  Hamilton-Jacobi (HJ) reachability analysis is an important formal
verification method for guaranteeing performance and safety properties of
dynamical control systems. Its advantages include compatibility with general
nonlinear system dynamics, formal treatment of bounded disturbances, and the
ability to deal with state and input constraints. However, it involves solving
a PDE, whose computational and memory complexity scales exponentially with
respect to the number of state variables, limiting its direct use to
small-scale systems. We propose DeepReach, a method that leverages new
developments in sinusoidal networks to develop a neural PDE solver for
high-dimensional reachability problems. The computational requirements of
DeepReach do not scale directly with the state dimension, but rather with the
complexity of the underlying reachable tube. DeepReach achieves comparable
results to the state-of-the-art reachability methods, does not require any
explicit supervision for the PDE solution, can easily handle external
disturbances, adversarial inputs, and system constraints, and also provides a
safety controller for the system. We demonstrate DeepReach on a 9D
multi-vehicle collision problem, and a 10D narrow passage problem, motivated by
autonomous driving applications.
