HighDimPDE.jl

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

$$\begin{array}{rl}({\partial }_{t}u)(t,x)& ={\int }_{\mathrm{\Omega }}f(t,x,\mathbf{x},u(t,x),u(t,\mathbf{x}),({\mathrm{\nabla }}_{x}u)(t,x),({\mathrm{\nabla }}_{x}u)(t,\mathbf{x}))d\mathbf{x}\\ & +⟨\mu (t,x),({\mathrm{\nabla }}_{x}u)(t,x)⟩+\frac{1}{2}\text{Trace}(\sigma (t,x)[\sigma (t,x){]}^{\ast }({\text{Hess}}_{x}u)(t,x)).\end{array}$$

where $u:[0,T]\times \mathrm{\Omega }\to \mathbb{R},\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ 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)