mlsanityData to 7 Oct 2026

Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness

arXiv 2104.05942 ↗First posted 13 Apr 2021cs.LG cs.SY eess.SYcited by 43

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Abstract (arXiv)

This paper introduces recurrent equilibrium networks (RENs), a new class of nonlinear dynamical models for applications in machine learning, system identification and control. The new model class admits ``built in'' behavioural guarantees of stability and robustness. All models in the proposed class are contracting -- a strong form of nonlinear stability -- and models can satisfy prescribed incremental integral quadratic constraints (IQC), including Lipschitz bounds and incremental passivity. RENs are otherwise very flexible: they can represent all stable linear systems, all previously-known sets of contracting recurrent neural networks and echo state networks, all deep feedforward neural networks, and all stable Wiener/Hammerstein models, and can approximate all fading-memory and contracting nonlinear systems. RENs are parameterized directly by a vector in RN, i.e. stability and robustness are ensured without parameter constraints, which simplifies learning since HLgeneric methods for unconstrained optimization such as stochastic gradient descent and its variants can be used. The performance and robustness of the new model set is evaluated on benchmark nonlinear system identification problems, and the paper also presents applications in data-driven nonlinear observer design and control with stability guarantees.

Key facts. Cited by 43 papers in the map (data through 7 Oct 2026). Builds on 34 of its 96 references that are in the map. First posted on arXiv 13 Apr 2021.

Builds on

34 of 96 references.

References · 34

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  1. 2020Monotone operator equilibrium networkscited by 100
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 1; 2023: 10; 2024: 12; 2025: 9; 2026: 8.

43 papers.

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  1. 2023Direct Parameterization of Lipschitz-Bounded Deep Networkscited by 49
  2. 2021Learning over All Stabilizing Nonlinear Controllers for a Partially-Observed Linear Systemcited by 13
  3. 2021Youla-REN: Learning Nonlinear Feedback Policies with Robust Stability Guaranteescited by 10
  4. 2022Neural System Level Synthesis: Learning over All Stabilizing Policies for Nonlinear Systemscited by 10
  5. 2023Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramiancited by 9
  6. 2024Monotone, Bi-Lipschitz, and Polyak-Lojasiewicz Networkscited by 9
  7. 2023Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equationscited by 8
  8. 2021Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativitycited by 7
  9. 2024Learning to optimize with convergence guarantees using nonlinear system theorycited by 7
  10. 2024Learning to Boost the Performance of Stable Nonlinear Systemscited by 7
  11. 2025Predictability Enables Parallelization of Nonlinear State Space Modelscited by 7
  12. 2023Learning Over Contracting and Lipschitz Closed-Loops for Partially-Observed Nonlinear Systems (Extended Version)cited by 6
31 more
  1. 2024Model order reduction of deep structured state-space models: A system-theoretic approachcited by 6
  2. 2023Convolutional Neural Networks as 2-D systemscited by 5
  3. 2024Contractive Dynamical Imitation Policies for Efficient Out-of-Sample Recoverycited by 5
  4. 2023Stable Linear Subspace Identification: A Machine Learning Approachcited by 4
  5. 2024Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guaranteescited by 4
  6. 2024State space representations of the Roesser type for convolutional layerscited by 3
  7. 2026Quantization Robustness of Monotone Operator Equilibrium Networkscited by 3
  8. 2024Learning Stable and Passive Neural Differential Equationscited by 2
  9. 2024LipKernel: Lipschitz-Bounded Convolutional Neural Networks via Dissipative Layerscited by 2
  10. 2025R2DN: Scalable Parameterization of Contracting and Lipschitz Recurrent Deep Networkscited by 2
  11. 2023Learning Stable and Robust Linear Parameter-Varying State-Space Modelscited by 1
  12. 2023A modular framework for stabilizing deep reinforcement learning controlcited by 1
  13. 2023Behavior quantification as the missing link between fields: Tools for digital psychiatry and their role in the future of neurobiologycited by 1
  14. 2023RobustNeuralNetworks.jl: a Package for Machine Learning and Data-Driven Control with Certified Robustnesscited by 1
  15. 2024Mapping back and forth between model predictive control and neural networkscited by 1
  16. 2024On Dissipativity of Cross-Entropy Loss in Training ResNetscited by 1
  17. 2025L2RU: a Structured State Space Model with prescribed L2-boundcited by 1
  18. 2025Robustly Invertible Nonlinear Dynamics and the BiLipREN: Contracting Neural Models with Contracting Inversescited by 1
  19. 2025React to Surprises: Stable-by-Design Neural Feedback Control and the Youla-RENcited by 1
  20. 2025Circuit realization and hardware linearization of monotone operator equilibrium networkscited by 1
  21. 2024Convex neural network synthesis for robustness in the 1-normcited by 0
  22. 2025Negative Imaginary Neural ODEs: Learning to Control Mechanical Systems with Stability Guaranteescited by 0
  23. 2025S-Crescendo: A Nested Transformer Weaving Framework for Scalable Nonlinear System in S-Domain Representationcited by 0
  24. 2025Fine-Tuning Flow Matching via Maximum Likelihood Estimation of Reconstructionscited by 0
  25. 2026Stability-Preserving Online Adaptation of Neural Closed-loop Mapscited by 0
  26. 2026Demystifying Lipschitz verification: positive matrices, negative resultscited by 0
  27. 2026Robustly Invertible Nonlinear Dynamics and the BiLipREN: From Inversion-Based Control to Generative Trajectory Modellingcited by 0
  28. 2026Can a Dynamic Internal Field Govern a Transformer's Cognition? Certifiability, not Superiority, in Homeostatic Compute Controlcited by 0
  29. 2026Scalable Incremental Robustness Analysis of Neural Network Feedback Systemscited by 0
  30. 2026Data-driven discrete-time deep recurrent neural network-based modeling for dissipative systemscited by 0
  31. 2026LipSSM: Structurally Lipschitz-Bounded Cascaded State-Space Model via Metric Transfer between Consecutive SSM Layerscited by 0