# DeepKriging: Spatially Dependent Deep Neural Networks for Spatial Prediction

arXiv: https://arxiv.org/abs/2007.11972 · HTML: https://mlsanity.com/paper/2007.11972/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jul 2020.

## Built on by (most cited first, 21 of 21)

- [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 networks for geospatial data](https://mlsanity.com/paper/2304.09157/) (2023)
- [Spatial Conformal Inference through Localized Quantile Regression](https://mlsanity.com/paper/2412.01098/) (2024)
- [Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint](https://mlsanity.com/paper/2409.00458/) (2024)
- [Kriging prior Regression: A Case for Kriging-Based Spatial Features with TabPFN in Soil Mapping](https://mlsanity.com/paper/2509.09408/) (2025)
- [Bivariate DeepKriging for Large-scale Spatial Interpolation of Wind Fields](https://mlsanity.com/paper/2307.08038/) (2023)
- [Spherical Poisson Point Process Intensity Function Modeling and Estimation with Measure Transport](https://mlsanity.com/paper/2201.09485/) (2022)
- [Random forests for binary geospatial data](https://mlsanity.com/paper/2302.13828/) (2023)
- [Spatial Bayesian Neural Networks](https://mlsanity.com/paper/2311.09491/) (2023)
- [A Hybrid Framework for Spatial Interpolation: Merging Data-driven with Domain Knowledge](https://mlsanity.com/paper/2409.00125/) (2024)
- [A2 Copula-Driven Spatial Bayesian Neural Network For Modeling Non-Gaussian Dependence: A Simulation Study](https://mlsanity.com/paper/2505.24006/) (2025)
- [Spatial Transfer Learning with Simple MLP](https://mlsanity.com/paper/2405.03720/) (2024)
- [A Subsampling Based Neural Network for Spatial Data](https://mlsanity.com/paper/2411.03620/) (2024)
- [Summary Statistics of Large-scale Model Outputs for Observation-corrected Outputs](https://mlsanity.com/paper/2506.15845/) (2025)
- [Modeling nonstationary spatial processes with normalizing flows](https://mlsanity.com/paper/2509.12884/) (2025)
- [Leveraging Teleconnections with Physics-Informed Graph Attention Networks for Long-Range Extreme Rainfall Forecasting in Thailand](https://mlsanity.com/paper/2510.12328/) (2025)
- [Deep classifier kriging for probabilistic spatial prediction of air quality index](https://mlsanity.com/paper/2512.23474/) (2025)
- [Permutation-preserving Functions and Neural Vecchia Covariance Kernels](https://mlsanity.com/paper/2605.05523/) (2026)
- [Zhinv: Real-time hub-height wind field reconstruction using only local sparse observations](https://mlsanity.com/paper/2607.25298/) (2026)
- [Interpretable Graph Partitioning for Spatial Prediction](https://mlsanity.com/paper/2610.16063/) (2026)

## Abstract (arXiv)

In spatial statistics, a common objective is to predict values of a spatial process at unobserved locations by exploiting spatial dependence. Kriging provides the best linear unbiased predictor using covariance functions and is often associated with Gaussian processes. However, when considering non-linear prediction for non-Gaussian and categorical data, the Kriging prediction is no longer optimal, and the associated variance is often overly optimistic. Although deep neural networks (DNNs) are widely used for general classification and prediction, they have not been studied thoroughly for data with spatial dependence. In this work, we propose a novel DNN structure for spatial prediction, where the spatial dependence is captured by adding an embedding layer of spatial coordinates with basis functions. We show in theory and simulation studies that the proposed DeepKriging method has a direct link to Kriging in the Gaussian case, and it has multiple advantages over Kriging for non-Gaussian and non-stationary data, i.e., it provides non-linear predictions and thus has smaller approximation errors, it does not require operations on covariance matrices and thus is scalable for large datasets, and with sufficiently many hidden neurons, it provides the optimal prediction in terms of model capacity. We further explore the possibility of quantifying prediction uncertainties based on density prediction without assuming any data distribution. Finally, we apply the method to predicting PM2.5 concentrations across the continental United States.