  We present a simple algorithm to approximate the viscosity solution of
Hamilton-Jacobi (HJ) equations by means of an artificial deep neural network.
The algorithm uses a stochastic gradient descent-based method to minimize the
least square principle defined by a monotone, consistent numerical scheme. We
analyze the least square principle's critical points and derive conditions that
guarantee that any critical point approximates the sought viscosity solution.
The use of a deep artificial neural network on a finite difference scheme lifts
the restriction of conventional finite difference methods that rely on
computing functions on a fixed grid. This feature makes it possible to solve HJ
equations posed in higher dimensions where conventional methods are infeasible.
We demonstrate the efficacy of our algorithm through numerical studies on
various canonical HJ equations across different dimensions, showcasing its
potential and versatility.
