# Explainable Tensorized Neural Ordinary Differential Equations forArbitrary-step Time Series Prediction

arXiv: https://arxiv.org/abs/2011.13174 · HTML: https://mlsanity.com/paper/2011.13174/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Nov 2020.

## Built on by (most cited first, 8 of 8)

- [Explainable Artificial Intelligence (XAI) on TimeSeries Data: A Survey](https://mlsanity.com/paper/2104.00950/) (2021)
- [Attentive Neural Controlled Differential Equations for Time-series Classification and Forecasting](https://mlsanity.com/paper/2109.01876/) (2021)
- [ACE-NODE: Attentive Co-Evolving Neural Ordinary Differential Equations](https://mlsanity.com/paper/2105.14953/) (2021)
- [Learning Informative Representation for Fairness-aware Multivariate Time-series Forecasting: A Group-based Perspective](https://mlsanity.com/paper/2301.11535/) (2023)
- [EgPDE-Net: Building Continuous Neural Networks for Time Series Prediction with Exogenous Variables](https://mlsanity.com/paper/2208.01913/) (2022)
- [GARNN: An Interpretable Graph Attentive Recurrent Neural Network for Predicting Blood Glucose Levels via Multivariate Time Series](https://mlsanity.com/paper/2402.16230/) (2024)
- [Probabilistic Learning of Multivariate Time Series with Temporal Irregularity](https://mlsanity.com/paper/2306.09147/) (2023)
- [DeLELSTM: Decomposition-based Linear Explainable LSTM to Capture Instantaneous and Long-term Effects in Time Series](https://mlsanity.com/paper/2308.13797/) (2023)

## Abstract (arXiv)

We propose a continuous neural network architecture, termed Explainable Tensorized Neural Ordinary Differential Equations (ETN-ODE), for multi-step time series prediction at arbitrary time points. Unlike the existing approaches, which mainly handle univariate time series for multi-step prediction or multivariate time series for single-step prediction, ETN-ODE could model multivariate time series for arbitrary-step prediction. In addition, it enjoys a tandem attention, w.r.t. temporal attention and variable attention, being able to provide explainable insights into the data. Specifically, ETN-ODE combines an explainable Tensorized Gated Recurrent Unit (Tensorized GRU or TGRU) with Ordinary Differential Equations (ODE). The derivative of the latent states is parameterized with a neural network. This continuous-time ODE network enables a multi-step prediction at arbitrary time points. We quantitatively and qualitatively demonstrate the effectiveness and the interpretability of ETN-ODE on five different multi-step prediction tasks and one arbitrary-step prediction task. Extensive experiments show that ETN-ODE can lead to accurate predictions at arbitrary time points while attaining best performance against the baseline methods in standard multi-step time series prediction.