arxiv: 2509.20599 · v2 · submitted 2025-09-24 · 💻 cs.LG · cs.NA· math.NA

Explicit and Effectively Symmetric Schemes for Neural SDEs on Lie Groups

Daniil Shmelev , Luke Thompson , Cristopher Salvi This is my paper

Pith reviewed 2026-05-18 13:36 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA

keywords neural SDEsLie groupsreversible integratorsstochastic differential equationsmanifold-valued problemsRunge-Kutta methodsadjoint methods

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The pith

Explicit and effectively symmetric schemes enable stable near-reversible integration of neural SDEs on Lie groups.

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

The paper develops extensions of effectively symmetric schemes to stochastic differential equations and then lifts those schemes to Lie groups and homogeneous spaces. This addresses the memory and stability problems that arise when training neural models whose states live on manifolds. Prior reversible solvers either become unstable with stiff drifts or large steps or cannot be adapted to group-valued problems without losing their reversible property. The resulting integrators support accurate gradients at constant memory cost while preserving explicitness and near-reversibility.

Core claim

EES schemes are extended from ODEs to SDEs, shown to admit an efficient Williamson 2N-storage realisation, and lifted to arbitrary Lie groups via Bazavov's commutator-free construction, producing the first explicit near-reversible integrator for manifold-valued neural SDEs and thereby unlocking the reversible adjoint approach in this setting.

What carries the argument

Explicit and Effectively Symmetric (EES) schemes, which are stable near-reversible explicit Runge-Kutta methods that admit efficient storage and admit a commutator-free lift to Lie groups.

If this is right

Training of neural SDEs on manifolds can now use reversible adjoint methods that deliver both constant memory and accurate gradients.
Stability holds under stiff drift terms and large integration steps where earlier reversible methods break down.
Memory cost on manifold-valued problems drops by up to an order of magnitude relative to existing baselines.

Where Pith is reading between the lines

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

The same lifting strategy could be applied to other classes of geometric integrators used in machine learning on manifolds.
The storage-efficient realisation may generalise to higher-order schemes or to related classes of stochastic equations.
Applications in robotics or physics simulation that already model states on Lie groups could adopt these integrators for memory-constrained training.

Load-bearing premise

The schemes remain stable and near-reversible when extended from ODEs to SDEs and then lifted to Lie groups under the stiff drifts and large step sizes used in neural SDE training.

What would settle it

A direct comparison on standard Euclidean and manifold neural SDE benchmarks in which the new schemes fail to improve stability over other reversible solvers or fail to reduce memory usage by a substantial factor would disprove the central claim.

Figures

Figures reproduced from arXiv: 2509.20599 by Cristopher Salvi, Daniil Shmelev, Luke Thompson.

**Figure 1.** Figure 1: Stability domain for EES(2, 5; 1/10) compared to Kutta’s RK3 and RK4. The stability region for EES(2, 5) schemes is significantly larger than those of the ALF and Reversible Heun integrators, and is comparable to classical methods such as Kutta’s RK4, as shown in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Cross sections of the mean-square stability domains of EESR(2, 5), R(RK3) and R(RK4). 3.5 BACKPROPAGATION THROUGH EXPLICIT RUNGE–KUTTA METHODS The algorithm for backpropagation through an explicit Runge–Kutta scheme Υ of the form in equation 6 is given in Algorithm 1. We assume the solver is applied to a (neural) RDE of the form dyh t = f(y h t ; θ)dXh t , (9) where θ are learnable parameters requiring bac… view at source ↗

**Figure 3.** Figure 3: Training MSE for OU dynamics with ν = 0.2, µ = 0.1 and σ = 2. where h is a learnable affine function of the input data x = {xn}n≥0, xn ∈ R 2 , sampled from the true OU dynamics, and g, f are neural networks parametrised by θg, θf respectively. We choose the dimension of the latent representation dz = 32, and parametrise f, g as 2-layer neural networks of width 32 with LipSwish activations. The SDEs are int… view at source ↗

**Figure 4.** Figure 4: Training loss for GBM with r = 0.5 and σ = 1.5. dSθ t = g(t, Sθ t ; θg)dt + f(t, Sθ t ; θf ) ◦ dWt, where g, f are neural networks parametrised by θg, θf respectively. For simplicity, we use Stratonovich integration and rely on the Neural SDE to learn the required Itô correction term. As in the previous example, we increase the difficulty of the integration by considering extreme dynamics with parameters … view at source ↗

**Figure 5.** Figure 5: Convergence rates for H = 0.4 4.0 3.5 3.0 2.5 2.0 1.5 log10 (h) 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 log10 ( (h)) (h) for EES (2, 5) log10 ( (h)) 0.5x+c 4.0 3.5 3.0 2.5 2.0 1.5 log10 (h) 10 9 8 7 6 5 4 log10 ( (h)) (h) for EES (2, 5) log10 ( (h)) 2.0x+c 4.0 3.5 3.0 2.5 2.0 1.5 log10 (h) 2.2 2.0 1.8 1.6 1.4 1.2 1.0 log10 ( (h)) (h) for EES (2, 7) log10 ( (h)) 0.5x+c 4.0 3.5 3.0 2.5 2.0 1.5 log10 (h) 14 12 10 8 6… view at source ↗

**Figure 6.** Figure 6: Convergence rates for H = 0.5 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

**Figure 7.** Figure 7: Convergence rates for H = 0.6 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

read the original abstract

Backpropagation through (neural) SDE solvers is traditionally approached in two ways: discretise-then-optimise, which offers accurate gradients but incurs prohibitive memory costs; and optimise-then-discretise, which achieves constant memory cost by solving an auxiliary backward SDE, but suffers from slower evaluation and gradient approximation errors. Algebraically reversible solvers promise both memory efficiency and gradient accuracy, yet existing methods such as Reversible Heun are often unstable under complex models and large step sizes, and their non-standard auxiliary-state structure obstructs extension to manifold-valued SDEs. Building on the recently introduced Explicit and Effectively Symmetric (EES) schemes - a class of stable, near-reversible explicit Runge--Kutta methods - we address both limitations of existing schemes. We extend EES schemes from ODEs to SDEs and show that they admit an efficient Williamson 2N-storage realisation. Bazavov's commutator-free construction then lifts these schemes to arbitrary Lie groups and homogeneous spaces. To our knowledge, this is the first explicit (near-)reversible integrator in this setting, unlocking the reversible adjoint approach for manifold-valued problems. On Euclidean neural SDE benchmarks, our schemes improve stability under stiff drift and large steps compared with other reversible solvers, while the commutator-free lift reduces memory by up to an order of magnitude on manifold-valued problems versus other baselines. These results establish effectively symmetric integration as a unified, geometry-aware foundation for memory-efficient and stable training of neural SDEs.

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

The paper extends EES schemes to SDEs then lifts them via Bazavov's commutator-free method to get an explicit near-reversible integrator for manifold neural SDEs, with reported stability and memory wins but limited isolation of the symmetry property.

read the letter

Colleague, the main point here is that the authors have taken the existing EES schemes, extended them from ODEs to SDEs with a Williamson 2N-storage realization, and then applied Bazavov's commutator-free construction to lift the result to arbitrary Lie groups and homogeneous spaces. This produces what they present as the first explicit near-reversible integrator for manifold-valued neural SDEs, which in principle unlocks the reversible adjoint approach for backpropagation without the usual memory blow-up or the instability problems seen in Reversible Heun under stiff drifts and large steps. On the Euclidean benchmarks they show clearer stability advantages over other reversible solvers, and the manifold experiments report memory reductions up to an order of magnitude versus baselines. That is concrete and useful numerical work for people who actually train these models. The construction itself looks grounded in prior numerical analysis rather than invented from scratch. The soft spot is whether the effective symmetry survives the extensions in the regimes that matter. The original EES schemes had a proven near-reversibility for ODEs, but once stochastic integrals enter and the commutator-free lift is applied, it is not obvious that the forward-backward composition still approximates the identity to the expected order, especially at the step sizes and stiffness levels common in neural SDE training. The abstract and reported gains are consistent with the claim, yet the paper would be stronger if it isolated this property more explicitly, either through targeted numerical checks or a short error argument that accounts for the new terms. Without that, the central promise for manifold problems rests partly on extrapolation. This is aimed at researchers working on geometric deep learning or continuous-time models where the state lives on a manifold. A reader who cares about practical integrators for SDEs will find the storage trick and the lift worth looking at. It is solid enough on its own terms to deserve a serious referee, with the main request being tighter verification of the symmetry after the SDE and group extensions. I would send it to review.

Referee Report

2 major / 2 minor

Summary. The paper extends Explicit and Effectively Symmetric (EES) schemes from ODEs to SDEs, demonstrates an efficient Williamson 2N-storage realization, and applies Bazavov's commutator-free construction to lift the schemes to arbitrary Lie groups and homogeneous spaces. It positions this as the first explicit near-reversible integrator for manifold-valued neural SDEs, enabling the reversible adjoint method, with reported gains in stability under stiff drifts/large steps on Euclidean benchmarks and up to an order-of-magnitude memory reduction on manifold problems versus baselines.

Significance. If the preservation of near-reversibility and stability under the targeted regimes is rigorously established, the work would supply a geometry-aware, memory-efficient foundation for training neural SDEs on manifolds, directly addressing shortcomings of Reversible Heun and related methods. The combination of explicitness, effective symmetry, and manifold compatibility could enable broader adoption of reversible-adjoint techniques in geometric deep learning.

major comments (2)

[Abstract and §3] Abstract and §3 (SDE extension): the central claim that EES schemes remain near-reversible after extension to SDEs and the Bazavov lift rests on the assertion that stochastic integrals and commutator-free updates preserve the algebraic symmetry property proven for the original ODE schemes, yet no explicit verification, order analysis, or counter-example checks are provided for the stiff-drift/large-step-size regime typical of neural SDE training; this directly underpins the 'unlocking the reversible adjoint approach' statement.
[§5] §5 (Experiments): the stability and memory benchmarks are reported, but the protocol does not isolate whether the 2N-storage SDE realization or the Lie-algebra lift introduces O(h) asymmetry; without such isolation the cross-regime claim that the schemes improve upon Reversible Heun under the conditions where the latter is unstable cannot be fully evaluated.

minor comments (2)

[§3] Notation for the stochastic integrals in the SDE extension should be clarified with respect to the original EES Butcher tableau to avoid ambiguity when readers compare with prior ODE work.
[§4] The manuscript would benefit from an explicit statement of the precise order of near-reversibility retained after the lift, together with a short table comparing storage and symmetry error against Reversible Heun and other baselines.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points regarding the rigor of the near-reversibility claim and the experimental isolation of contributions. We address each major comment below and will incorporate the suggested clarifications and additions in the revised manuscript.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (SDE extension): the central claim that EES schemes remain near-reversible after extension to SDEs and the Bazavov lift rests on the assertion that stochastic integrals and commutator-free updates preserve the algebraic symmetry property proven for the original ODE schemes, yet no explicit verification, order analysis, or counter-example checks are provided for the stiff-drift/large-step-size regime typical of neural SDE training; this directly underpins the 'unlocking the reversible adjoint approach' statement.

Authors: We agree that the manuscript would benefit from an explicit verification of the preservation of the algebraic symmetry property under the SDE extension and the Bazavov commutator-free lift. In the revised version we will add a dedicated paragraph in §3 that derives the preservation of effective symmetry for the stochastic integrals and shows that the commutator-free updates do not introduce additional asymmetry terms. We will also include a brief order analysis together with targeted numerical checks in the stiff-drift and large-step-size regime to directly support the applicability of the reversible adjoint method. revision: yes
Referee: [§5] §5 (Experiments): the stability and memory benchmarks are reported, but the protocol does not isolate whether the 2N-storage SDE realization or the Lie-algebra lift introduces O(h) asymmetry; without such isolation the cross-regime claim that the schemes improve upon Reversible Heun under the conditions where the latter is unstable cannot be fully evaluated.

Authors: We acknowledge that the current experimental protocol does not separately quantify any O(h) asymmetry that might arise from the 2N-storage realisation versus the Lie-algebra lift. In the revision we will augment §5 with two targeted ablation experiments: one that applies the 2N-storage SDE scheme on Euclidean problems while keeping the integrator fixed, and another that isolates the commutator-free lift on manifold problems. These additions will allow a clearer assessment of each component’s contribution to any residual asymmetry and will strengthen the comparison against Reversible Heun. revision: yes

Circularity Check

0 steps flagged

EES extension to SDEs and Bazavov lift to Lie groups rests on independently grounded prior constructions

full rationale

The derivation begins from the established EES schemes for ODEs and Bazavov's commutator-free method, both treated as external inputs with separate numerical-analysis foundations. The paper then performs an explicit extension to SDEs (including 2N-storage realisation) followed by the lift to Lie groups, without any equation or claim reducing the new stability or reversibility properties to a quantity fitted or defined inside this manuscript. No self-definitional loop, fitted-input prediction, or load-bearing self-citation chain appears; the central claim of unlocking reversible adjoints on manifolds follows directly from the cited constructions plus the described extension steps. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central contribution rests on the assumption that EES stability properties transfer to SDEs and Lie groups without new instabilities, plus standard background results from numerical SDE theory and Lie group integration.

axioms (2)

domain assumption EES schemes from ODEs extend to SDEs while preserving near-reversibility and stability under stiff drift and large steps.
Invoked when the paper states the extension and reports improved stability on Euclidean benchmarks.
domain assumption Bazavov's commutator-free construction lifts the schemes to arbitrary Lie groups without losing explicitness or reversibility.
Central to the manifold extension described in the abstract.

pith-pipeline@v0.9.0 · 5804 in / 1390 out tokens · 42122 ms · 2026-05-18T13:36:46.957877+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

EES(n,m) schemes: order n with recovery of initial condition up to order m via Φ_{-h} ∘ Φ_h = id + O(h^{m+1}); extended to R(Φ) for RDEs with local error O(h^{(n+1)α}) and global recovery O(h^{(m+1)α-1})

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · 3 internal anchors

[1]

Sig-sdes model for quantitative finance

Imanol Perez Arribas, Cristopher Salvi, and Lukasz Szpruch. Sig-sdes model for quantitative finance. In ACM International Conference on AI in Finance, 2020

work page 2020
[2]

Sigdiffusions: Score-based diffusion models for time series via log-signature embeddings

Barbora Barancikova, Zhuoyue Huang, and Cristopher Salvi. Sigdiffusions: Score-based diffusion models for time series via log-signature embeddings. In The Thirteenth International Conference on Learning Representations, 2024

work page 2024
[3]

Order conditions of stochastic runge--kutta methods by b-series

Kevin Burrage and Pamela M Burrage. Order conditions of stochastic runge--kutta methods by b-series. SIAM Journal on Numerical Analysis, 38 0 (5): 0 1626--1646, 2000

work page 2000
[4]

High strong order explicit runge-kutta methods for stochastic ordinary differential equations

Kevin Burrage and Pamela Marion Burrage. High strong order explicit runge-kutta methods for stochastic ordinary differential equations. Applied Numerical Mathematics, 22 0 (1-3): 0 81--101, 1996

work page 1996
[5]

General order conditions for stochastic runge-kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

Kevin Burrage and PM Burrage. General order conditions for stochastic runge-kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems. Applied Numerical Mathematics, 28 0 (2-4): 0 161--177, 1998

work page 1998
[6]

Symmetric general linear methods

JC Butcher, AT Hill, and TJT Norton. Symmetric general linear methods. BIT Numerical Mathematics, 56: 0 1189--1212, 2016

work page 2016
[7]

B-series: algebraic analysis of numerical methods, volume 55

John C Butcher. B-series: algebraic analysis of numerical methods, volume 55. Springer, 2021

work page 2021
[8]

On the B -series composition theorem

John C Butcher, Taketomo Mitsui, Yuto Miyatake, and Shun Sato. On the B -series composition theorem. arXiv preprint arXiv:2409.08533, 2024

work page arXiv 2024
[9]

Numerical methods for ordinary differential equations

John Charles Butcher. Numerical methods for ordinary differential equations. John Wiley & Sons, 2016

work page 2016
[10]

Lecture Notes on Rough Paths and Applications to Machine Learning , 2024

Thomas Cass and Cristopher Salvi. Lecture Notes on Rough Paths and Applications to Machine Learning , 2024

work page 2024
[11]

Neural signature kernels as infinite-width-depth-limits of controlled resnets

Nicola Muca Cirone, Maud Lemercier, and Cristopher Salvi. Neural signature kernels as infinite-width-depth-limits of controlled resnets. In International Conference on Machine Learning, pp.\ 25358--25425. PMLR, 2023

work page 2023
[12]

Hopf algebras, renormalization and noncommutative geometry

Alain Connes and Dirk Kreimer. Hopf algebras, renormalization and noncommutative geometry. In Quantum field theory: perspective and prospective, pp.\ 59--109. Springer, 1999

work page 1999
[13]

A milstein-type scheme without l \'e vy area terms for sdes driven by fractional brownian motion

Aur \'e lien Deya, Andreas Neuenkirch, and Samy Tindel. A milstein-type scheme without l \'e vy area terms for sdes driven by fractional brownian motion. In Annales de l'IHP Probabilit \'e s et statistiques , volume 48, pp.\ 518--550, 2012

work page 2012
[14]

Computer simulations of multiplicative stochastic differential equations

PD Drummond and IK Mortimer. Computer simulations of multiplicative stochastic differential equations. Journal of computational physics, 93 0 (1): 0 144--170, 1991

work page 1991
[15]

New directions in the applications of rough path theory

Adeline Fermanian, Terry Lyons, James Morrill, and Cristopher Salvi. New directions in the applications of rough path theory. IEEE BITS the Information Theory Magazine, 3 0 (2): 0 41--53, 2023

work page 2023
[16]

Convergence rates for the full gaussian rough paths

Peter Friz and Sebastian Riedel. Convergence rates for the full gaussian rough paths. In Annales de l'IHP Probabilit \'e s et statistiques , volume 50, pp.\ 154--194, 2014

work page 2014
[17]

Multidimensional stochastic processes as rough paths: theory and applications, volume 120

Peter K Friz and Nicolas B Victoir. Multidimensional stochastic processes as rough paths: theory and applications, volume 120. Cambridge University Press, 2010

work page 2010
[18]

Controlling rough paths

Massimiliano Gubinelli. Controlling rough paths. Journal of Functional Analysis, 216 0 (1): 0 86--140, 2004

work page 2004
[19]

Ramification of rough paths

Massimiliano Gubinelli. Ramification of rough paths. Journal of Differential Equations, 248 0 (4): 0 693--721, 2010

work page 2010
[20]

Hairer and G

E. Hairer and G. Wanner. On the B utcher group and general multi-value methods. Computing (Arch. Elektron. Rechnen), 13 0 (1): 0 1--15, 1974. ISSN 0010-485X,1436-5057. doi:10.1007/bf02268387. URL https://doi.org/10.1007/bf02268387

work page doi:10.1007/bf02268387 1974
[21]

Geometric numerical integration

Ernst Hairer, Marlis Hochbruck, Arieh Iserles, and Christian Lubich. Geometric numerical integration. Oberwolfach Reports, 3 0 (1): 0 805--882, 2006

work page 2006
[22]

Geometric versus non-geometric rough paths

Martin Hairer and David Kelly. Geometric versus non-geometric rough paths. In Annales de l'IHP Probabilit \'e s et statistiques , volume 51, pp.\ 207--251, 2015

work page 2015
[23]

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

Diego Bricio Hernandez and Renato Spigler. Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise. BIT Numerical Mathematics, 33 0 (4): 0 654--669, 1993

work page 1993
[24]

Neural jump ordinary differential equations: Consistent continuous-time prediction and filtering

Calypso Herrera, Florian Krach, and Josef Teichmann. Neural jump ordinary differential equations: Consistent continuous-time prediction and filtering. arXiv preprint arXiv:2006.04727, 2020

work page arXiv 2006
[25]

Mean-square and asymptotic stability of the stochastic theta method

Desmond J Higham. Mean-square and asymptotic stability of the stochastic theta method. SIAM journal on numerical analysis, 38 0 (3): 0 753--769, 2000

work page 2000
[26]

Michael E. Hoffman. Combinatorics of rooted trees and H opf algebras. Trans. Amer. Math. Soc., 355 0 (9): 0 3795--3811, 2003. ISSN 0002-9947,1088-6850

work page 2003
[27]

Exact gradients for stochastic spiking neural networks driven by rough signals

Christian Holberg and Cristopher Salvi. Exact gradients for stochastic spiking neural networks driven by rough signals. Advances in Neural Information Processing Systems, 37: 0 31907--31939, 2024

work page 2024
[28]

Non-adversarial training of neural sdes with signature kernel scores

Zacharia Issa, Blanka Horvath, Maud Lemercier, and Cristopher Salvi. Non-adversarial training of neural sdes with signature kernel scores. Advances in Neural Information Processing Systems, 36: 0 11102--11126, 2023

work page 2023
[29]

Neural jump stochastic differential equations

Junteng Jia and Austin R Benson. Neural jump stochastic differential equations. Advances in Neural Information Processing Systems, 32, 2019

work page 2019
[30]

Deep signature transforms

Patrick Kidger, Patric Bonnier, Imanol Perez Arribas, Cristopher Salvi, and Terry Lyons. Deep signature transforms. Advances in neural information processing systems, 32, 2019

work page 2019
[31]

Neural controlled differential equations for irregular time series

Patrick Kidger, James Morrill, James Foster, and Terry Lyons. Neural controlled differential equations for irregular time series. Advances in neural information processing systems, 33: 0 6696--6707, 2020

work page 2020
[32]

Neural sdes as infinite-dimensional gans

Patrick Kidger, James Foster, Xuechen Li, and Terry J Lyons. Neural sdes as infinite-dimensional gans. In International conference on machine learning, pp.\ 5453--5463. PMLR, 2021 a

work page 2021
[33]

Efficient and accurate gradients for neural sdes

Patrick Kidger, James Foster, Xuechen Chen Li, and Terry Lyons. Efficient and accurate gradients for neural sdes. Advances in Neural Information Processing Systems, 34: 0 18747--18761, 2021 b

work page 2021
[34]

Kernels for sequentially ordered data

Franz J Kir \'a ly and Harald Oberhauser. Kernels for sequentially ordered data. Journal of Machine Learning Research, 20 0 (31): 0 1--45, 2019

work page 2019
[35]

Some issues in discrete approximate solution for stochastic differential equations

Y Komori, Y Saito, and T Mitsui. Some issues in discrete approximate solution for stochastic differential equations. Computers & Mathematics with Applications, 28 0 (10-12): 0 269--278, 1994

work page 1994
[36]

Stable row-type weak scheme for stochastic differential equations

Yoshio Komori and Taketomo Mitsui. Stable row-type weak scheme for stochastic differential equations. Monte Carlo Methods and Applications, 1: 0 279--300, 01 1995

work page 1995
[37]

Distribution regression for sequential data

Maud Lemercier, Cristopher Salvi, Theodoros Damoulas, Edwin Bonilla, and Terry Lyons. Distribution regression for sequential data. In International Conference on Artificial Intelligence and Statistics, pp.\ 3754--3762. PMLR, 2021

work page 2021
[38]

Log-pde methods for rough signature kernels

Maud Lemercier, Terry Lyons, and Cristopher Salvi. Log-pde methods for rough signature kernels. arXiv preprint arXiv:2404.02926, 2024

work page arXiv 2024
[39]

Scalable gradients and variational inference for stochastic differential equations

Xuechen Li, Ting-Kam Leonard Wong, Ricky TQ Chen, and David K Duvenaud. Scalable gradients and variational inference for stochastic differential equations. In Symposium on Advances in Approximate Bayesian Inference, pp.\ 1--28. PMLR, 2020

work page 2020
[40]

Differential equations driven by rough signals

Terry J Lyons. Differential equations driven by rough signals. Revista Matem \'a tica Iberoamericana , 14 0 (2): 0 215--310, 1998

work page 1998
[41]

Hopf algebras, from basics to applications to renormalization

Dominique Manchon. Hopf algebras, from basics to applications to renormalization. arXiv preprint math/0408405, 2004

work page internal anchor Pith review Pith/arXiv arXiv 2004
[42]

Efficient, accurate and stable gradients for neural odes

Sam McCallum and James Foster. Efficient, accurate and stable gradients for neural odes. arXiv preprint arXiv:2410.11648, 2024

work page arXiv 2024
[43]

Butcher series: A story of rooted trees and numerical methods for evolution equations

Robert I McLachlan, Klas Modin, Hans Munthe-Kaas, and Olivier Verdier. Butcher series: a story of rooted trees and numerical methods for evolution equations. arXiv preprint arXiv:1512.00906, 2015

work page internal anchor Pith review Pith/arXiv arXiv 2015
[44]

Neural rough differential equations for long time series

James Morrill, Cristopher Salvi, Patrick Kidger, and James Foster. Neural rough differential equations for long time series. In International Conference on Machine Learning, pp.\ 7829--7838. PMLR, 2021

work page 2021
[45]

Parallelflow: Parallelizing linear transformers via flow discretization, 2025 a

Nicola Mu c a Cirone and Cristopher Salvi. Parallelflow: Parallelizing linear transformers via flow discretization, 2025 a

work page 2025
[46]

Rough kernel hedging, 2025 b

Nicola Mu c a Cirone and Cristopher Salvi. Rough kernel hedging, 2025 b

work page 2025
[47]

Theoretical foundations of deep selective state-space models, 2024

Nicola Mu c a Cirone, Antonio Orvieto, Benjamin Walker, Cristopher Salvi, and Terry Lyons. Theoretical foundations of deep selective state-space models, 2024

work page 2024
[48]

Stable neural stochastic differential equations in analyzing irregular time series data

YongKyung Oh, Dong-Young Lim, and Sungil Kim. Stable neural stochastic differential equations in analyzing irregular time series data. arXiv preprint arXiv:2402.14989, 2024

work page arXiv 2024
[49]

A path-dependent pde solver based on signature kernels, 2024

Alexandre Pannier and Cristopher Salvi. A path-dependent pde solver based on signature kernels, 2024

work page 2024
[50]

A general implicit splitting for stabilizing numerical simulations of it \^o stochastic differential equations

WP Petersen. A general implicit splitting for stabilizing numerical simulations of it \^o stochastic differential equations. SIAM journal on numerical analysis, 35 0 (4): 0 1439--1451, 1998

work page 1998
[51]

Runge-kutta methods for rough differential equations

Martin Redmann and Sebastian Riedel. Runge-kutta methods for rough differential equations. arXiv preprint arXiv:2003.12626, 2020

work page arXiv 2003
[52]

Runge-kutta methods for rough differential equations

Martin Redmann and Sebastian Riedel. Runge-kutta methods for rough differential equations. Journal of Stochastic Analysis, 3 0 (4): 0 6, 2022

work page 2022
[53]

T-stability of numerial scheme for stochastic differential equations

Yoshihiro Saito and Taketomo Mitsui. T-stability of numerial scheme for stochastic differential equations. In Contributions in numerical mathematics, pp.\ 333--344. World Scientific, 1993

work page 1993
[54]

Stability analysis of numerical schemes for stochastic differential equations

Yoshihiro Saito and Taketomo Mitsui. Stability analysis of numerical schemes for stochastic differential equations. SIAM Journal on Numerical Analysis, 33 0 (6): 0 2254--2267, 1996

work page 1996
[55]

The signature kernel is the solution of a goursat pde

Cristopher Salvi, Thomas Cass, James Foster, Terry Lyons, and Weixin Yang. The signature kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science, 3 0 (3): 0 873--899, 2021 a

work page 2021
[56]

Higher order kernel mean embeddings to capture filtrations of stochastic processes

Cristopher Salvi, Maud Lemercier, Chong Liu, Blanka Horvath, Theodoros Damoulas, and Terry Lyons. Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems, 34: 0 16635--16647, 2021 b

work page 2021
[57]

Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise

Henri Schurz. Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise. Stochastic Analysis and Applications, 14 0 (3): 0 313--353, 1996

work page 1996
[58]

Explicit and Effectively Symmetric Runge-Kutta Methods

Daniil Shmelev, Kurusch Ebrahimi-Fard, Nikolas Tapia, and Cristopher Salvi. Explicit and effectively symmetric runge-kutta methods. arXiv preprint arXiv:2507.21006, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[59]

Structured linear cdes: Maximally expressive and parallel-in-time sequence models

Benjamin Walker, Lingyi Yang, Nicola Muca Cirone, Cristopher Salvi, and Terry Lyons. Structured linear cdes: Maximally expressive and parallel-in-time sequence models. arXiv preprint arXiv:2505.17761, 2025

work page arXiv 2025
[60]

Path integral sampler: a stochastic control approach for sampling

Qinsheng Zhang and Yongxin Chen. Path integral sampler: a stochastic control approach for sampling. arXiv preprint arXiv:2111.15141, 2021

work page arXiv 2021
[61]

Mali: A memory efficient and reverse accurate integrator for neural odes

Juntang Zhuang, Nicha C Dvornek, Sekhar Tatikonda, and James S Duncan. Mali: A memory efficient and reverse accurate integrator for neural odes. arXiv preprint arXiv:2102.04668, 2021

work page arXiv 2021
[62]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

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[63]

@esa (Ref

\@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later remove the command n...

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[64]

\@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@first@sw \@firstoftwo \@ifundefined NAT@b*@#2 \@firstoftwo @num @NAT@ctr \@secondoft...

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[65]

@open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibsetup #1 @NAT@ctr @ @openbib .11em \@plus.33em \@minus.07em 4000 4000 `\.\@m @bibit...

work page