Discussion Papers 2021

CIRJE-F-1167	"Asymptotic Expansion and Deep Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Kolmogorov Partial Differential Equations with Nonlinear Coefficients"
Author Name	Takahashi, Akihiko and Toshihiro Yamada
Date	May 2021
Full Paper
Remarks	Revised in November 2022. Subsequently published as "Solving Kolmogorov PDEs without the Curse of Dimensionality via Deep Learning and Asymptotic Expansion with Malliavin Calculus" in Partial Differential Equations and Applications Vol.4, June 2023.

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, and a numerical example for a 100 dimensional Kolmogorov PDE shows effectiveness of our method.
Keywords. Asymptotic expansion, Deep learning, Kolmogorov PDEs, Malliavin calculus, Curse of dimensionality

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

Keywords. Asymptotic expansion, Deep learning, Kolmogorov PDEs, Malliavin calculus, Curse of dimensionality