# LSTM Neural Networks: Input to State Stability and Probabilistic Safety Verification

arXiv: https://arxiv.org/abs/1912.04377 · HTML: https://mlsanity.com/paper/1912.04377/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Dec 2019.

## Built on by (most cited first, 13 of 13)

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- [On Recurrent Neural Networks for learning-based control: recent results and ideas for future developments](https://mlsanity.com/paper/2111.13557/) (2021)
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- [Recurrent Neural Network Training with Convex Loss and Regularization Functions by Extended Kalman Filtering](https://mlsanity.com/paper/2111.02673/) (2021)
- [Modern Machine Learning Tools for Monitoring and Control of Industrial Processes: A Survey](https://mlsanity.com/paper/2209.11123/) (2022)
- [Recurrent Neural Network-based Internal Model Control design for stable nonlinear systems](https://mlsanity.com/paper/2108.04585/) (2021)
- [Input-to-State Stable Neural Ordinary Differential Equations with Applications to Transient Modeling of Circuits](https://mlsanity.com/paper/2202.06453/) (2022)
- [Deep Long-Short Term Memory networks: Stability properties and Experimental validation](https://mlsanity.com/paper/2304.02975/) (2023)
- [Towards lifelong learning of Recurrent Neural Networks for control design](https://mlsanity.com/paper/2208.03980/) (2022)
- [Enhancing AI System Resiliency: Formulation and Guarantee for LSTM Resilience Based on Control Theory](https://mlsanity.com/paper/2505.17696/) (2025)
- [Latent-Lagrangian Neural Networks for Reduced Order Modeling of Non-autonomous Nonlinear Dynamical Systems](https://mlsanity.com/paper/2610.04723/) (2026)

## Abstract (arXiv)

The goal of this paper is to analyze Long Short Term Memory (LSTM) neural networks from a dynamical system perspective. The classical recursive equations describing the evolution of LSTM can be recast in state space form, resulting in a time-invariant nonlinear dynamical system. A sufficient condition guaranteeing the Input-to-State (ISS) stability property of this class of systems is provided. The ISS property entails the boundedness of the output reachable set of the LSTM. In light of this result, a novel approach for the safety verification of the network, based on the Scenario Approach, is devised. The proposed method is eventually tested on a pH neutralization process.