# Physics-guided Deep Markov Models for Learning Nonlinear Dynamical Systems with Uncertainty

arXiv: https://arxiv.org/abs/2110.08607 · HTML: https://mlsanity.com/paper/2110.08607/

## Key facts

- Cited by 11 papers in the map (data through 7 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 16 Oct 2021.

## Builds on (most cited first, 19 of 19)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation](https://mlsanity.com/paper/1609.08144/) (2016)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [A Recurrent Latent Variable Model for Sequential Data](https://mlsanity.com/paper/1506.02216/) (2015)
- [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)
- [Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data](https://mlsanity.com/paper/1605.06432/) (2016)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [A Disentangled Recognition and Nonlinear Dynamics Model for Unsupervised Learning](https://mlsanity.com/paper/1710.05741/) (2017)
- [Motion Planning Among Dynamic, Decision-Making Agents with Deep Reinforcement Learning](https://mlsanity.com/paper/1805.01956/) (2018)
- [Learning Stochastic Recurrent Networks](https://mlsanity.com/paper/1411.7610/) (2014)
- [Deep State Space Models for Nonlinear System Identification](https://mlsanity.com/paper/2003.14162/) (2020)
- [Sockeye: A Toolkit for Neural Machine Translation](https://mlsanity.com/paper/1712.05690/) (2017)
- [Deep learning for affective computing: text-based emotion recognition in decision support](https://mlsanity.com/paper/1803.06397/) (2018)
- [Knowledge Transfer between Buildings for Seismic Damage Diagnosis through Adversarial Learning](https://mlsanity.com/paper/2002.09513/) (2020)
- [Probabilistic Inference for Structural Health Monitoring: New Modes of Learning from Data](https://mlsanity.com/paper/2103.01676/) (2021)

## Built on by (most cited first, 11 of 11)

- [Neural Extended Kalman Filters for Learning and Predicting Dynamics of Structural Systems](https://mlsanity.com/paper/2210.04165/) (2022)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [Neural modal ordinary differential equations: Integrating physics-based modeling with neural ordinary differential equations for modeling high-dimensional monitored structures](https://mlsanity.com/paper/2207.07883/) (2022)
- [Grammar-based Ordinary Differential Equation Discovery](https://mlsanity.com/paper/2504.02630/) (2025)
- [Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics](https://mlsanity.com/paper/2506.18339/) (2025)
- [Deep Belief Markov Models for POMDP Inference](https://mlsanity.com/paper/2503.13438/) (2025)
- [A Mechanistic Analysis of Transformers for Dynamical Systems](https://mlsanity.com/paper/2512.21113/) (2025)
- [Variational Rank Reduction Autoencoders for Generative Thermal Design](https://mlsanity.com/paper/2509.08515/) (2025)
- [PiGGO: Physics-Guided Learnable Graph Kalman Filters for Virtual Sensing of Nonlinear Dynamic Structures under Uncertainty](https://mlsanity.com/paper/2604.26593/) (2026)
- [Navigating Uncertainties in Machine Learning for Structural Dynamics: A Comprehensive Survey of Probabilistic and Non-Probabilistic Approaches in Forward and Inverse Problems](https://mlsanity.com/paper/2408.08629/) (2024)
- [Hybrid twinning using PBDW and DeepONet for the effective state estimation and prediction on partially known systems](https://mlsanity.com/paper/2512.11834/) (2025)

## Abstract (arXiv)

In this paper, we propose a probabilistic physics-guided framework, termed Physics-guided Deep Markov Model (PgDMM). The framework targets the inference of the characteristics and latent structure of nonlinear dynamical systems from measurement data, where exact inference of latent variables is typically intractable. A recently surfaced option pertains to leveraging variational inference to perform approximate inference. In such a scheme, transition and emission functions of the system are parameterized via feed-forward neural networks (deep generative models). However, due to the generalized and highly versatile formulation of neural network functions, the learned latent space often lacks physical interpretation and structured representation. To address this, we bridge physics-based state space models with Deep Markov Models, thus delivering a hybrid modeling framework for unsupervised learning and identification of nonlinear dynamical systems. The proposed framework takes advantage of the expressive power of deep learning, while retaining the driving physics of the dynamical system by imposing physics-driven restrictions on the side of the latent space. We demonstrate the benefits of such a fusion in terms of achieving improved performance on illustrative simulation examples and experimental case studies of nonlinear systems. Our results indicate that the physics-based models involved in the employed transition and emission functions essentially enforce a more structured and physically interpretable latent space, which is essential for enhancing and generalizing the predictive capabilities of deep learning-based models.