Solving Kolmogorov PDEs without the curse of dimensionality via deep learning and asymptotic expansion with Malliavin calculus (Forthcoming in “Partial Differential Equations and Applications”)(Revised version of CARF-F-547)

Abstract

This paper proposes a new spatial approximation method without the curse of dimensionality for solving high-dimensional partial differential equations (PDEs) by using an asymptotic expansion method with a deep learning-based algorithm. In particular, the mathematical justification on the spatial approximation is provided. Numerical examples for high-dimensional Kolmogorov PDEs show effectiveness of our method.