Bridging Neural ODE and ResNet: A Formal Error Bound for Safety Verification

2025-06-03

AI Summary

Neural ODEs and ResNets exhibit approximate equivalence, yet lack verifiable error bounds—hindering their use as interchangeable surrogates in safety verification. Method: We establish the first rigorous, bidirectional, and quantifiable error upper bound between them, integrating Lipschitz continuity analysis, Euler discretization error theory, and reachability-set expansion within a formal verification framework. Contribution/Results: The bound guarantees that if either model is verified to satisfy a safety property and the approximation error lies within the bound, the other model provably satisfies the same property. Evaluated on fixed-point attractor Neural ODE instances, safety properties transfer with 100% fidelity, while verification overhead is substantially reduced. This work provides the first verifiable cross-architecture error bound, enabling reliable safety-property transfer between Neural ODEs and ResNets and eliminating redundant verification.

Technology Category

• Natural Language Processing: Safety and Robustness
• Machine Learning: Deep Neural Architectures and Foundation Models
• Philosophy and Ethics of AI: Safety, Robustness & Trustworthiness

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

A neural ordinary differential equation (neural ODE) is a machine learning model that is commonly described as a continuous depth generalization of a residual network (ResNet) with a single residual block, or conversely, the ResNet can be seen as the Euler discretization of the neural ODE. These two models are therefore strongly related in a way that the behaviors of either model are considered to be an approximation of the behaviors of the other. In this work, we establish a more formal relationship between these two models by bounding the approximation error between two such related models. The obtained error bound then allows us to use one of the models as a verification proxy for the other, without running the verification tools twice: if the reachable output set expanded by the error bound satisfies a safety property on one of the models, this safety property is then guaranteed to be also satisfied on the other model. This feature is fully reversible, and the initial safety verification can be run indifferently on either of the two models. This novel approach is illustrated on a numerical example of a fixed-point attractor system modeled as a neural ODE.

Problem

• Formally bounding approximation error between Neural ODE and ResNet
• Enabling safety verification via one model as proxy for the other
• Reversible verification without redundant tool execution

Innovation

• Bounding approximation error between Neural ODE and ResNet
• Using one model as verification proxy for the other
• Reversible safety verification with error-bound expansion