# Explicit and Effectively Symmetric Schemes for Neural SDEs on Lie Groups

arXiv: https://arxiv.org/abs/2509.20599 · HTML: https://mlsanity.com/paper/2509.20599/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 91 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 24 of 24)

- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [MALI: A memory efficient and reverse accurate integrator for Neural ODEs](https://mlsanity.com/paper/2102.04668/) (2021)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Data](https://mlsanity.com/paper/2402.14989/) (2024)
- [Neural Jump Ordinary Differential Equations: Consistent Continuous-Time Prediction and Filtering](https://mlsanity.com/paper/2006.04727/) (2020)
- [Distribution Regression for Sequential Data](https://mlsanity.com/paper/2006.05805/) (2020)
- [Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes](https://mlsanity.com/paper/2109.03582/) (2021)
- [PPFlow: Target-aware Peptide Design with Torsional Flow Matching](https://mlsanity.com/paper/2405.06642/) (2024)
- [Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups](https://mlsanity.com/paper/2405.16381/) (2024)
- [Latent SDEs on Homogeneous Spaces](https://mlsanity.com/paper/2306.16248/) (2023)
- [EquiJump: Protein Dynamics Simulation via SO(3)-Equivariant Stochastic Interpolants](https://mlsanity.com/paper/2410.09667/) (2024)
- [Efficient, Accurate and Stable Gradients for Neural ODEs](https://mlsanity.com/paper/2410.11648/) (2024)
- [pySigLib -- Fast Signature-Based Computations on CPU and GPU](https://mlsanity.com/paper/2509.10613/) (2025)
- [Continuous-Time SO(3) Forecasting with Savitzky--Golay Neural Controlled Differential Equations](https://mlsanity.com/paper/2506.06780/) (2025)
- [Self-Organized Synchronization and Voltage Stability in Networks of Synchronous Machines](https://mlsanity.com/paper/1307.2748/) (2013)

## Built on by (most cited first, 1 of 1)

- [Learning Manifold and Ito Dynamics with Branched Neural Rough Differential Equations](https://mlsanity.com/paper/2606.05272/) (2026)

## Abstract (arXiv)

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.