We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, through a sequence of learning problems obtained from the minimization of suitable quadratic loss functions and training simulations. This methodology extends to the fully non- linear case the approach recently proposed in (Huré, Pham, Warin, 2019) for semi-linear Read more [...]