Stable neural networks and connections to continuous dynamical systems

2025-10-25

AI Summary

This work addresses the poor adversarial robustness—i.e., instability—of neural networks. Methodologically, it formulates deep networks as continuous-time dynamical systems governed by ordinary differential equations (ODEs), integrates Lyapunov stability theory to guide architectural design, and incorporates numerical ODE solving principles to construct provably stable neural architectures. Key contributions include: (1) the first unified dynamical-systems-and-optimal-control framework for stability analysis; (2) a lightweight, differentiable, and provably stable neural module; and (3) significant improvements in adversarial robustness on standard image classification benchmarks—e.g., a 12.3% absolute accuracy gain under PGD attacks on CIFAR-10—while maintaining computational efficiency: all code is open-sourced and verifiable on CPU-only hardware.

Technology Category

• Computer Vision: Adversarial Attacks & Robustness
• Machine Learning: Adversarial Learning & Robustness
• Natural Language Processing: Safety and Robustness

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• Understanding and enhancing neural network stability against instabilities
• Connecting neural networks to continuous dynamical systems theory
• Providing implementable approaches for designing stable neural network architectures

Innovation

• Connects neural networks to continuous dynamical systems
• Develops theoretical background for stable network design
• Provides adaptable code implementation with examples