Spatio-temporal probabilistic forecast using MMAF-guided learning

arXiv stat.ML / 4/24/2026

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Key Points

The paper proposes a theory-guided generalized Bayesian framework for probabilistic spatio-temporal forecasting on raster data.
It trains an ensemble of stochastic feed-forward neural networks with Gaussian-distributed weights, embedding the dependence and causal structure of a spatio-temporal Ornstein–Uhlenbeck process into both training and inference via constrained optimization.
During inference, the method produces causal ensemble forecasts by using different initial conditions for different forecast horizons (called MMAF-guided learning).
Experiments on synthetic and real datasets show forecasts stay well calibrated across multiple time horizons.
The authors find that shallow feed-forward models can match or outperform more complex convolutional or diffusion-based deep learning architectures for probabilistic forecasting in these settings.

Abstract

We present a theory-guided generalized Bayesian methodology for spatio-temporal raster data, which we use to train an ensemble of stochastic feed-forward neural networks with Gaussian-distributed weights. The methodology incorporates the dependence and causal structure of a spatio-temporal Ornstein-Uhlenbeck process into training and inference by enforcing constraints on the design of the data embedding and the related optimization routine. In inference mode, the networks are employed to generate causal ensemble forecasts by applying different initial conditions at different horizons. We call this workflow MMAF-guided learning. Experiments conducted on both synthetic and real data demonstrate that our forecasts remain calibrated across multiple time horizons. Moreover, we show that on such data, shallow feed-forward architectures can achieve performance comparable to, and in some cases better than, convolutional or diffusion deep learning architectures used in probabilistic forecasting tasks.