README.md

HighDimPDE.jl

HighDimPDE.jl is a Julia package to solve Highly Dimensional non-local, non-linear PDEs of the form

$$ \begin{aligned} (\partial_t u)(t,x) &= \int_{\Omega} f\big(t,x,{\bf x}, u(t,x),u(t,{\bf x}), ( \nabla_x u )(t,x ),( \nabla_x u )(t,{\bf x} ) \big) d{\bf x} \\ & \quad + \big\langle \mu(t,x), ( \nabla_x u )( t,x ) \big\rangle + \tfrac{1}{2} \text{Trace} \big(\sigma(t,x) [ \sigma(t,x) ]^* ( \text{Hess}_x u)(t, x ) \big). \end{aligned} $$

where $u \colon [0,T] \times \Omega \to \mathbb{R}, \Omega \subseteq \mathbb{R}^{d}$ is subject to initial and boundary conditions, and where $d$ is large.

Tutorials and Documentation

For information on using the package, see the stable documentation. Use the in-development documentation for the version of the documentation, which contains the unreleased features.

Installation

Open Julia and type the following

using Pkg;
Pkg.add("HighDimPDE.jl")

This will download the latest version from the git repo and download all dependencies.

Getting started

See documentation and test folders.

Reference

• Boussange, V., Becker, S., Jentzen, A., Kuckuck, B., Pellissier, L., Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. arXiv (2022)