Spatial prediction without retraining per data model: one process model handles counts, censoring and areal aggregates

Spatial prediction can now train a process model once and attach different data models at prediction time, instead of retraining a conditional generative network for each observation type. Neural conditional simulation fixes the data model at training time, so a changed or misspecified data model forces the network to be retrained. On the Ginzburg–Landau amplitude field, one network estimating the process-model score was paired with areal-aggregate and upper-tail-censoring likelihoods, and both gave predictive diagnostics; a neural conditional simulation network trained on missing-at-random Gaussian data degraded under censoring.

Guided diffusion splits the predictive distribution into a process-model score and a data-model likelihood gradient, which are combined step by step during reverse diffusion to sample from the conditional distribution. An unconditional diffusion model learns only the process model, and conditioning enters at sampling time through the likelihood gradient rather than during training. Across Gaussian-process and Poisson-count, areal-aggregate, right-censored and area-exceedance data models, diffusion posterior sampling and pseudoinverse-guided diffusion each drew samples with 500 discretization steps and were checked against exact Gaussian predictive distributions or elliptical slice sampling.

In methane-flux inversion, guided diffusion produced predictive mean and standard-deviation maps close to Hamiltonian Monte Carlo, with per-sample cost growing linearly in the latent dimension. MCMC is feasible for this inversion but factorizes large matrices at every iteration, so its cost rises sharply with spatial resolution. In an observing system simulation experiment over the UK and Ireland, the flux field covered land cells and observations came from simulated mole fractions at Mace Head, Ridge Hill, Tacolneston and Angus; across three resolutions MCMC was roughly eleven times more expensive per effective sample at the higher resolution and no longer practical.

The method applies to spatial prediction problems where samples can be simulated from the process model and a differentiable likelihood can be built from the data model, covering direct partial observations, linear and areal aggregates, counts, right censoring and area-exceedance functionals. Demonstrations span a Gaussian process, a Ginzburg–Landau amplitude field, and an observing system simulation experiment of methane-flux inversion over the UK and Ireland. Guided diffusion does not return a posterior distribution over the process-model parameters themselves.

Relative to MCMC, diffusion posterior sampling and pseudoinverse-guided diffusion are both slightly under-dispersed and rely on several approximations that, unlike MCMC, carry no notion of asymptotic exactness. The non-Gaussian case needs enough simulated realizations from the process model to train the score network, which may be prohibitive where simulation is expensive. The process model, its prior distributions and the gridding are frozen, and no posterior over process-model parameters is returned.