Solving PDEs Using Neural Networks

Abstract

Lately, there has been a lot of research on using deep learning as an alternative method to solve PDEs. The major benefit is being able to solve in higher dimensions without using a full grid of mesh points. Here we try to implement the algorithm, without utilising any pre-existing machine learning packages, to understand how the process is done. We will try to see if we can extend solutions to higher dimensions than with an explicit finite difference scheme, even with a simple feedforward neural network. A natural application will be the Black--Scholes equation with multiple underlying assets.