# Modeling nonstationary spatial processes with normalizing flows

arXiv: https://arxiv.org/abs/2509.12884 · HTML: https://mlsanity.com/paper/2509.12884/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 47 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 19 of 19)

- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets](https://mlsanity.com/paper/1406.7343/) (2014)
- [A general framework for Vecchia approximations of Gaussian processes](https://mlsanity.com/paper/1708.06302/) (2017)
- [Permutation and Grouping Methods for Sharpening Gaussian Process Approximations](https://mlsanity.com/paper/1609.05372/) (2016)
- [DeepKriging: Spatially Dependent Deep Neural Networks for Spatial Prediction](https://mlsanity.com/paper/2007.11972/) (2020)
- [Deep Compositional Spatial Models](https://mlsanity.com/paper/1906.02840/) (2019)
- [Non-Stationary Spatial Modeling](https://mlsanity.com/paper/2212.08043/) (2022)
- [Spatio-temporal DeepKriging for Interpolation and Probabilistic Forecasting](https://mlsanity.com/paper/2306.11472/) (2023)
- [Review: Nonstationary Spatial Modeling, with Emphasis on Process Convolution and Covariate-Driven Approaches](https://mlsanity.com/paper/1610.02447/) (2016)
- [Bayesian Monotone Regression using Gaussian Process Projection](https://mlsanity.com/paper/1306.4041/) (2013)
- [Modeling Nonstationary and Asymmetric Multivariate Spatial Covariances via Deformations](https://mlsanity.com/paper/2004.08724/) (2020)
- [Estimation of Spatial Deformation for Nonstationary Processes via Variogram Alignment](https://mlsanity.com/paper/1911.02249/) (2019)
- [Bivariate DeepKriging for Large-scale Spatial Interpolation of Wind Fields](https://mlsanity.com/paper/2307.08038/) (2023)
- [Spatial Deep Convolutional Neural Networks](https://mlsanity.com/paper/2409.07559/) (2024)
- [Efficient Large-scale Nonstationary Spatial Covariance Function Estimation Using Convolutional Neural Networks](https://mlsanity.com/paper/2306.11487/) (2023)
- [A Statistician's Overview of Physics-Informed Neural Networks for Spatio-Temporal Data](https://mlsanity.com/paper/2507.14336/) (2025)

## Abstract (arXiv)

Nonstationary spatial processes can often be represented as stationary processes on a warped spatial domain. Selecting an appropriate spatial warping function for a given application is often difficult and, as a result of this, warping methods have largely been limited to two-dimensional spatial domains. In this paper, we introduce a novel approach to modeling nonstationary, anisotropic spatial processes using neural autoregressive flows (NAFs), a class of invertible mappings capable of generating complex, high-dimensional warpings. Through simulation studies we demonstrate that a NAF-based model has greater representational capacity than other commonly used spatial process models. We apply our proposed modeling framework to a subset of the 3D Argo Floats dataset, highlighting the utility of our framework in real-world applications.