Title: Solving high-dimensional partial differential equations using deep learning,
Abstract: Developing algorithms for solving high-dimensional partial differential
equations (PDEs) has been an exceedingly difficult task for a long time, due to
the notoriously difficult problem known as the "curse of dimensionality". This
paper introduces a deep learning-based approach that can handle general
high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using
backward stochastic differential equations and the gradient of the unknown
solution is approximated by neural networks, very much in the spirit of deep
reinforcement learning with the gradient acting as the policy function.
Numerical results on examples including the nonlinear Black-Scholes equation,
the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation suggest that
the proposed algorithm is quite effective in high dimensions, in terms of both
accuracy and cost. This opens up new possibilities in economics, finance,
operational research, and physics, by considering all participating agents,
assets, resources, or particles together at the same time, instead of making ad
hoc assumptions on their inter-relationships.