# Log Neural Controlled Differential Equations: The Lie Brackets Make a Difference

arXiv: https://arxiv.org/abs/2402.18512 · HTML: https://mlsanity.com/paper/2402.18512/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 33 references that are in the map.
- First posted on arXiv 28 Feb 2024.

## Builds on (most cited first, 10 of 10)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Spectral Norm Regularization for Improving the Generalizability of Deep Learning](https://mlsanity.com/paper/1705.10941/) (2017)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)

## Built on by (most cited first, 30 of 35)

- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [Oscillatory State-Space Models](https://mlsanity.com/paper/2410.03943/) (2024)
- [Comprehensive Review of Neural Differential Equations for Time Series Analysis](https://mlsanity.com/paper/2502.09885/) (2025)
- [Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models](https://mlsanity.com/paper/2505.17761/) (2025)
- [Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patching](https://mlsanity.com/paper/2405.20799/) (2024)
- [Structured Sparse Transition Matrices to Enable State Tracking in State-Space Models](https://mlsanity.com/paper/2509.22284/) (2025)
- [Parallelization of Non-linear State-Space Models: Scaling Up Liquid-Resistance Liquid-Capacitance Networks for Efficient Sequence Modeling](https://mlsanity.com/paper/2505.21717/) (2025)
- [Learning to Dissipate Energy in Oscillatory State-Space Models](https://mlsanity.com/paper/2505.12171/) (2025)
- [Learning long range dependencies through time reversal symmetry breaking](https://mlsanity.com/paper/2506.05259/) (2025)
- [Strong Stochastic Flow Maps](https://mlsanity.com/paper/2606.01086/) (2026)
- [Weight-Space Linear Recurrent Neural Networks](https://mlsanity.com/paper/2506.01153/) (2025)
- [Random Feature Representation Boosting](https://mlsanity.com/paper/2501.18283/) (2025)
- [Permutation Equivariant Neural Controlled Differential Equations for Dynamic Graph Representation Learning](https://mlsanity.com/paper/2506.20324/) (2025)
- [Data-Driven Open-Loop Simulation for Digital-Twin Operator Decision Support in Wastewater Treatment](https://mlsanity.com/paper/2604.20935/) (2026)
- [Autonomous-Flow-Based Generation](https://mlsanity.com/paper/2511.09902/) (2025)
- [Efficient Neural Controlled Differential Equations via Attentive Kernel Smoothing](https://mlsanity.com/paper/2602.02157/) (2026)
- [Why Depth Matters in Parallelizable Sequence Models: A Lie Algebraic View](https://mlsanity.com/paper/2603.05573/) (2026)
- [BioOSS: A Bio-Inspired Oscillatory State System with Spatio-Temporal Dynamics](https://mlsanity.com/paper/2510.10790/) (2025)
- [ASecond-Order SpikingSSM for Wearables](https://mlsanity.com/paper/2510.14386/) (2025)
- [Rotation-free Online Handwritten Character Recognition Using Linear Recurrent Units](https://mlsanity.com/paper/2602.01533/) (2026)
- [U-Former ODE: Fast Probabilistic Forecasting of Irregular Time Series](https://mlsanity.com/paper/2602.11738/) (2026)
- [From Path Signatures to Sequential Modeling: Incremental Signature Contributions for Offline RL](https://mlsanity.com/paper/2602.11805/) (2026)
- [Extending SSMs with the Exponentially Weighted Signature](https://mlsanity.com/paper/2603.19198/) (2026)
- [Scalable Memristive-Friendly Reservoir Computing for Time Series Classification](https://mlsanity.com/paper/2604.19343/) (2026)
- [TIDES: Implicit Time-Awareness in Selective State Space Models](https://mlsanity.com/paper/2605.09742/) (2026)
- [Reshape and Recur: Improving SSMs with Input Reshaping and Depth Recurrence](https://mlsanity.com/paper/2605.16048/) (2026)
- [Flash PD-SSM: Memory-Optimized Structured Sparse State-Space Models](https://mlsanity.com/paper/2605.19150/) (2026)
- [Faithful Embeddings of Irregular and Asynchronous Data for Online Log-NCDEs](https://mlsanity.com/paper/2605.30213/) (2026)
- [Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solvers](https://mlsanity.com/paper/2606.02623/) (2026)
- [Learning Manifold and Ito Dynamics with Branched Neural Rough Differential Equations](https://mlsanity.com/paper/2606.05272/) (2026)

## Abstract (arXiv)

The vector field of a controlled differential equation (CDE) describes the relationship between a control path and the evolution of a solution path. Neural CDEs (NCDEs) treat time series data as observations from a control path, parameterise a CDE's vector field using a neural network, and use the solution path as a continuously evolving hidden state. As their formulation makes them robust to irregular sampling rates, NCDEs are a powerful approach for modelling real-world data. Building on neural rough differential equations (NRDEs), we introduce Log-NCDEs, a novel, effective, and efficient method for training NCDEs. The core component of Log-NCDEs is the Log-ODE method, a tool from the study of rough paths for approximating a CDE's solution. Log-NCDEs are shown to outperform NCDEs, NRDEs, the linear recurrent unit, S5, and MAMBA on a range of multivariate time series datasets with up to 50,000 observations.