# Almost Surely Stable Deep Dynamics

arXiv: https://arxiv.org/abs/2103.14722 · HTML: https://mlsanity.com/paper/2103.14722/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 43 references that are in the map.
- First posted on arXiv 26 Mar 2021.

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [World Models](https://mlsanity.com/paper/1803.10122/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [A Lyapunov-based Approach to Safe Reinforcement Learning](https://mlsanity.com/paper/1805.07708/) (2018)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Gradient Descent Learns Linear Dynamical Systems](https://mlsanity.com/paper/1609.05191/) (2016)
- [Reversible Architectures for Arbitrarily Deep Residual Neural Networks](https://mlsanity.com/paper/1709.03698/) (2017)
- [Stable Recurrent Models](https://mlsanity.com/paper/1805.10369/) (2018)
- [Neural Networks with Cheap Differential Operators](https://mlsanity.com/paper/1912.03579/) (2019)
- [Stochastic Gradient Descent Learns State Equations with Nonlinear Activations](https://mlsanity.com/paper/1809.03019/) (2018)
- [Lyapunov Theory for Discrete Time Systems](https://mlsanity.com/paper/1809.05289/) (2018)
- [LSTM Neural Networks: Input to State Stability and Probabilistic Safety Verification](https://mlsanity.com/paper/1912.04377/) (2019)
- [Implicitly Defined Layers in Neural Networks](https://mlsanity.com/paper/2003.01822/) (2020)
- [Equilibrium Distributions and Stability Analysis of Gaussian Process State Space Models](https://mlsanity.com/paper/1811.06653/) (2018)
- [Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation](https://mlsanity.com/paper/1902.04332/) (2019)
- [Stability of Gaussian Process State Space Models](https://mlsanity.com/paper/1811.06646/) (2018)

## Built on by (most cited first, 16 of 16)

- [Stability Verification in Stochastic Control Systems via Neural Network Supermartingales](https://mlsanity.com/paper/2112.09495/) (2021)
- [Learning Control Policies for Stochastic Systems with Reach-avoid Guarantees](https://mlsanity.com/paper/2210.05308/) (2022)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Scalable and Efficient Continual Learning from Demonstration via a Hypernetwork-generated Stable Dynamics Model](https://mlsanity.com/paper/2311.03600/) (2023)
- [No More Pesky Hyperparameters: Offline Hyperparameter Tuning for RL](https://mlsanity.com/paper/2205.08716/) (2022)
- [Stabilizing reinforcement learning control: A modular framework for optimizing over all stable behavior](https://mlsanity.com/paper/2310.14098/) (2023)
- [MPCritic: A plug-and-play MPC architecture for reinforcement learning](https://mlsanity.com/paper/2504.01086/) (2025)
- [Learning Stabilizable Deep Dynamics Models](https://mlsanity.com/paper/2203.09710/) (2022)
- [Learning Deep Input-Output Stable Dynamics](https://mlsanity.com/paper/2206.13093/) (2022)
- [Stable Autonomous Flow Matching](https://mlsanity.com/paper/2402.05774/) (2024)
- [The Underlying Correlated Dynamics in Neural Training](https://mlsanity.com/paper/2212.09040/) (2022)
- [A Policy Optimization Method Towards Optimal-time Stability](https://mlsanity.com/paper/2301.00521/) (2023)
- [A modular framework for stabilizing deep reinforcement learning control](https://mlsanity.com/paper/2304.03422/) (2023)
- [Efficient Exploration Using Extra Safety Budget in Constrained Policy Optimization](https://mlsanity.com/paper/2302.14339/) (2023)
- [LILAD: Learning In-context Lyapunov-stable Adaptive Dynamics Models](https://mlsanity.com/paper/2511.21846/) (2025)
- [Atoms to Processes: The Role of Artificial Intelligence and Machine Learning in Chemical Engineering](https://mlsanity.com/paper/2610.02014/) (2026)

## Abstract (arXiv)

We introduce a method for learning provably stable deep neural network based dynamic models from observed data. Specifically, we consider discrete-time stochastic dynamic models, as they are of particular interest in practical applications such as estimation and control. However, these aspects exacerbate the challenge of guaranteeing stability. Our method works by embedding a Lyapunov neural network into the dynamic model, thereby inherently satisfying the stability criterion. To this end, we propose two approaches and apply them in both the deterministic and stochastic settings: one exploits convexity of the Lyapunov function, while the other enforces stability through an implicit output layer. We demonstrate the utility of each approach through numerical examples.