  A method based on deep artificial neural networks and empirical risk
minimization is developed to calculate the boundary separating the stopping and
continuation regions in optimal stopping. The algorithm parameterizes the
stopping boundary as the graph of a function and introduces relaxed stopping
rules based on fuzzy boundaries to facilitate efficient optimization. Several
financial instruments, some in high dimensions, are analyzed through this
method, demonstrating its effectiveness. The existence of the stopping boundary
is also proved under natural structural assumptions.
