Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning

TL;DR

This paper introduces a deep learning approach that efficiently solves entire families of high-dimensional Kolmogorov PDEs simultaneously, avoiding the curse of dimensionality and applicable to models like heat equations and Black-Scholes options.

Contribution

The paper presents a novel deep learning method reformulating the PDE family solution as a single statistical learning problem, demonstrating efficiency and scalability in high dimensions.

Findings

Single neural network learns entire PDE family solutions

Method avoids curse of dimensionality

Effective for heat and Black-Scholes models

Abstract

We present a deep learning algorithm for the numerical solution of parametric families of high-dimensional linear Kolmogorov partial differential equations (PDEs). Our method is based on reformulating the numerical approximation of a whole family of Kolmogorov PDEs as a single statistical learning problem using the Feynman-Kac formula. Successful numerical experiments are presented, which empirically confirm the functionality and efficiency of our proposed algorithm in the case of heat equations and Black-Scholes option pricing models parametrized by affine-linear coefficient functions. We show that a single deep neural network trained on simulated data is capable of learning the solution functions of an entire family of PDEs on a full space-time region. Most notably, our numerical observations and theoretical results also demonstrate that the proposed method does not suffer from the curse of dimensionality, distinguishing it from almost all standard numerical methods for PDEs.