# Irregularly-Sampled Time Series Modeling with Spline Networks

arXiv: https://arxiv.org/abs/2210.10630 · HTML: https://mlsanity.com/paper/2210.10630/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 35 references that are in the map.
- First posted on arXiv 19 Oct 2022.

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [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)
- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [Multi-Time Attention Networks for Irregularly Sampled Time Series](https://mlsanity.com/paper/2101.10318/) (2021)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Neural Flows: Efficient Alternative to Neural ODEs](https://mlsanity.com/paper/2110.13040/) (2021)
- [Phased LSTM: Accelerating Recurrent Network Training for Long or Event-based Sequences](https://mlsanity.com/paper/1610.09513/) (2016)
- [Interpolation-Prediction Networks for Irregularly Sampled Time Series](https://mlsanity.com/paper/1909.07782/) (2019)
- [Learning Long-Term Dependencies in Irregularly-Sampled Time Series](https://mlsanity.com/paper/2006.04418/) (2020)
- [Processing of missing data by neural networks](https://mlsanity.com/paper/1805.07405/) (2018)
- [On Tractable Computation of Expected Predictions](https://mlsanity.com/paper/1910.02182/) (2019)
- [Generalised Interpretable Shapelets for Irregular Time Series](https://mlsanity.com/paper/2005.13948/) (2020)

## Built on by (most cited first, 2 of 2)

- [Universal Time-Series Representation Learning: A Survey](https://mlsanity.com/paper/2401.03717/) (2024)
- [BSAT: B-Spline Adaptive Tokenizer for Long-Term Time Series Forecasting](https://mlsanity.com/paper/2601.00698/) (2026)

## Abstract (arXiv)

Observations made in continuous time are often irregular and contain the missing values across different channels. One approach to handle the missing data is imputing it using splines, by fitting the piecewise polynomials to the observed values. We propose using the splines as an input to a neural network, in particular, applying the transformations on the interpolating function directly, instead of sampling the points on a grid. To do that, we design the layers that can operate on splines and which are analogous to their discrete counterparts. This allows us to represent the irregular sequence compactly and use this representation in the downstream tasks such as classification and forecasting. Our model offers competitive performance compared to the existing methods both in terms of the accuracy and computation efficiency.