Discussion Papers 2023

CIRJE-F-1212	"Solving Kolmogorov PDEs without the Curse of Dimensionality via Deep Learning and Asymptotic Expansion with Malliavin Calculus"
Author Name	Takahashi, Akihiko and Toshihiro Yamada
Date	April 2023
Full Paper	PDF file
Remarks	Published 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 justication on the spatial approximation is provided. Numerical examples for high-dimensional Kolmogorov PDEs show 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 justication
on the spatial approximation is provided. Numerical examples for high-dimensional Kolmogorov PDEs show effectiveness of our method.

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