JSoC Final Report

Julia Season of Contribution 2019 has come to an end. I worked on implementation the library for solve high-dimensional PDEs using neural networks. You can find the project in the repository for this link - NeuralNetDiffEq.jl. This post is about a summary result of my work in JSoC.

Achievements

Earlier I wrote posts about the work done. For details about the work you can visit the previous blog posts and also I breaf describe what was done:

First of all, as preparation for the main work, I implemented the prototype algorithm for solve ODE used neural network.
Second, It was necessary to re-write outdated code, thus support the work of the library with the latest version of Julia and perform migrate from Knet and ForwardDiff libraries to the Flux library.
The third point, which took the most time and, in general, was the main goal of my project. It is implementation of algorithm for solve PDEs using neural network and backward stochastic differential equations (BSDEs).
And last, it was improving the method that use neural sde for solve PDEs.

In addition to what has been described above, there has also been wrote many tests for each methods (you can find them in the repo). Each test represent a numerical solve for semi-liner high-dimensional differential equation that to compare with right answer.

There was written tests for next differential equations:

Heat equation
Black Scholes Barenblatt equation
Allen-Cahn equation
Hamilton Jacobi Bellman equation
Nonlinear Black-Scholes equation with Default Risk

In addition, besides everything that is describe above we started writing an article using the results of work in the project at JSoC

Future issues

Anyway, there is a lot more stuff to be done, from solving small bugs to implementing major features.

For example:

Optimal stopping problems through neural networks
Kolmogorov equations through neural nets
Deep learning for random PDEs
Deep learning methods for stochastic elliptic PDEs
Deep learning methods for free-boundary PDEs
GPU and TPU support
Option that allow get solution of differential equation on every time step

Final word

It's been exciting be part of all that. I had fun and very productive summer at JSoC. Moreover, I would be exactly glad to continue working on NeuralNetDiffEq project.

It was a pleasure working under my mentor Chris Rackauckas, who has helped me tremendously throughout this journey.

Thanks to the Julia community for this marvelous opportunity to participate in the project.