# Deep Gaussian Markov Random Fields for Graph-Structured Dynamical Systems

arXiv: https://arxiv.org/abs/2306.08445 · HTML: https://mlsanity.com/paper/2306.08445/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 66 references that are in the map.
- First posted on arXiv 14 Jun 2023.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Composing graphical models with neural networks for structured representations and fast inference](https://mlsanity.com/paper/1603.06277/) (2016)
- [A Disentangled Recognition and Nonlinear Dynamics Model for Unsupervised Learning](https://mlsanity.com/paper/1710.05741/) (2017)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Recurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces](https://mlsanity.com/paper/1905.07357/) (2019)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Spatio-Temporal Variational Gaussian Processes](https://mlsanity.com/paper/2111.01732/) (2021)
- [Black box variational inference for state space models](https://mlsanity.com/paper/1511.07367/) (2015)
- [Variational Auto-encoded Deep Gaussian Processes](https://mlsanity.com/paper/1511.06455/) (2015)
- [Fast Variational Learning in State-Space Gaussian Process Models](https://mlsanity.com/paper/2007.04731/) (2020)
- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [Doubly Sparse Variational Gaussian Processes](https://mlsanity.com/paper/2001.05363/) (2020)
- [EM-like Learning Chaotic Dynamics from Noisy and Partial Observations](https://mlsanity.com/paper/1903.10335/) (2019)
- [Non-separable Spatio-temporal Graph Kernels via SPDEs](https://mlsanity.com/paper/2111.08524/) (2021)
- [Sparse Algorithms for Markovian Gaussian Processes](https://mlsanity.com/paper/2103.10710/) (2021)
- [Dynamic Network Models for Forecasting](https://mlsanity.com/paper/1303.5396/) (2013)
- [Deep Gaussian Markov Random Fields](https://mlsanity.com/paper/2002.07467/) (2020)
- [Efficient Covariance Approximations for Large Sparse Precision Matrices](https://mlsanity.com/paper/1705.08656/) (2017)
- [Scalable Deep Gaussian Markov Random Fields for General Graphs](https://mlsanity.com/paper/2206.05032/) (2022)

## Abstract (arXiv)

Probabilistic inference in high-dimensional state-space models is computationally challenging. For many spatiotemporal systems, however, prior knowledge about the dependency structure of state variables is available. We leverage this structure to develop a computationally efficient approach to state estimation and learning in graph-structured state-space models with (partially) unknown dynamics and limited historical data. Building on recent methods that combine ideas from deep learning with principled inference in Gaussian Markov random fields (GMRF), we reformulate graph-structured state-space models as Deep GMRFs defined by simple spatial and temporal graph layers. This results in a flexible spatiotemporal prior that can be learned efficiently from a single time sequence via variational inference. Under linear Gaussian assumptions, we retain a closed-form posterior, which can be sampled efficiently using the conjugate gradient method, scaling favourably compared to classical Kalman filter based approaches