Mixed Monotonicity Reachability Analysis of Neural ODE: A Trade-Off Between Tightness and Efficiency
Abdelrahman Sayed Sayed, Pierre-Jean Meyer, Mohamed Ghazel
TL;DR
The paper tackles the verification of neural ODEs by introducing an interval-based reachability framework grounded in continuous-time mixed monotonicity. By embedding the neural ODE dynamics into a monotone system and exploiting a homeomorphism between initial-state boundaries and reachable sets, it enables sound over-approximations that are efficient to compute. The authors implement three propagation modes in TIRA—single-step, incremental, and boundary-based—and compare against CORA and NNV2.0 across spiral (2D) and FPA (5D) benchmarks, highlighting substantial gains in computational efficiency at the cost of some tightness. This work broadens scalable verification opportunities for neural ODEs, particularly in real-time and safety-critical contexts, and suggests future avenues for combining boundary methods with incremental steps and partitioning to tighten bounds further.
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.