Data-Driven Reachability Analysis via Diffusion Models with PAC Guarantees

2026-03-31

AI Summary

This work proposes a data-driven reachability analysis method for nonlinear dynamical systems without explicit models, leveraging denoising diffusion probabilistic models. The approach learns the system’s state distribution from trajectory data and constructs reachable sets as sublevel sets of a non-conformity score derived from reconstruction errors. By integrating the Learn Then Test framework, it calibrates the decision threshold to guarantee a Probably Approximately Correct (PAC) error bound. This is the first method to combine diffusion models with PAC guarantees, requiring no prior knowledge of the system dynamics and scaling effectively to high-dimensional settings. Experiments on the Duffing oscillator, a planar quadrotor, and a high-dimensional reaction-diffusion system demonstrate that the empirical miss rate consistently remains below the theoretical PAC bound, significantly outperforming conventional grid-based or polynomial methods.

Abstract

We present a data-driven framework for reachability analysis of nonlinear dynamical systems that requires no explicit model. A denoising diffusion probabilistic model learns the time-evolving state distribution of a dynamical system from trajectory data alone. The predicted reachable set takes the form of a sublevel set of a nonconformity score derived from the reconstruction error, with the threshold calibrated via the Learn Then Test procedure so that the probability of excluding a reachable state is bounded with high probability. Experiments on three nonlinear systems, a forced Duffing oscillator, a planar quadrotor, and a high-dimensional reaction-diffusion system, confirm that the empirical miss rate remains below the Probably Approximately Correct (PAC) bound while scaling to state dimensions beyond the reach of classical grid-based and polynomial methods.

Problem

• Reachability Analysis
• Data-Driven
• Nonlinear Dynamical Systems
• PAC Guarantees
• Diffusion Models

Innovation

• diffusion models
• reachability analysis
• PAC guarantees
• data-driven dynamics
• nonconformity score