# Fast covariance parameter estimation of spatial Gaussian process models using neural networks

arXiv: https://arxiv.org/abs/2012.15339 · HTML: https://mlsanity.com/paper/2012.15339/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Dec 2020.

## Built on by (most cited first, 25 of 25)

- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [Likelihood-Free Parameter Estimation with Neural Bayes Estimators](https://mlsanity.com/paper/2208.12942/) (2022)
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- [Simulator-Based Inference with Waldo: Confidence Regions by Leveraging Prediction Algorithms and Posterior Estimators for Inverse Problems](https://mlsanity.com/paper/2205.15680/) (2022)
- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [Neural Bayes Estimators for Irregular Spatial Data using Graph Neural Networks](https://mlsanity.com/paper/2310.02600/) (2023)
- [Likelihood-Free Frequentist Inference: Bridging Classical Statistics and Machine Learning for Reliable Simulator-Based Inference](https://mlsanity.com/paper/2107.03920/) (2021)
- [Neural Likelihood Surfaces for Spatial Processes with Computationally Intensive or Intractable Likelihoods](https://mlsanity.com/paper/2305.04634/) (2023)
- [Fast parameter estimation of Generalized Extreme Value distribution using Neural Networks](https://mlsanity.com/paper/2305.04341/) (2023)
- [Towards black-box parameter estimation](https://mlsanity.com/paper/2303.15041/) (2023)
- [Neural Bayes estimators for censored inference with peaks-over-threshold models](https://mlsanity.com/paper/2306.15642/) (2023)
- [Demonstrating the power and flexibility of variational assumptions for amortized neural posterior estimation in environmental applications](https://mlsanity.com/paper/2404.10899/) (2024)
- [Theoretical guarantees for neural estimators in parametric statistics](https://mlsanity.com/paper/2506.18508/) (2025)
- [Spherical Poisson Point Process Intensity Function Modeling and Estimation with Measure Transport](https://mlsanity.com/paper/2201.09485/) (2022)
- [Efficient Large-scale Nonstationary Spatial Covariance Function Estimation Using Convolutional Neural Networks](https://mlsanity.com/paper/2306.11487/) (2023)
- [Modeling Spatial Extremes using Non-Gaussian Spatial Autoregressive Models via Convolutional Neural Networks](https://mlsanity.com/paper/2505.03034/) (2025)
- [LatticeVision: Image to Image Networks for Modeling Non-Stationary Spatial Data](https://mlsanity.com/paper/2505.09803/) (2025)
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- [Latent Variable Sequence Identification for Cognitive Models with Neural Network Estimators](https://mlsanity.com/paper/2406.14742/) (2024)
- [Modeling Spatial Extremal Dependence of Precipitation Using Distributional Neural Networks](https://mlsanity.com/paper/2407.08668/) (2024)
- [A Subsampling Based Neural Network for Spatial Data](https://mlsanity.com/paper/2411.03620/) (2024)
- [When the whole is greater than the sum of its parts: Scaling black-box inference to large data settings through divide-and-conquer](https://mlsanity.com/paper/2412.20323/) (2024)
- [Fast Likelihood-Free Parameter Estimation for L'evy Processes](https://mlsanity.com/paper/2505.01639/) (2025)
- [A Framework for Nonstationary Gaussian Processes with Neural Network Parameters](https://mlsanity.com/paper/2507.12262/) (2025)
- [Keeping Score: Adaptive, Tuning-Free Loss Weighting for Score-Augmented Neural Ratio Estimation](https://mlsanity.com/paper/2605.12118/) (2026)

## Abstract (arXiv)

Gaussian processes (GPs) are a popular model for spatially referenced data and allow descriptive statements, predictions at new locations, and simulation of new fields. Often a few parameters are sufficient to parameterize the covariance function, and maximum likelihood (ML) methods can be used to estimate these parameters from data. ML methods, however, are computationally demanding. For example, in the case of local likelihood estimation, even fitting covariance models on modest size windows can overwhelm typical computational resources for data analysis. This limitation motivates the idea of using neural network (NN) methods to approximate ML estimates. We train NNs to take moderate size spatial fields or variograms as input and return the range and noise-to-signal covariance parameters. Once trained, the NNs provide estimates with a similar accuracy compared to ML estimation and at a speedup by a factor of 100 or more. Although we focus on a specific covariance estimation problem motivated by a climate science application, this work can be easily extended to other, more complex, spatial problems and provides a proof-of-concept for this use of machine learning in computational statistics.