  Hamilton-Jacobi (HJ) reachability analysis is a verification tool that
provides safety and performance guarantees for autonomous systems. It is widely
adopted because of its ability to handle nonlinear dynamical systems with
bounded adversarial disturbances and constraints on states and inputs. However,
it involves solving a PDE to compute a safety value function, whose
computational and memory complexity scales exponentially with the state
dimension, making its direct usage in large-scale systems intractable.
Recently, a learning-based approach called DeepReach, has been proposed to
approximate high-dimensional reachable tubes using neural networks. While
DeepReach has been shown to be effective, the accuracy of the learned solution
decreases with the increase in system complexity. One of the reasons for this
degradation is the inexact imposition of safety constraints during the learning
process, which corresponds to the PDE's boundary conditions. Specifically,
DeepReach imposes boundary conditions as soft constraints in the loss function,
which leaves room for error during the value function learning. Moreover, one
needs to carefully adjust the relative contributions from the imposition of
boundary conditions and the imposition of the PDE in the loss function. This,
in turn, induces errors in the overall learned solution. In this work, we
propose a variant of DeepReach that exactly imposes safety constraints during
the learning process by restructuring the overall value function as a weighted
sum of the boundary condition and neural network output. This eliminates the
need for a boundary loss during training, thus bypassing the need for loss
adjustment. We demonstrate the efficacy of the proposed approach in
significantly improving the accuracy of learned solutions for challenging
high-dimensional reachability tasks, such as rocket-landing and multivehicle
collision-avoidance problems.
