# Neural Conditional Simulation for Complex Spatial Processes

## TL;DR

This paper introduces neural conditional simulation (NCS), a neural diffusion-based method for efficient and accurate spatial process simulation conditioned on observations, applicable to complex models where traditional methods are intractable.

## Contribution

The paper presents a novel neural diffusion model for conditional spatial simulation that is amortized and flexible, outperforming traditional methods like MCMC in efficiency and accuracy.

## Key findings

- NCS accurately replicates true Gaussian process conditional distributions.
- NCS outperforms MCMC in efficiency and accuracy for spatial extremes.
- The method is broadly applicable to various spatial process models.

## Abstract

A key objective in spatial statistics is to simulate from the distribution of a spatial process at a selection of unobserved locations conditional on observations (i.e., a predictive distribution) to enable spatial prediction and uncertainty quantification. However, exact conditional simulation from this predictive distribution is intractable or inefficient for many spatial process models. In this paper, we propose neural conditional simulation (NCS), a general method for spatial conditional simulation that is based on neural diffusion models. Specifically, using spatial masks, we implement a conditional score-based diffusion model that evolves Gaussian noise into samples from a predictive distribution when given a partially observed spatial field and spatial process parameters as inputs. The diffusion model relies on a neural network that only requires unconditional samples from the spatial process for training. Once trained, the diffusion model is amortized with respect to the observations in the partially observed field, the number and locations of those observations, and the spatial process parameters, and can therefore be used to conditionally simulate from a broad class of predictive distributions without retraining the neural network. We assess the NCS-generated simulations against simulations from the true conditional distribution of a Gaussian process model, and against Markov chain Monte Carlo (MCMC) simulations from a Brown--Resnick process model for spatial extremes. In the latter case, we show that it is more efficient and accurate to conditionally simulate using NCS than classical MCMC techniques implemented in standard software. We conclude that NCS enables efficient and accurate conditional simulation from spatial predictive distributions that are challenging to sample from using traditional methods.