Solving differential equations with machine learning

Description

The recent explosion of attention for neural network methods has led to a wealth of new results in computer vision, language processing and all sorts of classification problems. In essence, however, neural networks remain nothing but very advanced data inter/extrapolators. Viewed from this perspective, it is natural to look for applications of neural networks which are outside the `classic' areas mentioned above.

One non-standard area in which neural networks have been used is the numerical solution of differential equations. Essentially, these networks produce approximations to the solutions of the differential equation by a complicated superposition of nonlinear functions. 'Training' the network consists of finding the superposition which is optimal in the sense of leading to the most accurate solution. Both ordinary and partial differential equations can be tackled with this method.

In this project, you will investigate how to solve differential equations using neural network techniques, and compare these new methods with established numerical integration methods.

Prerequisites

You need to have good Python skills.

Some background material

• A fairly readable introduction to this topic is "Artificial Neural Networks for Solving Ordinary and Partial Differential Equations" by Lagaris et al., physics/9705023.
• Also see the MSc thesis by Thomas Harper, "A comparative study of function approximators involving neural networks".