  We propose a new probabilistic scheme which combines deep learning techniques
with high order schemes for backward stochastic differential equations
belonging to the class of Runge-Kutta methods to solve high-dimensional
semi-linear parabolic partial differential equations. Our approach notably
extends the one introduced in [Hure Pham Warin 2020] for the implicit Euler
scheme to schemes which are more efficient in terms of discrete-time error. We
establish some convergence results for our implemented schemes under classical
regularity assumptions. We also illustrate the efficiency of our method for
different schemes of order one, two and three. Our numerical results indicate
that the Crank-Nicolson schemes is a good compromise in terms of precision,
computational cost and numerical implementation.
