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Data-Driven Reachability Analysis via Diffusion Models with PAC Guarantees

Yanliang Huang, Peng Xie, Wenyuan Wu, Zhuoqi Zeng, Amr Alanwar

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.

Data-Driven Reachability Analysis via Diffusion Models with PAC Guarantees

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.

Paper Structure

This paper contains 18 sections, 3 theorems, 22 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contribution
  4. Preliminaries and Problem Statement
  5. Dynamical Systems and Reachable Sets
  6. Problem Statement
  7. Method
  8. Reachability Analysis via Diffusion Models
  9. Conformal Prediction via LTT
  10. From Distributional to Geometric Coverage
  11. Experiments
  12. Experimental Setup
  13. Score Design Choices
  14. Baseline Selection
  15. Duffing Oscillator
  16. ...and 3 more sections

Key Result

Proposition 1

Let $\bm{\epsilon}^*$ be the Bayes-optimal denoiser and assume Gaussian parameterization at all diffusion steps. Define the ideal score $s^*(\bm{x},k) = \sum_{\tau=1}^{T} \gamma_\tau \lVert\bm{\epsilon}^*(\bm{x}_\tau,\tau,k) - \bm{\epsilon}_\tau\rVert^2$, where $\gamma_\tau$ are the loss weights in where $C$ is a constant independent of $\bm{x}$ and $\bar{\alpha}_T = \prod_{s=1}^{T}\alpha_s$. $\b

Figures (5)

  • Figure 3: Pipeline overview for the data-driven reachability analysis via diffusion models. From left to right: (1) trajectory data is collected from the nonlinear dynamical system at discrete time steps; (2) a time-conditioned DDPM is trained to denoise states, with the reconstruction error serving as a nonconformity score; (3) the LTT procedure selects a threshold $q_k$ for the nonconformity score that controls the false negative rate with PAC guarantee; (4) the predicted reachable set is the sublevel set of all states whose score does not exceed $q_k$.
  • Figure 4: Predicted reachable sets for the Duffing oscillator at six time steps $k$.
  • Figure 5: Ablation visualizations at $k{=}149$. Top row: effect of training data size $N$ (model $2048{\times}12$). Bottom row: effect of model capacity (hidden dim ${\times}$ layers, $N{=}10^6$).
  • Figure 6: Predicted reachable sets for the planar quadrotor projected onto $(x,h)$ at $t{=}5.0$.
  • Figure 7: Sensitivity analysis for the Gray-Scott system at $t{=}29$. (a) Acceptance rate under Gaussian perturbation. (b),(c) Score distributions for clean in-set, perturbed in-set, and different-parameter samples.

Theorems & Definitions (9)

  • Definition 1: Forward Reachable Set
  • Proposition 1: Score--likelihood correspondence
  • proof
  • Remark 1
  • Theorem 1: PAC Reachability Coverage
  • proof
  • Proposition 2: Set-theoretic volume bound
  • proof
  • Remark 2: Geometric coverage for dissipative systems