Working Papers

Quantitative Finance




Asymptotic expansion and deep neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with nonlinear coefficients

Author:Akihiko Takahashi, Toshihiro Yamada


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, and a numerical example for a 100 dimensional Kolmogorov PDE shows effectiveness of our method.