HighDimPDE.jl
HighDimPDE.jl is a Julia package to solve Highly Dimensional non-linear, non-local PDEs of the forms:
1. Partial Integro Differential Equations:
\[\begin{aligned} (\partial_t u)(t,x) &= \int_{\Omega} f\big(t,x,{\bf x}, u(t,x),u(t,{\bf x}), ( \nabla_x u )(t,x ),( \nabla_x u )(t,{\bf x} ) \big) \, d{\bf x} \\ & \quad + \big\langle \mu(t,x), ( \nabla_x u )( t,x ) \big\rangle + \tfrac{1}{2} \text{Trace} \big(\sigma(t,x) [ \sigma(t,x) ]^* ( \text{Hess}_x u)(t, x ) \big). \end{aligned}\]
where $u \colon [0,T] \times \Omega \to \R$, $\Omega \subseteq \R^d$ is subject to initial and boundary conditions, and where $d$ is large. These equations are defined using the PIDEProblem
2. Parabolic Partial Differential Equations:
\[\begin{aligned} (\partial_t u)(t,x) &= f\big(t,x, u(t,x), ( \nabla_x u )(t,x )\big) + \big\langle \mu(t,x), ( \nabla_x u )( t,x ) \big\rangle + \tfrac{1}{2} \text{Trace} \big(\sigma(t,x) [ \sigma(t,x) ]^* ( \text{Hess}_x u)(t, x ) \big). \end{aligned}\]
where $u \colon [0,T] \times \Omega \to \R$, $\Omega \subseteq \R^d$ is subject to initial and boundary conditions, and where $d$ is large. These equations are defined using the ParabolicPDEProblem
The difference between the two problems is that in Partial Integro Differential Equations, the integrand is integrated over x, while in Parabolic Integro Differential Equations, the function f is just evaluated for x.
HighDimPDE.jl implements solver algorithms that break down the curse of dimensionality, including
NNKolmogorov and NNParamKolmogorov schemes for Kolmogorov PDEs.
[NNStopping] scheme for solving optimal stopping time problem.
To make the most out of HighDimPDE.jl, we advise to first have a look at the
as all solver algorithms heavily rely on it.
Algorithm overview
| Features | DeepSplitting | MLP | DeepBSDE |
|---|---|---|---|
| Time discretization free | ❌ | ✅ | ❌ |
| Mesh-free | ✅ | ✅ | ✅ |
| Single point $x \in \R^d$ approximation | ✅ | ✅ | ✅ |
| $d$-dimensional cube $[a,b]^d$ approximation | ✅ | ❌ | ✔️ |
| GPU | ✅ | ❌ | ✅ |
| Gradient non-linearities | ✔️ | ❌ | ✅ |
✔️ : will be supported in the future
Reproducibility
The documentation of this SciML package was built using these direct dependencies,
Status `/var/lib/buildkite-agent/builds/gpuci-14/julialang/highdimpde-dot-jl/docs/Project.toml`
[052768ef] CUDA v5.4.3
[e30172f5] Documenter v1.5.0
[587475ba] Flux v0.14.16
[57c578d5] HighDimPDE v2.0.0 `/var/lib/buildkite-agent/builds/gpuci-14/julialang/highdimpde-dot-jl`
[789caeaf] StochasticDiffEq v6.67.0
[02a925ec] cuDNN v1.3.2
[37e2e46d] LinearAlgebraand using this machine and Julia version.
Julia Version 1.10.4
Commit 48d4fd48430 (2024-06-04 10:41 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 48 × AMD EPYC 7402 24-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 2 virtual cores)
Environment:
JULIA_CPU_THREADS = 2
JULIA_DEPOT_PATH = /root/.cache/julia-buildkite-plugin/depots/018d1fa6-388a-44ce-8188-c97dc1207ef6
LD_LIBRARY_PATH = /usr/local/nvidia/lib:/usr/local/nvidia/lib64
JULIA_PKG_SERVER =A more complete overview of all dependencies and their versions is also provided.
Status `/var/lib/buildkite-agent/builds/gpuci-14/julialang/highdimpde-dot-jl/docs/Manifest.toml`
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`You can also download the manifest file and the project file.