mlsanityData to 7 Oct 2026

Stable neural networks and connections to continuous dynamical systems

arXiv 2510.22299 ↗First posted 25 Oct 2025math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

The existence of instabilities, for example in the form of adversarial examples, has given rise to a highly active area of research concerning itself with understanding and enhancing the stability of neural networks. We focus on a popular branch within this area which draws on connections to continuous dynamical systems and optimal control, giving a bird's eye view of this area. We identify and describe the fundamental concepts that underlie much of the existing work in this area. Following this, we go into more detail on a specific approach to designing stable neural networks, developing the theoretical background and giving a description of how these networks can be implemented. We provide code that implements the approach that can be adapted and extended by the reader. The code further includes a notebook with a fleshed-out toy example on adversarial robustness of image classification that can be run without heavy requirements on the reader's computer. We finish by discussing this toy example so that the reader can interactively follow along on their computer. This work will be included as a chapter of a book on scientific machine learning, which is currently under revision and aimed at students.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 22 of its 37 references that are in the map. First posted on arXiv 25 Oct 2025.

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References · 22

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
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  3. 2017Towards Deep Learning Models Resistant to Adversarial Attackscited by 6,326
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  5. 2018Neural Ordinary Differential Equationscited by 3,733
  6. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,618
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  9. 2019On instabilities of deep learning in image reconstruction - Does AI come at a cost?cited by 211
  10. 2018Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networkscited by 193
  11. 2020Learning Stable Deep Dynamics Modelscited by 110
  12. 2021Orthogonalizing Convolutional Layers with the Cayley Transformcited by 105
10 more
  1. 2021Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attackscited by 91
  2. 2021A Dynamical System Perspective for Lipschitz Neural Networkscited by 59
  3. 2020Convolutional Proximal Neural Networks and Plug-and-Play Algorithmscited by 36
  4. 2021CLIP: Cheap Lipschitz Training of Neural Networkscited by 29
  5. 2021Hamiltonian Deep Neural Networks Guaranteeing Non-vanishing Gradients by Designcited by 18
  6. 20231-Lipschitz Layers Compared: Memory, Speed, and Certifiable Robustnesscited by 15
  7. 2024Learning Neural Contracting Dynamics: Extended Linearization and Global Guaranteescited by 11
  8. 2022Dynamical systems' based neural networkscited by 10
  9. 2023Designing Stable Neural Networks using Convex Analysis and ODEscited by 10
  10. 2022FI-ODE: Certifiably Robust Forward Invariance in Neural ODEscited by 3

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