Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lipschitz) operators, as long as the spectral norms of the weights are appropriately constrained. This is to be contrasted with the ordinary ResNet architecture which, even if the spectral norms of the weights are constrained, has a Lipschitz constant that, in the worst case, grows exponentially with the depth of the network. Further analysis of the proposed architecture shows that the spectral norms of the weights can be further constrained to ensure that the network is an averaged operator, making it a natural candidate for a learned denoiser in Plug-and-Play algorithms. Using a novel adaptive way of enforcing the spectral norm constraints, we show that, even with these constraints, it is possible to train performant networks. The proposed architecture is applied to the problem of adversarially robust image classification, to image denoising, and finally to the inverse problem of deblurring.

Original languageEnglish
Article number134159
JournalPhysica D: Nonlinear Phenomena
Early online date15 Apr 2024
Publication statusE-pub ahead of print - 15 Apr 2024

Data Availability Statement

All data is in the form of publicly available datasets, and the code for the paper has been published at https://github.com/fsherry/non-expansive-odes.


  • Convex analysis
  • Deep learning
  • Inverse problems
  • Monotone operator theory
  • Numerical integration of ODEs

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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