mlsanityData to 7 Oct 2026

Learning Density Distribution of Reachable States for Autonomous Systems

arXiv 2109.06728 ↗First posted 14 Sep 2021cs.AI cs.LGcited by 11

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Abstract (arXiv)

State density distribution, in contrast to worst-case reachability, can be leveraged for safety-related problems to better quantify the likelihood of the risk for potentially hazardous situations. In this work, we propose a data-driven method to compute the density distribution of reachable states for nonlinear and even black-box systems. Our semi-supervised approach learns system dynamics and the state density jointly from trajectory data, guided by the fact that the state density evolution follows the Liouville partial differential equation. With the help of neural network reachability tools, our approach can estimate the set of all possible future states as well as their density. Moreover, we could perform online safety verification with probability ranges for unsafe behaviors to occur. We use an extensive set of experiments to show that our learned solution can produce a much more accurate estimate on density distribution, and can quantify risks less conservatively and flexibly comparing with worst-case analysis.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 41 of its 85 references that are in the map. First posted on arXiv 14 Sep 2021.

Builds on

41 of 85 references.

References · 41

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29 more
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Cited by year

Citing papers per year: 2021: 1; 2022: 5; 2023: 1; 2024: 3; 2025: 1.

11 papers.

  1. 2022The Third International Verification of Neural Networks Competition (VNN-COMP 2022): Summary and Resultscited by 39
  2. 2022Statistical Safety and Robustness Guarantees for Feedback Motion Planning of Unknown Underactuated Stochastic Systemscited by 11
  3. 2022Reachable Polyhedral Marching (RPM): An Exact Analysis Tool for Deep-Learned Control Systemscited by 6
  4. 2023NeuroCodeBench: a plain C neural network benchmark for software verificationcited by 3
  5. 2024Automated Repair of AI Code with Large Language Models and Formal Verificationcited by 3
  6. 2024Transport of Algebraic Structure to Latent Embeddingscited by 2
  7. 2025Floating-Point Neural Network Verification at the Software Levelcited by 2
  8. 2022Case Studies for Computing Density of Reachable States for Safe Autonomous Motion Planningcited by 1
  9. 2021Data-driven Set-based Estimation of Polynomial Systems with Application to SIR Epidemicscited by 0
  10. 2022NExG: Provable and Guided State Space Exploration of Neural Network Control Systems using Sensitivity Approximationcited by 0
  11. 2024Predicting AI Agent Behavior through Approximation of the Perron-Frobenius Operatorcited by 0