We present a method for computing exact reachable sets for deep neural networks with rectified linear unit (ReLU) activation. Our method is well-suited for use in rigorous safety analysis of robotic perception and control systems with deep neural network components. Our algorithm can compute both forward and backward reachable sets for a ReLU network iterated over multiple time steps, as would be found in a perception-action loop in a robotic system. Our algorithm is unique in that it builds the reachable sets by expanding a front of polyhedral cells in the input space, rather than iterating layer-by-layer through the network as in other methods. If an unsafe cell is found, our algorithm can return this result without completing the full reachability computation, thus giving an anytime property that accelerates safety verification. We demonstrate our algorithm on safety verification of the ACAS Xu aircraft advisory system. We find unsafe actions many times faster than the fastest existing method and certify no unsafe actions exist in about twice the time of the existing method. We also compute forward and backward reachable sets for a learned model of pendulum dynamics over a 50 time step horizon in 87s on a laptop computer. Source code for our algorithm can be found at https://github.com/StanfordMSL/Neural-Network-Reach.

中文翻译：

---

可达多面行进（RPM）：具有深层神经网络组件的机器人系统的安全验证算法

我们提出了一种用于计算具有校正线性单元（ReLU）激活的深层神经网络的精确可达集的方法。我们的方法非常适合用于具有深层神经网络组件的机器人感知和控制系统的严格安全性分析。我们的算法可以计算在多个时间步长上迭代的ReLU网络的正向和反向可达集，这将在机器人系统的感知行为循环中找到。我们的算法的独特之处在于，它通过在输入空间中扩展多面体单元的前面来构建可到达的集合，而不是像其他方法一样通过网络逐层迭代。如果找到了不安全的小区，我们的算法可以返回此结果，而无需完成完整的可达性计算，因此可以提供随时可用的属性，从而可以加快安全性验证。我们演示了ACAS Xu飞机咨询系统的安全验证算法。我们发现不安全的动作比最快的现有方法快许多倍，并且证明不存在不安全的动作的时间是现有方法的两倍。我们还为便携式计算机在87s内的50个时间步长范围内的钟摆动力学学习模型计算了向前和向后可到达的集合。可以在https://github.com/StanfordMSL/Neural-Network-Reach中找到我们算法的源代码。我们还为便携式计算机在87s内的50个时间步长范围内的钟摆动力学学习模型计算了向前和向后可到达的集合。可以在https://github.com/StanfordMSL/Neural-Network-Reach中找到我们算法的源代码。我们还为便携式计算机在87s内的50个时间步长范围内的钟摆动力学学习模型计算了向前和向后可到达的集合。可以在https://github.com/StanfordMSL/Neural-Network-Reach中找到我们算法的源代码。