# On Recurrent Neural Networks for learning-based control: recent results and ideas for future developments

arXiv: https://arxiv.org/abs/2111.13557 · HTML: https://mlsanity.com/paper/2111.13557/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 107 references that are in the map.
- First posted on arXiv 26 Nov 2021.

## Builds on (most cited first, 30 of 36)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [A Survey on Neural Network Interpretability](https://mlsanity.com/paper/2012.14261/) (2020)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [A Dual Approach to Scalable Verification of Deep Networks](https://mlsanity.com/paper/1803.06567/) (2018)
- [Reachability Analysis of Deep Neural Networks with Provable Guarantees](https://mlsanity.com/paper/1805.02242/) (2018)
- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [On Interpretability of Artificial Neural Networks: A Survey](https://mlsanity.com/paper/2001.02522/) (2020)
- [Output Reachable Set Estimation and Verification for Multi-Layer Neural Networks](https://mlsanity.com/paper/1708.03322/) (2017)
- [Physics-based Deep Learning](https://mlsanity.com/paper/2109.05237/) (2021)
- [Learning an Approximate Model Predictive Controller with Guarantees](https://mlsanity.com/paper/1806.04167/) (2018)
- [Nonlinear System Identification: A User-Oriented Roadmap](https://mlsanity.com/paper/1902.00683/) (2019)
- [On Generalization Bounds of a Family of Recurrent Neural Networks](https://mlsanity.com/paper/1910.12947/) (2019)
- [POPQORN: Quantifying Robustness of Recurrent Neural Networks](https://mlsanity.com/paper/1905.07387/) (2019)
- [Data-Driven Model Predictive Control with Stability and Robustness Guarantees](https://mlsanity.com/paper/1906.04679/) (2019)
- [Physics-Guided Architecture (PGA) of Neural Networks for Quantifying Uncertainty in Lake Temperature Modeling](https://mlsanity.com/paper/1911.02682/) (2019)
- [Verifying Recurrent Neural Networks using Invariant Inference](https://mlsanity.com/paper/2004.02462/) (2020)
- [Multi-Step Prediction of Dynamic Systems with Recurrent Neural Networks](https://mlsanity.com/paper/1806.00526/) (2018)
- [Performance-oriented model learning for data-driven MPC design](https://mlsanity.com/paper/1904.10839/) (2019)
- [On the stability properties of Gated Recurrent Units neural networks](https://mlsanity.com/paper/2011.06806/) (2020)
- [Model structures and fitting criteria for system identification with neural networks](https://mlsanity.com/paper/1911.13034/) (2019)
- [LSTM Neural Networks: Input to State Stability and Probabilistic Safety Verification](https://mlsanity.com/paper/1912.04377/) (2019)
- [Nonlinear MPC for Offset-Free Tracking of systems learned by GRU Neural Networks](https://mlsanity.com/paper/2103.02383/) (2021)
- [Recurrent Neural Network-based Model Predictive Control for Continuous Pharmaceutical Manufacturing](https://mlsanity.com/paper/1807.09556/) (2018)
- [Stability of discrete-time feed-forward neural networks in NARX configuration](https://mlsanity.com/paper/2012.03741/) (2020)
- [Deep Energy-Based NARX Models](https://mlsanity.com/paper/2012.04136/) (2020)

## Built on by (most cited first, 14 of 14)

- [From Google Gemini to OpenAI Q* (Q-Star): A Survey of Reshaping the Generative Artificial Intelligence (AI) Research Landscape](https://mlsanity.com/paper/2312.10868/) (2023)
- [Generative AI and Process Systems Engineering: The Next Frontier](https://mlsanity.com/paper/2402.10977/) (2024)
- [Nonlinear MPC design for incrementally ISS systems with application to GRU networks](https://mlsanity.com/paper/2309.16428/) (2023)
- [Structured state-space models are deep Wiener models](https://mlsanity.com/paper/2312.06211/) (2023)
- [Deep Operator Neural Network Model Predictive Control](https://mlsanity.com/paper/2505.18008/) (2025)
- [An Offset-Free Nonlinear MPC scheme for systems learned by Neural NARX models](https://mlsanity.com/paper/2203.16290/) (2022)
- [Deep Long-Short Term Memory networks: Stability properties and Experimental validation](https://mlsanity.com/paper/2304.02975/) (2023)
- [L2RU: a Structured State Space Model with prescribed L2-bound](https://mlsanity.com/paper/2503.23818/) (2025)
- [Tuning the burn-in phase in training recurrent neural networks improves their performance](https://mlsanity.com/paper/2602.10911/) (2026)
- [A critical look at deep neural network for dynamic system modeling](https://mlsanity.com/paper/2301.11604/) (2023)
- [Incorporating Recurrent Reinforcement Learning into Model Predictive Control for Adaptive Control in Autonomous Driving](https://mlsanity.com/paper/2301.13313/) (2023)
- [Recurrent neural network-based robust control systems with regional properties and application to MPC design](https://mlsanity.com/paper/2506.20334/) (2025)
- [Lipschitz-Based Robustness Certification for Recurrent Neural Networks via Convex Relaxation](https://mlsanity.com/paper/2509.17898/) (2025)
- [Towards Safe Learning-Based Non-Linear Model Predictive Control through Recurrent Neural Network Modeling](https://mlsanity.com/paper/2603.24503/) (2026)

## Abstract (arXiv)

This paper aims to discuss and analyze the potentialities of Recurrent Neural Networks (RNN) in control design applications. The main families of RNN are considered, namely Neural Nonlinear AutoRegressive eXogenous, (NNARX), Echo State Networks (ESN), Long Short Term Memory (LSTM), and Gated Recurrent Units (GRU). The goal is twofold. Firstly, to survey recent results concerning the training of RNN that enjoy Input-to-State Stability (ISS) and Incremental Input-to-State Stability (δISS) guarantees. Secondly, to discuss the issues that still hinder the widespread use of RNN for control, namely their robustness, verifiability, and interpretability. The former properties are related to the so-called generalization capabilities of the networks, i.e. their consistency with the underlying real plants, even in presence of unseen or perturbed input trajectories. The latter is instead related to the possibility of providing a clear formal connection between the RNN model and the plant. In this context, we illustrate how ISS and δISS represent a significant step towards the robustness and verifiability of the RNN models, while the requirement of interpretability paves the way to the use of physics-based networks. The design of model predictive controllers with RNN as plant's model is also briefly discussed. Lastly, some of the main topics of the paper are illustrated on a simulated chemical system.