Neural networks provide quick approximations to complex functions, and have been increasingly used in perception as well as control tasks. For use in mission-critical and safety-critical applications, however, it is important to be able to analyze what a neural network can and cannot do. For feed-forward neural networks with ReLU activation functions, although exact analysis is NP-complete, recently-proposed verification methods can sometimes succeed. The main practical problem with neural network verification is excessive analysis runtime. Even on small networks, tools that are theoretically complete can sometimes run for days without producing a result. In this paper, we work to address the runtime problem by improving upon a recently-proposed geometric path enumeration method. Through a series of optimizations, several of which are new algorithmic improvements, we demonstrate significant speed improvement of exact analysis on the well-studied ACAS Xu benchmarks, sometimes hundreds of times faster than the original implementation. On more difficult benchmark instances, our optimized approach is often the fastest, even outperforming inexact methods that leverage overapproximation and refinement.

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More than three miles above the Arizona desert, an F-16 student pilot experienced a gravityinduced loss of consciousness (GLOC), passing out while turning at nearly 9Gs (nine times the force of gravity) flying over 400 knots (over 460 miles per hour). With its pilot unconscious, the aircraft turn devolved into a dive, dropping from over 17,000 feet to less than 8,000 feet in altitude in less than 10 seconds. An auditory warning in the cockpit called out to the pilot "altitude, altitude" just before he crossed through 11,000 feet, switching to a command to "pull up" around 8,000 feet. Meanwhile, the student's instructor was watching the event unfold from his own aircraft. As the student's aircraft passed through 12,500 feet, the instructor called over the radio "two recover," commanding the student ("two") to end the dive. As the student's aircraft passed through 11,000 feet the instructor's "two recover!" came with increased urgency. At 9,000 feet, and with terror rising in his voice the instructor yelled "TWO RECOVER!" Fortunately, at the same time as the instructor's third panicked radio call, a new Run Time Assurance (RTA) system kicked in to automatically recover the aircraft. The Automatic Ground Collision Avoidance System (Auto GCAS), an RTA system integrated on the jets less than two years earlier in the Fall of 2014, detected that the aircraft was about to collide, commanded a roll to wings level and pull up maneuver, and recovered the aircraft less than 3,000 feet above the