# Spatio-temporal probabilistic forecast using MMAF-guided learning

arXiv: https://arxiv.org/abs/2603.15055 · HTML: https://mlsanity.com/paper/2603.15055/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 56 references that are in the map.
- First posted on arXiv 16 Mar 2026.

## Builds on (most cited first, 23 of 23)

- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Neural networks for post-processing ensemble weather forecasts](https://mlsanity.com/paper/1805.09091/) (2018)
- [DiffSTG: Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models](https://mlsanity.com/paper/2301.13629/) (2023)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [A Survey on Diffusion Models for Time Series and Spatio-Temporal Data](https://mlsanity.com/paper/2404.18886/) (2024)
- [Simpler PAC-Bayesian Bounds for Hostile Data](https://mlsanity.com/paper/1610.07193/) (2016)
- [f-Divergence Variational Inference](https://mlsanity.com/paper/2009.13093/) (2020)
- [A Practical Probabilistic Benchmark for AI Weather Models](https://mlsanity.com/paper/2401.15305/) (2024)
- [On the Difficulty of Unbiased Alpha Divergence Minimization](https://mlsanity.com/paper/2010.09541/) (2020)
- [Dynamic Causal Explanation Based Diffusion-Variational Graph Neural Network for Spatio-temporal Forecasting](https://mlsanity.com/paper/2305.09703/) (2023)
- [Generalised Bayesian Inference for Discrete Intractable Likelihood](https://mlsanity.com/paper/2206.08420/) (2022)
- [Physics-Informed Variational State-Space Gaussian Processes](https://mlsanity.com/paper/2409.13876/) (2024)
- [Spatio-temporal Ornstein-Uhlenbeck processes: theory, simulation and statistical inference](https://mlsanity.com/paper/1504.08327/) (2015)
- [Bridging between short-range and long-range dependence with mixed spatio-temporal Ornstein-Uhlenbeck processes](https://mlsanity.com/paper/1704.04070/) (2017)
- [ConvGRU in Fine-grained Pitching Action Recognition for Action Outcome Prediction](https://mlsanity.com/paper/2008.07819/) (2020)
- [High-frequency Estimation of the L'evy-driven Graph Ornstein-Uhlenbeck process](https://mlsanity.com/paper/2008.10930/) (2020)
- [Mixed moving average field guided learning for spatio-temporal data](https://mlsanity.com/paper/2301.00736/) (2023)
- [Calibrated Generalized Bayesian Inference](https://mlsanity.com/paper/2311.15485/) (2023)

## Abstract (arXiv)

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.