mlsanityData to 7 Oct 2026

Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks

arXiv 2110.12976 ↗First posted 25 Oct 2021cs.LG cs.CR cs.CVcited by 91

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Abstract (arXiv)

Deep neural networks (DNNs) are well-known to be vulnerable to adversarial attacks, where malicious human-imperceptible perturbations are included in the input to the deep network to fool it into making a wrong classification. Recent studies have demonstrated that neural Ordinary Differential Equations (ODEs) are intrinsically more robust against adversarial attacks compared to vanilla DNNs. In this work, we propose a stable neural ODE with Lyapunov-stable equilibrium points for defending against adversarial attacks (SODEF). By ensuring that the equilibrium points of the ODE solution used as part of SODEF is Lyapunov-stable, the ODE solution for an input with a small perturbation converges to the same solution as the unperturbed input. We provide theoretical results that give insights into the stability of SODEF as well as the choice of regularizers to ensure its stability. Our analysis suggests that our proposed regularizers force the extracted feature points to be within a neighborhood of the Lyapunov-stable equilibrium points of the ODE. SODEF is compatible with many defense methods and can be applied to any neural network's final regressor layer to enhance its stability against adversarial attacks.

Key facts. Cited by 91 papers in the map (data through 7 Oct 2026). Builds on 27 of its 47 references that are in the map. First posted on arXiv 25 Oct 2021.

Builds on

27 of 47 references.

References · 27

  1. 2014Explaining and Harnessing Adversarial Examplescited by 7,287
  2. 2019EfficientNet: Rethinking Model Scaling for Convolutional Neural Networkscited by 6,415
  3. 2017Towards Deep Learning Models Resistant to Adversarial Attackscited by 6,326
  4. 2013Intriguing properties of neural networkscited by 5,114
  5. 2018Neural Ordinary Differential Equationscited by 3,733
  6. 2016Towards Evaluating the Robustness of Neural Networkscited by 2,976
  7. 2016A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networkscited by 2,362
  8. 2016Adversarial examples in the physical worldcited by 1,762
  9. 2020Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attackscited by 1,442
  10. 2015DeepFool: a simple and accurate method to fool deep neural networkscited by 1,315
  11. 2015The Limitations of Deep Learning in Adversarial Settingscited by 784
  12. 2019Square Attack: a query-efficient black-box adversarial attack via random searchcited by 682
15 more
  1. 2015Distillation as a Defense to Adversarial Perturbations against Deep Neural Networkscited by 675
  2. 2018Defense-GAN: Protecting Classifiers Against Adversarial Attacks Using Generative Modelscited by 362
  3. 2017Stable Architectures for Deep Neural Networkscited by 340
  4. 2019Minimally distorted Adversarial Examples with a Fast Adaptive Boundary Attackcited by 333
  5. 2017MagNet: a Two-Pronged Defense against Adversarial Examplescited by 268
  6. 2018Audio Adversarial Examples: Targeted Attacks on Speech-to-Textcited by 268
  7. 2020Bag of Tricks for Adversarial Trainingcited by 199
  8. 2019Adversarial Sensor Attack on LiDAR-based Perception in Autonomous Drivingcited by 151
  9. 2018Generating 3D Adversarial Point Cloudscited by 144
  10. 2018Adversarial Attacks Against Automatic Speech Recognition Systems via Psychoacoustic Hidingcited by 83
  11. 2019On Robustness of Neural Ordinary Differential Equationscited by 83
  12. 2016Globally Normalized Transition-Based Neural Networkscited by 26
  13. 2020Adversarial Robustness of Stabilized NeuralODEs Might be from Obfuscated Gradientscited by 22
  14. 2019Error-Correcting Output Codes with Ensemble Diversity for Robust Learning in Neural Networkscited by 11
  15. 2019Direction Concentration Learning: Enhancing Congruency in Machine Learningcited by 5

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Citing papers per year: 2022: 20; 2023: 25; 2024: 20; 2025: 16; 2026: 10.

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  1. 2022Diffusion Models for Adversarial Purificationcited by 469
  2. 2023Better Diffusion Models Further Improve Adversarial Trainingcited by 207
  3. 2023Robust Evaluation of Diffusion-Based Adversarial Purificationcited by 65
  4. 2023How Deep Learning Sees the World: A Survey on Adversarial Attacks & Defensescited by 65
  5. 2022Evaluating the Adversarial Robustness of Adaptive Test-time Defensescited by 64
  6. 2022Knowledge Graph Embedding: A Survey from the Perspective of Representation Spacescited by 41
  7. 2022Visual Prompting for Adversarial Robustnesscited by 33
  8. 2022DISCO: Adversarial Defense with Local Implicit Functionscited by 33
  9. 2022On the Robustness of Graph Neural Diffusion to Topology Perturbationscited by 30
  10. 2023Graph Neural Convection-Diffusion with Heterophilycited by 26
  11. 2023Adversarial Robustness in Graph Neural Networks: A Hamiltonian Approachcited by 24
  12. 2023Enhancing Adversarial Robustness via Score-Based Optimizationcited by 19
38 more
  1. 2022RobustLoc: Robust Camera Pose Regression in Challenging Driving Environmentscited by 18
  2. 2023Improving the Accuracy-Robustness Trade-Off of Classifiers via Adaptive Smoothingcited by 18
  3. 2023MimicDiffusion: Purifying Adversarial Perturbation via Mimicking Clean Diffusion Modelcited by 18
  4. 2024Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FRONDcited by 16
  5. 2025AutoAdvExBench: Benchmarking autonomous exploitation of adversarial example defensescited by 14
  6. 2022Robust Classification using Contractive Hamiltonian Neural ODEscited by 13
  7. 2022Towards Lightweight Black-Box Attacks against Deep Neural Networkscited by 13
  8. 2024SegStitch: Multidimensional Transformer for Robust and Efficient Medical Imaging Segmentationcited by 13
  9. 2023Node Embedding from Neural Hamiltonian Orbits in Graph Neural Networkscited by 12
  10. 2023Leveraging Label Non-Uniformity for Node Classification in Graph Neural Networkscited by 11
  11. 2023Lyapunov-Stable Deep Equilibrium Modelscited by 10
  12. 2024Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Studycited by 9
  13. 2023Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equationscited by 8
  14. 2022Adversarially Robust Prototypical Few-shot Segmentation with Neural-ODEscited by 6
  15. 2024Structure-Preserving Physics-Informed Neural Networks With Energy or Lyapunov Structurecited by 6
  16. 2024PointDifformer: Robust Point Cloud Registration With Neural Diffusion and Transformercited by 6
  17. 2024Adversarially Robust Out-of-Distribution Detection Using Lyapunov-Stabilized Embeddingscited by 6
  18. 2022Self-Healing Robust Neural Networks via Closed-Loop Controlcited by 5
  19. 2022Robust Prototypical Few-Shot Organ Segmentation with Regularized Neural-ODEscited by 5
  20. 2023RobustMat: Neural Diffusion for Street Landmark Patch Matching under Challenging Environmentscited by 5
  21. 2024DeNOTS: Stable Deep Neural ODEs for Time Seriescited by 5
  22. 2024ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergencecited by 5
  23. 2025Neural Variable-Order Fractional Differential Equation Networkscited by 5
  24. 2023Continual Adversarial Defensecited by 4
  25. 2024Robustness Reprogramming for Representation Learningcited by 4
  26. 2022Towards Alternative Techniques for Improving Adversarial Robustness: Analysis of Adversarial Training at a Spectrum of Perturbationscited by 3
  27. 2022FI-ODE: Certifiably Robust Forward Invariance in Neural ODEscited by 3
  28. 2023Efficient Certified Training and Robustness Verification of Neural ODEscited by 3
  29. 2023Robust Feature Inference: A Test-time Defense Strategy using Spectral Projectionscited by 3
  30. 2024Interpretability-Guided Test-Time Adversarial Defensecited by 3
  31. 2025Efficient Training of Neural Fractional-Order Differential Equation via Adjoint Backpropagationcited by 3
  32. 2025Neural Predictive Control to Coordinate Discrete- and Continuous-Time Models for Time-Series Analysis with Control-Theoretical Improvementscited by 3
  33. 2022Adversarially Robust Video Perception by Seeing Motioncited by 2
  34. 2023Eigen-informed NeuralODEs: Dealing with stability and convergence issues of NeuralODEscited by 2
  35. 2023Improving Adversarial Robustness of DEQs with Explicit Regulations Along the Neural Dynamicscited by 2
  36. 2024PosDiffNet: Positional Neural Diffusion for Point Cloud Registration in a Large Field of View with Perturbationscited by 2
  37. 2024Stable Autonomous Flow Matchingcited by 2
  38. 2024PRFusion: Toward Effective and Robust Multi-Modal Place Recognition with Image and Point Cloud Fusioncited by 2

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