# Deep Integro-Difference Equation Models for Spatio-Temporal Forecasting

arXiv: https://arxiv.org/abs/1910.13524 · HTML: https://mlsanity.com/paper/1910.13524/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2019.

## Built on by (most cited first, 15 of 15)

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- [Likelihood-Free Parameter Estimation with Neural Bayes Estimators](https://mlsanity.com/paper/2208.12942/) (2022)
- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [Spatio-temporal DeepKriging for Interpolation and Probabilistic Forecasting](https://mlsanity.com/paper/2306.11472/) (2023)
- [Neural Bayes Estimators for Irregular Spatial Data using Graph Neural Networks](https://mlsanity.com/paper/2310.02600/) (2023)
- [Scaling Up Bayesian Uncertainty Quantification for Inverse Problems using Deep Neural Networks](https://mlsanity.com/paper/2101.03906/) (2021)
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- [Efficient data-driven gap filling of satellite image time series using deep neural networks with partial convolutions](https://mlsanity.com/paper/2208.08781/) (2022)
- [Semiparametric Regression for Spatial Data via Deep Learning](https://mlsanity.com/paper/2301.03747/) (2023)
- [DeepMIDE: A Multi-Output Spatio-Temporal Method for Ultra-Scale Offshore Wind Energy Forecasting](https://mlsanity.com/paper/2410.20166/) (2024)
- [Spherical Poisson Point Process Intensity Function Modeling and Estimation with Measure Transport](https://mlsanity.com/paper/2201.09485/) (2022)
- [Spatio-temporal modeling and forecasting with Fourier neural operators](https://mlsanity.com/paper/2601.01813/) (2026)
- [Latent Variable Sequence Identification for Cognitive Models with Neural Network Estimators](https://mlsanity.com/paper/2406.14742/) (2024)
- [Fast Likelihood-Free Parameter Estimation for L'evy Processes](https://mlsanity.com/paper/2505.01639/) (2025)
- [Spatially-informed transformers: Injecting geostatistical covariance biases into self-attention for spatio-temporal forecasting](https://mlsanity.com/paper/2512.17696/) (2025)

## Abstract (arXiv)

Integro-difference equation (IDE) models describe the conditional dependence between the spatial process at a future time point and the process at the present time point through an integral operator. Nonlinearity or temporal dependence in the dynamics is often captured by allowing the operator parameters to vary temporally, or by re-fitting a model with a temporally-invariant linear operator in a sliding window. Both procedures tend to be excellent for prediction purposes over small time horizons, but are generally time-consuming and, crucially, do not provide a global prior model for the temporally-varying dynamics that is realistic. Here, we tackle these two issues by using a deep convolution neural network (CNN) in a hierarchical statistical IDE framework, where the CNN is designed to extract process dynamics from the process' most recent behaviour. Once the CNN is fitted, probabilistic forecasting can be done extremely quickly online using an ensemble Kalman filter with no requirement for repeated parameter estimation. We conduct an experiment where we train the model using 13 years of daily sea-surface temperature data in the North Atlantic Ocean. Forecasts are seen to be accurate and calibrated. A key advantage of our approach is that the CNN provides a global prior model for the dynamics that is realistic, interpretable, and computationally efficient. We show the versatility of the approach by successfully producing 10-minute nowcasts of weather radar reflectivities in Sydney using the same model that was trained on daily sea-surface temperature data in the North Atlantic Ocean.