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.

翻译：州密度分布(与最差的可达性不同)可以被安全相关问题所利用,以更好地量化潜在危险情况的风险可能性。 在这项工作中,我们提出一种数据驱动方法,用于计算非线性甚至黑箱系统的可达状态的密度分布。 我们的半监督方法从轨迹数据中学习系统动态和状态密度,其依据是州密度的演进遵循了Liouville部分差异方程式。在神经网络可达性工具的帮助下,我们的方法可以估算所有可能的未来状态及其密度。 此外,我们可以进行在线安全核查,并设定发生不安全行为的概率范围。 我们使用一系列广泛的实验来证明我们所学的解决方案能够产生对密度分布的更准确的估算,并且可以与最坏情况分析相比,以更保守和灵活的方式量化风险。