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