# Spatio-temporal DeepKriging for Interpolation and Probabilistic Forecasting

arXiv: https://arxiv.org/abs/2306.11472 · HTML: https://mlsanity.com/paper/2306.11472/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 83 references that are in the map.
- First posted on arXiv 20 Jun 2023.

## Builds on (most cited first, 13 of 13)

- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [An Introduction to Convolutional Neural Networks](https://mlsanity.com/paper/1511.08458/) (2015)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [DLIME: A Deterministic Local Interpretable Model-Agnostic Explanations Approach for Computer-Aided Diagnosis Systems](https://mlsanity.com/paper/1906.10263/) (2019)
- [Will Artificial Intelligence supersede Earth System and Climate Models?](https://mlsanity.com/paper/2101.09126/) (2021)
- [Application of Deep Convolutional Neural Networks for Detecting Extreme Weather in Climate Datasets](https://mlsanity.com/paper/1605.01156/) (2016)
- [Missing Data Reconstruction in Remote Sensing image with a Unified Spatial-Temporal-Spectral Deep Convolutional Neural Network](https://mlsanity.com/paper/1802.08369/) (2018)
- [Time Series Forecasting Using LSTM Networks: A Symbolic Approach](https://mlsanity.com/paper/2003.05672/) (2020)
- [DeepKriging: Spatially Dependent Deep Neural Networks for Spatial Prediction](https://mlsanity.com/paper/2007.11972/) (2020)
- [Valid model-free spatial prediction](https://mlsanity.com/paper/2006.15640/) (2020)
- [Deep Integro-Difference Equation Models for Spatio-Temporal Forecasting](https://mlsanity.com/paper/1910.13524/) (2019)
- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [An Ensemble Quadratic Echo State Network for Nonlinear Spatio-Temporal Forecasting](https://mlsanity.com/paper/1708.05094/) (2017)

## Built on by (most cited first, 13 of 13)

- [Transformer with Koopman-Enhanced Graph Convolutional Network for Spatiotemporal Dynamics Forecasting](https://mlsanity.com/paper/2507.03855/) (2025)
- [An Adaptive Approach for Probabilistic Wind Power Forecasting Based on Meta-Learning](https://mlsanity.com/paper/2308.07980/) (2023)
- [STACI: Spatio-Temporal Aleatoric Conformal Inference](https://mlsanity.com/paper/2505.21658/) (2025)
- [Spatio-temporal DeepKriging in PyTorch: A Supplementary Application to Precipitation Data for Interpolation and Probabilistic Forecasting](https://mlsanity.com/paper/2509.12708/) (2025)
- [Spatio-temporal modeling and forecasting with Fourier neural operators](https://mlsanity.com/paper/2601.01813/) (2026)
- [A Subsampling Based Neural Network for Spatial Data](https://mlsanity.com/paper/2411.03620/) (2024)
- [Modeling nonstationary spatial processes with normalizing flows](https://mlsanity.com/paper/2509.12884/) (2025)
- [Predicting and Interpolating Spatiotemporal Environmental Data: A Case Study of Groundwater Storage in Bangladesh](https://mlsanity.com/paper/2511.22378/) (2025)
- [Time-aware UNet and super-resolution deep residual networks for spatial downscaling](https://mlsanity.com/paper/2512.13753/) (2025)
- [Deep classifier kriging for probabilistic spatial prediction of air quality index](https://mlsanity.com/paper/2512.23474/) (2025)
- [GraphSVR: A Graph Convolutional Support Vector Regression Framework for Robust Spatiotemporal Air Pollution Forecasting](https://mlsanity.com/paper/2605.03795/) (2026)
- [Probabilistic Classification and Uncertainty Quantification of Sahara Desert Climate Using Feedforward Neural Networks](https://mlsanity.com/paper/2605.04286/) (2026)
- [TSCoNet: A Two-Stage Copula CNN-LSTM for Uncertainty-Aware Spatio-Temporal Forecasting](https://mlsanity.com/paper/2607.10410/) (2026)

## Abstract (arXiv)

Gaussian processes (GP) and Kriging are widely used in traditional spatio-temporal mod-elling and prediction. These techniques typically presuppose that the data are observed from a stationary GP with parametric covariance structure. However, processes in real-world applications often exhibit non-Gaussianity and nonstationarity. Moreover, likelihood-based inference for GPs is computationally expensive and thus prohibitive for large datasets. In this paper we propose a deep neural network (DNN) based two-stage model for spatio-temporal interpolation and forecasting. Interpolation is performed in the first step, which utilizes a dependent DNN with the embedding layer constructed with spatio-temporal basis functions. For the second stage, we use Long-Short Term Memory (LSTM) and convolutional LSTM to forecast future observations at a given location. We adopt the quantile-based loss function in the DNN to provide probabilistic forecasting. Compared to Kriging, the proposed method does not require specifying covariance functions or making stationarity assumption, and is computationally efficient. Therefore, it is suitable for large-scale prediction of complex spatio-temporal processes. We apply our method to monthly PM2.5 data at more than 200,000 space-time locations from January 1999 to December 2022 for fast imputation of missing values and forecasts with uncertainties.