We propose new machine learning schemes for solving high dimensional nonlinear partial di_erential equations (PDEs). Relying on the classical backward stochastic di_erential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions de_ned recursively by backward induction. The methodology is extended to variational inequalities arising Read more [...]