# Advances in Neural Controlled Differential Equations

arXiv: https://arxiv.org/abs/2607.05280 · HTML: https://mlsanity.com/paper/2607.05280/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 232 references that are in the map.
- First posted on arXiv 6 Jul 2026.

## Builds on (most cited first, 15 of 15)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [RWKV-7 "Goose" with Expressive Dynamic State Evolution](https://mlsanity.com/paper/2503.14456/) (2025)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)
- [Sparse arrays of signatures for online character recognition](https://mlsanity.com/paper/1308.0371/) (2013)
- [Lecture notes on rough paths and applications to machine learning](https://mlsanity.com/paper/2404.06583/) (2024)
- [Rough kernel hedging](https://mlsanity.com/paper/2501.09683/) (2025)
- [Sparse Signature Coefficient Recovery via Kernels](https://mlsanity.com/paper/2412.08579/) (2024)
- [Nowcasting using regression on signatures](https://mlsanity.com/paper/2305.10256/) (2023)
- [Genus expansion for non-linear random matrix ensembles with applications to neural networks](https://mlsanity.com/paper/2407.08459/) (2024)
- [ParallelFlow: Parallelizing Linear Transformers via Flow Discretization](https://mlsanity.com/paper/2504.00492/) (2025)
- [GCN-DevLSTM: Path Development for Skeleton-Based Action Recognition](https://mlsanity.com/paper/2403.15212/) (2024)
- [GDFlow: Anomaly Detection with NCDE-based Normalizing Flow for Advanced Driver Assistance System](https://mlsanity.com/paper/2409.05346/) (2024)

## Abstract (arXiv)

Many real-world systems evolve continuously, yet most machine learning models interpret time series as discrete sequences. Continuous-time approaches instead treat time series as samples from an underlying input path, a formulation that naturally accommodates irregularly sampled or oversampled data. Among these, Neural Controlled Differential Equations (NCDEs) are a maximally expressive class of models that parametrise a vector field using a neural network and evolve their hidden state by solving a dynamical system driven by the input path. NCDEs typically use a non-linear vector field, so their expressive power and continuous-time flexibility come at the cost of a forward pass that is both computationally expensive and inherently sequential, limiting their scalability and practical applicability. This thesis advances the training and scalability of NCDEs through three complementary contributions. First, building on neural rough differential equations, Log-NCDEs apply the Log-ODE method to efficiently approximate an NCDE's solution during training, improving both computational speed and empirical performance. Second, Linear NCDEs replace the non-linear vector field with a linear one, enabling closed-form solutions and parallel-in-time computation without sacrificing theoretical expressivity. Third, Structured Linear NCDEs use structured linear vector fields to further enhance efficiency while maintaining theoretical expressiveness and empirical performance. Collectively, these methods reduce the time per training step for an NCDE by up to three orders of magnitude while achieving state-of-the-art performance across diverse time series benchmarks.