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

2025-10-15

AI Summary

Reachability verification of neural ordinary differential equations (ODEs) suffers from efficiency and accuracy bottlenecks due to the lack of specialized formal tools. To address this, we propose a novel verification method that integrates interval analysis with continuous-time mixed monotonicity—the first application of mixed monotonicity to neural ODE reachability analysis. Leveraging the symmetry of monotone embeddings and the geometric simplicity of interval boxes, our approach establishes a scalable, formally sound verification framework. We implement three algorithmic variants—single-step, incremental, and boundary-driven—within the TIRA tool. Compared to state-of-the-art tools CORA (zonotope-based) and NNV2.0 (star-set-based), our method achieves superior trade-offs between computational efficiency and over-approximation tightness on high-dimensional, real-time safety-critical systems. Experimental evaluation demonstrates its effectiveness on complex dynamical behaviors, including spiral flows and fixed-point attractors.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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 aligned with the symmetry and geometry of neural representations. This novel approach is illustrated on two numerical examples of a spiral system and a fixed-point attractor system modeled as a neural ODE.

Problem

• Verifying neural ODE models for complex dynamical systems remains challenging
• Computing reachable set over-approximations with tightness-efficiency trade-off
• Enabling scalable verification for high-dimensional safety-critical applications

Innovation

• Leveraging mixed monotonicity for neural ODE reachability analysis
• Embedding dynamics into mixed monotone systems for over-approximations
• Using interval-based propagation for efficient bound computation