Mixed Monotonicity Reachability Analysis of Neural ODE: A Trade-Off Between Tightness and Efficiency

TL;DR

This paper introduces a novel interval-based reachability analysis method for neural ODEs using mixed monotonicity techniques, balancing tightness and computational efficiency for high-dimensional and safety-critical applications.

Contribution

It presents a new approach that embeds neural ODEs into mixed monotone systems to efficiently compute over-approximations of reachable sets, improving scalability and efficiency.

Findings

Provides sound over-approximations with improved efficiency

Balances tightness and computational speed for high-dimensional systems

Demonstrates effectiveness on numerical examples like spiral and attractor systems

Abstract

Neural ordinary differential equations (neural ODE) are powerful continuous-time machine learning models for depicting the behavior of complex dynamical systems, but their verification remains challenging due to limited reachability analysis tools adapted to them. We propose a novel interval-based reachability method that leverages continuous-time mixed monotonicity techniques for dynamical systems to compute an over-approximation for the neural ODE reachable sets. By exploiting the geometric structure of full initial sets and their boundaries via the homeomorphism property, our approach ensures efficient bound propagation. By embedding neural ODE dynamics into a mixed monotone system, our interval-based reachability approach, implemented in TIRA with single-step, incremental, and boundary-based approaches, provides sound and computationally efficient over-approximations compared with CORA's zonotopes and NNV2.0 star set representations, while trading tightness for efficiency. This trade-off makes our method particularly suited for high-dimensional, real-time, and safety-critical applications. Applying mixed monotonicity to neural ODE reachability analysis paves the way for lightweight formal analysis by leveraging the symmetric structure of monotone embeddings and the geometric simplicity of interval boxes, opening new avenues for scalable verification. This novel approach is illustrated on two numerical examples of a spiral system and a fixed-point attractor system modeled as a neural ODE.