dimensional motion forecasting is central to robotics and computer vision, with applications to pose prediction 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Schematic description of the sim-to-real dynamics training pipeline. Adapted from Bastian et al. (2025). (Yang et al., 2021; Dunkel et al. ¨ , 2024) and occlusion-robust perception (Di et al., 2021). We consider sim-to-real forecasting of rotational dynamics, previously out of reach for signature methods due to the lack of a manifold-compatible formulation. As a deterministic task without an Ito dynamic,… view at source ↗

**Figure 5.** Figure 5: Ground truth vs. B-NRDE eigenvalue trajectories. 5. Discussion In this work, we introduced B-NRDE, a generalisation of NRDE to branched rough paths, and a branched-signaturekernel training objective for neural stochastic differential equations (SDEs). Our method extends signature methods beyond the geometric/Stratonovich setting to Ito, manifold, ˆ and Ito-on-manifold dynamics, while remaining empirically… view at source ↗

**Figure 6.** Figure 6: Positioning of our method and a hypothetical planarregularity-structure extension. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: Milliseconds to compute log-signatures for H ∈ {H, HGL, HMKW} with path dimension d ∈ [2, 24] and signature truncation depth N ∈ {2, 3}. The custom CUDA kernels implemented in pySigLib do not support tree counts greater than 1024, producing the blank cells in Figure 7a and Figure 7b. D. Experimental Details D.1. Software and Hardware Details Software details All experiments were conducted on Python 3.13 u… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

B-NRDEs swap shuffle algebra for branched trees to reach Itô and manifold settings, but the exact preservation claim needs the derivations to hold up.

read the letter

The core move here is replacing the shuffle product in NRDEs with Grossman-Larson trees for Itô and Munthe-Kaas-Wright trees for manifolds, plus a branched signature kernel loss that surfaces quadratic variation. That directly targets the two limits the abstract names: missing Itô corrections and inability to stay on connection-equipped manifolds under coarse steps.

The paper does a clean job stating the algebraic mismatch and naming the right tree structures to fix it. The applications listed—rough Bergomi, SO(3) sim-to-real, and SPD covariance—are the right test beds, and claiming a unified treatment across them is reasonable if the algebra works.

The soft spot is the central preservation claim. The abstract says the branched log-ODE step yields dynamics that lie exactly on the manifold and recover the Itô terms without truncation error, but supplies no derivation showing the neural vector field commutes with the tree retraction or that the quadratic terms appear cleanly. The stress-test note on manifold drift is the one to check first; if the full text only asserts the match without bounds or a commuting diagram, that part stays unverified.

No obvious circularity or invented entities beyond the new name and objective. The citation pattern looks standard for rough-path extensions.

This is for readers already using NRDEs or neural SDEs who hit Itô or manifold constraints. It deserves a serious referee because the target problem is real and the proposed fix is specific, even if the proofs will need close reading and the experiments will need numbers to judge effect size.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces Branched Neural Rough Differential Equations (B-NRDEs), extending NRDEs via Hopf-algebraic branched signatures and rooted-tree structures (Grossman-Larson for Euclidean Itô, Munthe-Kaas-Wright for manifolds with connection, shuffle for Stratonovich). It recasts the log-ODE step as geometric integration to exactly preserve manifold constraints and introduces a branched signature-kernel objective that exposes quadratic-variation terms for Itô-consistent law matching. Experiments are reported on rough Bergomi volatility, sim-to-real SO(3) forecasting, and SPD covariance dynamics.

Significance. If the algebraic matching and exact-preservation claims hold, the work supplies a principled unification of neural rough-path methods across Itô, Stratonovich, and manifold settings, with potential impact on stochastic volatility modeling and geometric time-series tasks. The explicit use of tree algebras to surface quadratic variation during training is a concrete technical contribution.

major comments (2)

[Abstract] Abstract (final paragraph) and the central derivation of the log-ODE step: the claim that Munthe-Kaas-Wright / Grossman-Larson trees yield dynamics that lie exactly on the manifold (no drift off the constraint) and recover the Itô correction without truncation error at finite depth is asserted but not accompanied by an explicit commutativity argument between the neural vector field and the tree-based retraction, nor by an error analysis for the finite-depth branched signature. This is load-bearing for the headline claim of exact constraint preservation.
[Method / algebraic construction (likely §3)] The equivalence between the classical NRDE shuffle-algebra log-ODE and the new branched-tree integrators is stated as following directly from matching the driving algebra to the governing calculus, yet no intermediate algebraic identity or diagram is supplied showing how the neural vector field is lifted to the Grossman-Larson or Munthe-Kaas-Wright algebra while preserving the manifold embedding.

minor comments (1)

[Abstract / §4] Notation for the branched signature truncation depth and the precise form of the kernel objective should be introduced with an explicit equation rather than only in prose.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments both concern the need for more explicit algebraic derivations to support the exact manifold preservation and Itô-correction claims. We agree these clarifications will strengthen the manuscript and will supply them in revision.

read point-by-point responses

Referee: [Abstract] Abstract (final paragraph) and the central derivation of the log-ODE step: the claim that Munthe-Kaas-Wright / Grossman-Larson trees yield dynamics that lie exactly on the manifold (no drift off the constraint) and recover the Itô correction without truncation error at finite depth is asserted but not accompanied by an explicit commutativity argument between the neural vector field and the tree-based retraction, nor by an error analysis for the finite-depth branched signature. This is load-bearing for the headline claim of exact constraint preservation.

Authors: We accept that an explicit commutativity argument and finite-depth error analysis are required to make the exact-preservation claim fully rigorous. In the revised manuscript we will insert a new subsection (after the definition of the branched log-ODE step) that (i) proves the neural vector field commutes with the retraction induced by the Munthe-Kaas–Wright / Grossman–Larson rooted-tree series, thereby showing the integrated flow remains on the manifold by algebraic construction, and (ii) supplies a remainder estimate for the truncated branched signature that quantifies how the Itô quadratic-variation correction is recovered exactly in the infinite-depth limit and with an explicit O(2^{-N}) bound at depth N. These additions directly address the load-bearing claim. revision: yes
Referee: [Method / algebraic construction (likely §3)] The equivalence between the classical NRDE shuffle-algebra log-ODE and the new branched-tree integrators is stated as following directly from matching the driving algebra to the governing calculus, yet no intermediate algebraic identity or diagram is supplied showing how the neural vector field is lifted to the Grossman-Larson or Munthe-Kaas-Wright algebra while preserving the manifold embedding.

Authors: We agree that an explicit lifting diagram and intermediate identities would make the equivalence transparent. In the revised §3 we will add a commutative diagram (Figure 3) together with the corresponding algebraic identities that show how the neural vector field is lifted first to the Grossman–Larson algebra (Euclidean Itô), then to the Munthe-Kaas–Wright planar trees (manifold case), with the classical shuffle algebra recovered as the quotient by the ideal generated by the Lie bracket. The diagram will also indicate the embedding of the manifold constraint at each step. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation introduces independent algebraic extensions

full rationale

The paper presents B-NRDEs as a new Hopf-algebraic framework that recasts the NRDE log-ODE step via Grossman-Larson and Munthe-Kaas-Wright rooted trees to handle Itô dynamics and manifold constraints. The abstract asserts that matching the driving algebra (branched signatures) to the governing calculus yields intrinsic dynamics that exactly preserve manifold constraints and expose quadratic-variation terms. No equations or claims in the provided text reduce any prediction or result to a fitted input by construction, nor do they rely on self-citations for load-bearing uniqueness theorems or ansatzes. The central construction is positioned as an extension of existing shuffle-algebra NRDEs with new tree algebras, without tautological redefinitions or renaming of known results. This is self-contained against external benchmarks of rough-path theory and geometric integration.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Abstract-only review; specific free parameters and invented entities cannot be enumerated beyond the high-level framework. The approach rests on standard Hopf-algebra and rough-path background plus the modeling choice that tree structures will match the target calculus.

axioms (2)

standard math The shuffle algebra is the algebraic counterpart of Stratonovich calculus
Stated as the basis for classical NRDEs
domain assumption Hopf-algebraic rooted trees can recast log-ODE steps as geometric integration on manifolds
Core modeling assumption of the B-NRDE construction

invented entities (2)

Branched Neural Rough Differential Equations (B-NRDEs) no independent evidence
purpose: Unified handling of Itô and manifold dynamics
New framework introduced to overcome NRDE limitations
Branched signature-kernel objective no independent evidence
purpose: Make quadratic-variation terms visible during training
New training objective for Itô consistency

pith-pipeline@v0.9.1-grok · 5801 in / 1367 out tokens · 23096 ms · 2026-06-28T07:06:01.476200+00:00 · methodology

discussion (0)

Reference graph

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URL http://arxiv.org/abs/2506.067

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Bayer, C., Friz, P

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URL http://arxiv.org/abs/1310.4054 . arXiv:1310.4054 [math]. Gatheral, J., Jaisson, T., and Rosenbaum, M. V olatility is rough, October 2014. URL http://arxiv.org/ab s/1410.3394. arXiv:1410.3394 [q-fin]. Gierjatowicz, P., Sabate-Vidales, M., ˇSiˇska, D., Szpruch, L., and ˇZuriˇc, v. Robust pricing and hedging via neural SDEs, July 2020. URL http://arxiv.o...

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URL http://arxiv.org/abs/1303.5113 . arXiv:1303.5113 [math]. Hairer, M. and Kelly, D. Geometric versus non-geometric rough paths, January 2014. URL http://arxiv.or g/abs/1210.6294. arXiv:1210.6294 [math]. Hu, P., Meng, Q., Chen, B., Gong, S., Wang, Y ., Chen, W., Zhu, R., Ma, Z.-M., and Liu, T.-Y . Neural Operator with Regularity Structure for Modeling Dy...

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doi: 10.1016/B978-0-12-814725-2.00010-8. URL https://www.sciencedirect.com/scienc e/chapter/edited-volume/abs/pii/B9780 128147252000108. Rahm, L. Planar Regularity Structures, December 2022. URL http://arxiv.org/abs/2212.04856 . arXiv:2212.04856 [math]. Reiss, A., Indlekofer, I., Schmidt, P., and Van Laerhoven, K. Deep ppg: Large-scale heart rate estimati...

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ISSN 1424-8220. doi: 10.3390/s19143079. URL http://dx.doi.org/10.3390/s19143079. Salvi, C., Cass, T., Foster, J., Lyons, T., and Yang, W. The Signature Kernel is the solution of a Goursat PDE.SIAM Journal on Mathematics of Data Science, 3(3):873–899, January 2021. ISSN 2577-0187. doi: 10.1137/20M13667

work page doi:10.3390/s19143079 2021

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2018

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Branched neural rough differential equation (B-NRDE): multilayer perceptron (MLP) To save parameters in the readout MLP, all models output vectors inR6, which are transformed back into R3×3 rotation matrices via a Gram-Schmidt procedure (Zhou et al., 2020). D.4. Mean-reverting Dynamics over the SPD Manifold under Affine-invariant Geometry We simulate Equa...

2020