  Classical numerical methods for solving partial differential equations suffer
from the curse dimensionality mainly due to their reliance on meticulously
generated spatio-temporal grids. Inspired by modern deep learning based
techniques for solving forward and inverse problems associated with partial
differential equations, we circumvent the tyranny of numerical discretization
by devising an algorithm that is scalable to high-dimensions. In particular, we
approximate the unknown solution by a deep neural network which essentially
enables us to benefit from the merits of automatic differentiation. To train
the aforementioned neural network we leverage the well-known connection between
high-dimensional partial differential equations and forward-backward stochastic
differential equations. In fact, independent realizations of a standard
Brownian motion will act as training data. We test the effectiveness of our
approach for a couple of benchmark problems spanning a number of scientific
domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman
equations, both in 100-dimensions.
