# On the Stochastic Stability of Deep Markov Models

arXiv: https://arxiv.org/abs/2111.04601 · HTML: https://mlsanity.com/paper/2111.04601/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 73 references that are in the map.
- First posted on arXiv 8 Nov 2021.

## Builds on (most cited first, 30 of 34)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Deep Kalman Filters](https://mlsanity.com/paper/1511.05121/) (2015)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Sequential Neural Models with Stochastic Layers](https://mlsanity.com/paper/1605.07571/) (2016)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [GMNN: Graph Markov Neural Networks](https://mlsanity.com/paper/1905.06214/) (2019)
- [Recurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces](https://mlsanity.com/paper/1905.07357/) (2019)
- [Efficient Orthogonal Parametrisation of Recurrent Neural Networks Using Householder Reflections](https://mlsanity.com/paper/1612.00188/) (2016)
- [Limitations of the Lipschitz constant as a defense against adversarial examples](https://mlsanity.com/paper/1807.09705/) (2018)
- [Lyapunov spectra of chaotic recurrent neural networks](https://mlsanity.com/paper/2006.02427/) (2020)
- [Stabilizing Gradients for Deep Neural Networks via Efficient SVD Parameterization](https://mlsanity.com/paper/1803.09327/) (2018)
- [Identification of Gaussian Process State Space Models](https://mlsanity.com/paper/1705.10888/) (2017)
- [On Lyapunov Exponents for RNNs: Understanding Information Propagation Using Dynamical Systems Tools](https://mlsanity.com/paper/2006.14123/) (2020)
- [IMEXnet: A Forward Stable Deep Neural Network](https://mlsanity.com/paper/1903.02639/) (2019)
- [Unsupervised Neural Hidden Markov Models](https://mlsanity.com/paper/1609.09007/) (2016)
- [Deep Generative Markov State Models](https://mlsanity.com/paper/1805.07601/) (2018)
- [Human Pose Forecasting via Deep Markov Models](https://mlsanity.com/paper/1707.09240/) (2017)
- [Constrained Neural Ordinary Differential Equations with Stability Guarantees](https://mlsanity.com/paper/2004.10883/) (2020)
- [Factorized Inference in Deep Markov Models for Incomplete Multimodal Time Series](https://mlsanity.com/paper/1905.13570/) (2019)
- [NAIS-Net: Stable Deep Networks from Non-Autonomous Differential Equations](https://mlsanity.com/paper/1804.07209/) (2018)
- [Powering Hidden Markov Model by Neural Network based Generative Models](https://mlsanity.com/paper/1910.05744/) (2019)
- [Eigenvalue Decay Implies Polynomial-Time Learnability for Neural Networks](https://mlsanity.com/paper/1708.03708/) (2017)
- [Markov Chain Neural Networks](https://mlsanity.com/paper/1805.00784/) (2018)
- [Reinforced Deep Markov Models With Applications in Automatic Trading](https://mlsanity.com/paper/2011.04391/) (2020)

## Built on by (most cited first, 4 of 4)

- [A Multifidelity deep operator network approach to closure for multiscale systems](https://mlsanity.com/paper/2303.08893/) (2023)
- [Synthetic Time Series Forecasting with Transformer Architectures: Extensive Simulation Benchmarks](https://mlsanity.com/paper/2505.20048/) (2025)
- [Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control](https://mlsanity.com/paper/2602.02592/) (2026)
- [DeepKoopFormer: A Koopman Enhanced Transformer Based Architecture for Time Series Forecasting](https://mlsanity.com/paper/2508.02616/) (2025)

## Abstract (arXiv)

Deep Markov models (DMM) are generative models that are scalable and expressive generalization of Markov models for representation, learning, and inference problems. However, the fundamental stochastic stability guarantees of such models have not been thoroughly investigated. In this paper, we provide sufficient conditions of DMM's stochastic stability as defined in the context of dynamical systems and propose a stability analysis method based on the contraction of probabilistic maps modeled by deep neural networks. We make connections between the spectral properties of neural network's weights and different types of used activation functions on the stability and overall dynamic behavior of DMMs with Gaussian distributions. Based on the theory, we propose a few practical methods for designing constrained DMMs with guaranteed stability. We empirically substantiate our theoretical results via intuitive numerical experiments using the proposed stability constraints.