# EXIT: Extrapolation and Interpolation-based Neural Controlled Differential Equations for Time-series Classification and Forecasting

arXiv: https://arxiv.org/abs/2204.08771 · HTML: https://mlsanity.com/paper/2204.08771/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 45 references that are in the map.
- First posted on arXiv 19 Apr 2022.

## Builds on (most cited first, 21 of 21)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [DeepMind Control Suite](https://mlsanity.com/paper/1801.00690/) (2018)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [Deep learning for time series classification: a review](https://mlsanity.com/paper/1809.04356/) (2018)
- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Time Series Forecasting With Deep Learning: A Survey](https://mlsanity.com/paper/2004.13408/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [Financial Time Series Forecasting with Deep Learning : A Systematic Literature Review: 2005-2019](https://mlsanity.com/paper/1911.13288/) (2019)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Recurrent Neural Network Regularization](https://mlsanity.com/paper/1409.2329/) (2014)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [Neural Controlled Differential Equations for Online Prediction Tasks](https://mlsanity.com/paper/2106.11028/) (2021)
- [MALI: A memory efficient and reverse accurate integrator for Neural ODEs](https://mlsanity.com/paper/2102.04668/) (2021)
- [Attentive Neural Controlled Differential Equations for Time-series Classification and Forecasting](https://mlsanity.com/paper/2109.01876/) (2021)
- [Gated recurrent units viewed through the lens of continuous time dynamical systems](https://mlsanity.com/paper/1906.01005/) (2019)

## Built on by (most cited first, 22 of 22)

- [UniTime: A Language-Empowered Unified Model for Cross-Domain Time Series Forecasting](https://mlsanity.com/paper/2310.09751/) (2023)
- [ImDiffusion: Imputed Diffusion Models for Multivariate Time Series Anomaly Detection](https://mlsanity.com/paper/2307.00754/) (2023)
- [Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Data](https://mlsanity.com/paper/2402.14989/) (2024)
- [Universal Time-Series Representation Learning: A Survey](https://mlsanity.com/paper/2401.03717/) (2024)
- [Comprehensive Review of Neural Differential Equations for Time Series Analysis](https://mlsanity.com/paper/2502.09885/) (2025)
- [Unleashing The Power of Pre-Trained Language Models for Irregularly Sampled Time Series](https://mlsanity.com/paper/2408.08328/) (2024)
- [Signed Graph Neural Ordinary Differential Equation for Modeling Continuous-time Dynamics](https://mlsanity.com/paper/2312.11198/) (2023)
- [Addressing Prediction Delays in Time Series Forecasting: A Continuous GRU Approach with Derivative Regularization](https://mlsanity.com/paper/2407.01622/) (2024)
- [Time Series Forecasting with Hypernetworks Generating Parameters in Advance](https://mlsanity.com/paper/2211.12034/) (2022)
- [SeqLink: A Robust Neural-ODE Architecture for Modelling Partially Observed Time Series](https://mlsanity.com/paper/2212.03560/) (2022)
- [GDFlow: Anomaly Detection with NCDE-based Normalizing Flow for Advanced Driver Assistance System](https://mlsanity.com/paper/2409.05346/) (2024)
- [EgPDE-Net: Building Continuous Neural Networks for Time Series Prediction with Exogenous Variables](https://mlsanity.com/paper/2208.01913/) (2022)
- [SVTime: Small Time Series Forecasting Models Informed by "Physics" of Large Vision Model Forecasters](https://mlsanity.com/paper/2510.09780/) (2025)
- [FlowPath: Learning Data-Driven Manifolds with Invertible Flows for Robust Irregularly-sampled Time Series Classification](https://mlsanity.com/paper/2511.10841/) (2025)
- [Long-term Time Series Forecasting based on Decomposition and Neural Ordinary Differential Equations](https://mlsanity.com/paper/2311.04522/) (2023)
- [Bridging Dynamic Factor Models and Neural Controlled Differential Equations for Nowcasting GDP](https://mlsanity.com/paper/2409.08732/) (2024)
- [Efficient Neural Controlled Differential Equations via Attentive Kernel Smoothing](https://mlsanity.com/paper/2602.02157/) (2026)
- [MuSiCNet: A Gradual Coarse-to-Fine Framework for Irregularly Sampled Multivariate Time Series Analysis](https://mlsanity.com/paper/2412.01063/) (2024)
- [TANDEM: Temporal Attention-guided Neural Differential Equations for Missingness in Time Series Classification](https://mlsanity.com/paper/2508.17519/) (2025)
- [KAIROS: Unified Training for Universal Non-Autoregressive Time Series Forecasting](https://mlsanity.com/paper/2510.02084/) (2025)
- [SEDformer: Event-Synchronous Spiking Transformers for Irregular Telemetry Time Series Forecasting](https://mlsanity.com/paper/2602.02230/) (2026)
- [Faithful Embeddings of Irregular and Asynchronous Data for Online Log-NCDEs](https://mlsanity.com/paper/2605.30213/) (2026)

## Abstract (arXiv)

Deep learning inspired by differential equations is a recent research trend and has marked the state of the art performance for many machine learning tasks. Among them, time-series modeling with neural controlled differential equations (NCDEs) is considered as a breakthrough. In many cases, NCDE-based models not only provide better accuracy than recurrent neural networks (RNNs) but also make it possible to process irregular time-series. In this work, we enhance NCDEs by redesigning their core part, i.e., generating a continuous path from a discrete time-series input. NCDEs typically use interpolation algorithms to convert discrete time-series samples to continuous paths. However, we propose to i) generate another latent continuous path using an encoder-decoder architecture, which corresponds to the interpolation process of NCDEs, i.e., our neural network-based interpolation vs. the existing explicit interpolation, and ii) exploit the generative characteristic of the decoder, i.e., extrapolation beyond the time domain of original data if needed. Therefore, our NCDE design can use both the interpolated and the extrapolated information for downstream machine learning tasks. In our experiments with 5 real-world datasets and 12 baselines, our extrapolation and interpolation-based NCDEs outperform existing baselines by non-trivial margins.