In spatio-temporal data analysis, estimating covariance parameters for Gaussian random fields via maximum likelihood estimation (MLE) is computationally prohibitive for large datasets. Following recent advances in simulation-based inference techniques, we propose two neural network architectures - Neural Bayes Estimation (NBE) and Neural Ratio Estimation (NRE) - to learn the parameters of Gaussian spatio-temporal Gneiting-Mat{é}rn models with advection. The networks are trained on simulations generated directly from the underlying Gaussian random field model. A thorough simulation study demonstrates strong performance across one-dimensional time and both one-dimensional space and two-dimensional space, yielding high accuracy and reliable uncertainty quantification for both NBE and NRE. Through extensive offline training, the framework achieves fully amortized inference, replacing costly optimization with a single forward pass. Finally, we apply our framework to solar radiation prediction and short-term forecasting. By adopting a sliding-window approach, the model accounts for temporal non-stationarity while retaining this key advantage: achieving near-instantaneous parameter inference and forecasting.

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