In Geostatistics, Gaussian random fields have been extensively used to model complex spatial data. In such cases, non stationary Gaussian random fields offer the flexibility to handle spatially varying correlation structures observed in the data. But bayesian predictions based on such models, which require first the inference of the parameter posterior distributions, generally come at a high operational costs (computational and storage-wise), thus limiting the use of such models on large datasets. In this talk, we present an approach that builds on the recent advances in score-based generative-models to tackle this problem. Our contribution is two-fold, we first show that score-based models are able to faithfully reproduce the distribution of non-stationary Gaussian processes arising from SPDE models with spatially varying coefficients and then how recent advances in aposteriori conditioning of such models can be leveraged to offer an efficient framework for the prediction of partially observed fields. We apply this framework to the bayesian predictions of spatial data arising from climate applications.