Merge pull request #96 from ashutosh-b-b/docs

Fix and clear documentation

.buildkite/pipeline.yml

@@ -0,0 +1,36 @@
+steps:
+  - label: "Julia 1"
+    plugins:
+      - JuliaCI/julia#v1:
+          version: "1"
+      - JuliaCI/julia-test#v1:
+           coverage: false # 1000x slowdown
+    agents:
+      queue: "juliagpu"
+      cuda: "*"
+    env:
+      GROUP: 'GPU'
+    timeout_in_minutes: 60
+    # Don't run Buildkite if the commit message includes the text [skip tests]
+    if: build.message !~ /\[skip tests\]/
+
+  - label: "Documentation"
+    plugins:
+      - JuliaCI/julia#v1:
+          version: "1"
+    command: |
+      julia --project=docs -e '
+        println("--- :julia: Instantiating project")
+        using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()
+        println("+++ :julia: Building documentation")
+        include("docs/make.jl")'
+    agents:
+      queue: "juliagpu"
+      cuda: "*"
+    if: build.message !~ /\[skip docs\]/ && !build.pull_request.draft
+    timeout_in_minutes: 1000
+    env:
+      SECRET_DOCUMENTER_KEY: "v/2iYv+rW3iCgDU5Cp4DN1P51xQvXkC4aTO28pDLysKaNTVopjzsQNNtgJaswckg5ahdusdnAmlDRd8oMEobmyPM7PY7wppGdr6MHfUhwakJ7AAWAyAEIb+e+fNFuu+EIHS43bU+vkgzCgKojFEGwa5AFSW69rucnuwROXzIrQQ6GgXF4LayvDMDbUgYPjjnS2zAgfkWOR1L5UJ2ODuy3zK+VESS5YVFRNSmhAnT0EdS2AsenBwa25IgHsYkGO1fR/CL+1epFg7yzSOA+bExDWD0L4WkjKzU8qTlMAI2BAw+clJLhcpZyy4MdUG8hLmic0vvYjBOdjgqtl1f9xtMwQ==;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"
+
+env:
+  JULIA_PKG_SERVER: "" # it often struggles with our large artifacts

LICENSE.md

@@ -19,4 +19,3 @@ The HighDimPDE.jl package is licensed under the MIT "Expat" License:
 > LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 > OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 > SOFTWARE.
-> 

Project.toml

@@ -12,6 +12,7 @@ Functors = "d9f16b24-f501-4c13-a1f2-28368ffc5196"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
+SciMLBase = "0bca4576-84f4-4d90-8ffe-ffa030f20462"
 SciMLSensitivity = "1ed8b502-d754-442c-8d5d-10ac956f44a1"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
@@ -33,6 +34,7 @@ LinearAlgebra = "1.10"
 Random = "1.10"
 Reexport = "1.2.2"
 SafeTestsets = "0.1"
+SciMLBase = "2.42"
 SciMLSensitivity = "7.62"
 SparseArrays = "1.10"
 Statistics = "1.10"

README.md

@@ -4,7 +4,7 @@
 [![codecov](https://codecov.io/gh/SciML/HighDimPDE.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/SciML/HighDimPDE.jl)
 [![Build Status](https://github.com/SciML/HighDimPDE.jl/workflows/CI/badge.svg)](https://github.com/SciML/HighDimPDE.jl/actions?query=workflow%3ACI)
 
-[![ColPrac: Contributor's Guide on Collaborative Practices for Community Packages](https://img.shields.io/badge/ColPrac-Contributor's%20Guide-blueviolet)](https://github.com/SciML/ColPrac)
+[![ColPrac: Contributor's Guide on Collaborative Practices for Community Packages](https://img.shields.io/badge/ColPrac-Contributor%27s%20Guide-blueviolet)](https://github.com/SciML/ColPrac)
 [![SciML Code Style](https://img.shields.io/static/v1?label=code%20style&message=SciML&color=9558b2&labelColor=389826)](https://github.com/SciML/SciMLStyle)
 
 # HighDimPDE.jl
@@ -13,12 +13,13 @@
 
 $$
 \begin{aligned}
-    (\partial_t u)(t,x) &= \int_{\Omega} f\big(t,x,{\bf x}, u(t,x),u(t,{\bf x}), ( \nabla_x u )(t,x ),( \nabla_x u )(t,{\bf x} ) \big) d{\bf x} \\
-    & \quad + \big\langle \mu(t,x), ( \nabla_x u )( t,x ) \big\rangle + \tfrac{1}{2} \text{Trace} \big(\sigma(t,x) [ \sigma(t,x) ]^* ( \text{Hess}_x u)(t, x ) \big).
+(\partial_t u)(t,x) &= \int_{\Omega} f\big(t,x,{\bf x}, u(t,x),u(t,{\bf x}), ( \nabla_x u )(t,x ),( \nabla_x u )(t,{\bf x} ) \big) d{\bf x} \\
+& \quad + \big\langle \mu(t,x), ( \nabla_x u )( t,x ) \big\rangle + \tfrac{1}{2} \text{Trace} \big(\sigma(t,x) [ \sigma(t,x) ]^* ( \text{Hess}_x u)(t, x ) \big).
 \end{aligned}
 $$
 
 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,
@@ -27,22 +28,26 @@ For information on using the package,
 the documentation, which contains the unreleased features.
 
 ## Installation
+
 Open Julia and type the following
 
 ```julia
 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](https://arxiv.org/abs/2205.03672) (2022)
+
+  - Boussange, V., Becker, S., Jentzen, A., Kuckuck, B., Pellissier, L., Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. [arXiv](https://arxiv.org/abs/2205.03672) (2022)
+
 <!-- - Becker, S., Braunwarth, R., Hutzenthaler, M., Jentzen, A., von Wurstemberger, P., Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations. [arXiv](https://arxiv.org/abs/2005.10206) (2020)
 - Beck, C., Becker, S., Cheridito, P., Jentzen, A., Neufeld, A., Deep splitting method for parabolic PDEs. [arXiv](https://arxiv.org/abs/1907.03452) (2019)
 - Han, J., Jentzen, A., E, W., Solving high-dimensional partial differential equations using deep learning. [arXiv](https://arxiv.org/abs/1707.02568) (2018) -->
-
 <!-- ## Acknowledgements
 `HighDimPDE.jl` is inspired from Sebastian Becker's scripts in Python, TensorFlow, and C++. Pr. Arnulf Jentzen largely contributed to the theoretical developments of the solver algorithms implemented. -->

docs/Project.toml

@@ -1,10 +1,15 @@
 [deps]
 CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
+cuDNN = "02a925ec-e4fe-4b08-9a7e-0d78e3d38ccd"
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c"
 HighDimPDE = "57c578d5-59d4-4db8-a490-a9fc372d19d2"
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+StochasticDiffEq = "789caeaf-c7a9-5a7d-9973-96adeb23e2a0"
 
 [compat]
-CUDA = "5"
+CUDA = "5.4.2"
+cuDNN = "1.3.2"
 Documenter = "1"
-Flux = "0.13, 0.14"
+Flux = "0.14.16"
+StochasticDiffEq = "6.66"

docs/make.jl

@@ -6,7 +6,7 @@ cp("./docs/Project.toml", "./docs/src/assets/Project.toml", force = true)
 include("pages.jl")
 
 makedocs(sitename = "HighDimPDE.jl",
-    authors = "Victor Boussange",
+    authors = "#",
     pages = pages,
     clean = true, doctest = false, linkcheck = true,
     format = Documenter.HTML(assets = ["assets/favicon.ico"],

docs/pages.jl

@@ -5,7 +5,7 @@ pages = [
     "Solver Algorithms" => ["MLP.md",
         "DeepSplitting.md",
         "DeepBSDE.md",
-        "NNStopping.md", 
+        "NNStopping.md",
         "NNKolmogorov.md",
         "NNParamKolmogorov.md"],
     "Tutorials" => [
@@ -14,7 +14,13 @@ pages = [
         "tutorials/mlp.md",
         "tutorials/nnstopping.md",
         "tutorials/nnkolmogorov.md",
-        "tutorials/nnparamkolmogorov.md",
+        "tutorials/nnparamkolmogorov.md"
     ],
-    "Feynman Kac formula" => "Feynman_Kac.md",
+    "Extended Examples" => [
+        "examples/blackscholes.md"
+    ],
+    "Solver Algorithms" => ["MLP.md",
+        "DeepSplitting.md",
+        "DeepBSDE.md"],
+    "Feynman Kac formula" => "Feynman_Kac.md"
 ]

docs/src/DeepBSDE.md

@@ -1,16 +1,19 @@
 # [The `DeepBSDE` algorithm](@id deepbsde)
 
 ### Problems Supported:
-1. [`ParabolicPDEProblem`](@ref)
+
+ 1. [`ParabolicPDEProblem`](@ref)
 
 ```@autodocs
 Modules = [HighDimPDE]
 Pages   = ["DeepBSDE.jl", "DeepBSDE_Han.jl"]
 ```
 
 ## The general idea 💡
-The `DeepBSDE` algorithm is similar in essence to the `DeepSplitting` algorithm, with the difference that 
+
+The `DeepBSDE` algorithm is similar in essence to the `DeepSplitting` algorithm, with the difference that
 it uses two neural networks to approximate both the the solution and its gradient.
 
 ## References
-- Han, J., Jentzen, A., E, W., Solving high-dimensional partial differential equations using deep learning. [arXiv](https://arxiv.org/abs/1707.02568) (2018)
+
+  - Han, J., Jentzen, A., E, W., Solving high-dimensional partial differential equations using deep learning. [arXiv](https://arxiv.org/abs/1707.02568) (2018)

docs/src/DeepSplitting.md

@@ -1,8 +1,9 @@
 # [The `DeepSplitting` algorithm](@id deepsplitting)
 
 ### Problems Supported:
-1. [`PIDEProblem`](@ref)
-2. [`ParabolicPDEProblem`](@ref)
+
+ 1. [`PIDEProblem`](@ref)
+ 2. [`ParabolicPDEProblem`](@ref)
 
 ```@autodocs
 Modules = [HighDimPDE]
@@ -13,76 +14,93 @@ The `DeepSplitting` algorithm reformulates the PDE as a stochastic learning prob
 
 The algorithm relies on two main ideas:
 
-- The approximation of the solution $u$ by a parametric function $\bf u^\theta$. This function is generally chosen as a (Feedforward) Neural Network, as it is a [universal approximator](https://en.wikipedia.org/wiki/Universal_approximation_theorem).
+  - The approximation of the solution $u$ by a parametric function $\bf u^\theta$. This function is generally chosen as a (Feedforward) Neural Network, as it is a [universal approximator](https://en.wikipedia.org/wiki/Universal_approximation_theorem).
 
-- The training of $\bf u^\theta$ by simulated stochastic trajectories of particles, through the link between linear PDEs and the expected trajectory of associated Stochastic Differential Equations (SDEs), explicitly stated by the [Feynman Kac formula](https://en.wikipedia.org/wiki/Feynman–Kac_formula).
+  - The training of $\bf u^\theta$ by simulated stochastic trajectories of particles, through the link between linear PDEs and the expected trajectory of associated Stochastic Differential Equations (SDEs), explicitly stated by the [Feynman Kac formula](https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula).
 
 ## The general idea 💡
+
 Consider the PDE
+
 ```math
 \partial_t u(t,x) = \mu(t, x) \nabla_x u(t,x) + \frac{1}{2} \sigma^2(t, x) \Delta_x u(t,x) + f(x, u(t,x)) \tag{1}
 ```
-with initial conditions $u(0, x) = g(x)$, where $u \colon \R^d \to \R$. 
+
+with initial conditions $u(0, x) = g(x)$, where $u \colon \R^d \to \R$.
 
 ### Local Feynman Kac formula
+
 `DeepSplitting` solves the PDE iteratively over small time intervals by using an approximate [Feynman-Kac representation](@ref feynmankac) locally.
 
 More specifically, considering a small time step $dt = t_{n+1} - t_n$ one has that
+
 ```math
 u(t_{n+1}, X_{T - t_{n+1}}) \approx \mathbb{E} \left[ f(t, X_{T - t_{n}}, u(t_{n},X_{T - t_{n}}))(t_{n+1} - t_n) + u(t_{n}, X_{T - t_{n}}) | X_{T - t_{n+1}}\right] \tag{3}.
 ```
+
 One can therefore use Monte Carlo integrations to approximate the expectations
+
 ```math
 u(t_{n+1}, X_{T - t_{n+1}}) \approx \frac{1}{\text{batch\_size}}\sum_{j=1}^{\text{batch\_size}} \left[ u(t_{n}, X_{T - t_{n}}^{(j)}) + (t_{n+1} - t_n)\sum_{k=1}^{K} \big[ f(t_n, X_{T - t_{n}}^{(j)}, u(t_{n},X_{T - t_{n}}^{(j)})) \big] \right]
 ```
 
-
 ### Reformulation as a learning problem
+
 The `DeepSplitting` algorithm approximates $u(t_{n+1}, x)$ by a parametric function ${\bf u}^\theta_n(x)$. It is advised to let this function be a neural network ${\bf u}_\theta \equiv NN_\theta$ as they are universal approximators.
 
-For each time step $t_n$, the `DeepSplitting` algorithm 
+For each time step $t_n$, the `DeepSplitting` algorithm
 
-1. Generates the particle trajectories $X^{x, (j)}$ satisfying [Eq. (2)](@ref feynmankac) over the whole interval $[0,T]$.
+ 1. Generates the particle trajectories $X^{x, (j)}$ satisfying [Eq. (2)](@ref feynmankac) over the whole interval $[0,T]$.
 
-2. Seeks ${\bf u}_{n+1}^{\theta}$  by minimizing the loss function
+ 2. Seeks ${\bf u}_{n+1}^{\theta}$  by minimizing the loss function
 
 ```math
 L(\theta) = ||{\bf u}^\theta_{n+1}(X_{T - t_n}) - \left[ f(t, X_{T - t_{n-1}}, {\bf u}_{n-1}(X_{T - t_{n-1}}))(t_{n} - t_{n-1}) + {\bf u}_{n-1}(X_{T - t_{n-1}}) \right] ||
 ```
+
 This way, the PDE approximation problem is decomposed into a sequence of separate learning problems.
 In `HighDimPDE.jl` the right parameter combination $\theta$ is found by iteratively minimizing $L$ using **stochastic gradient descent**.
 
 !!! tip
+
     To solve with `DeepSplitting`, one needs to provide to `solve`
-    - `dt`
-    - `batch_size`
-    - `maxiters`: the number of iterations for minimizing the loss function
-    - `abstol`: the absolute tolerance for the loss function
-    - `use_cuda`: if you have a Nvidia GPU, recommended.
+
+      - `dt`
+      - `batch_size`
+      - `maxiters`: the number of iterations for minimizing the loss function
+      - `abstol`: the absolute tolerance for the loss function
+      - `use_cuda`: if you have a Nvidia GPU, recommended.
 
 ## Solving point-wise or on a hypercube
 
 ### Pointwise
+
 `DeepSplitting` allows obtaining $u(t,x)$ on a single point  $x \in \Omega$ with the keyword $x$.
 
 ```julia
-prob = PIDEProblem(μ, σ, x, tspan, g, f,)
+prob = PIDEProblem(μ, σ, x, tspan, g, f)
 ```
 
 ### Hypercube
+
 Yet more generally, one wants to solve Eq. (1) on a $d$-dimensional cube $[a,b]^d$. This is offered by `HighDimPDE.jl` with the keyword `x0_sample`.
 
 ```julia
 prob = PIDEProblem(μ, σ, x, tspan, g, f; x0_sample = x0_sample)
 ```
+
 Internally, this is handled by assigning a random variable as the initial point of the particles, i.e.
+
 ```math
 X_t^\xi = \int_0^t \mu(X_s^x)ds + \int_0^t\sigma(X_s^x)dB_s + \xi,
 ```
+
 where $\xi$ a random variable uniformly distributed over $[a,b]^d$. This way, the neural network is trained on the whole interval $[a,b]^d$ instead of a single point.
 
 ## Non-local PDEs
+
 `DeepSplitting` can solve for non-local reaction diffusion equations of the type
+
 ```math
 \partial_t u = \mu(x) \nabla_x u + \frac{1}{2} \sigma^2(x) \Delta u + \int_{\Omega}f(x,y, u(t,x), u(t,y))dy
 ```
@@ -93,13 +111,14 @@ The non-localness is handled by a Monte Carlo integration.
 u(t_{n+1}, X_{T - t_{n+1}}) \approx \sum_{j=1}^{\text{batch\_size}} \left[ u(t_{n}, X_{T - t_{n}}^{(j)}) + \frac{(t_{n+1} - t_n)}{K}\sum_{k=1}^{K} \big[ f(t, X_{T - t_{n}}^{(j)}, Y_{X_{T - t_{n}}^{(j)}}^{(k)}, u(t_{n},X_{T - t_{n}}^{(j)}), u(t_{n},Y_{X_{T - t_{n}}^{(j)}}^{(k)})) \big] \right]
 ```
 
-!!! tip 
-    In practice, if you have a non-local model, you need to provide the sampling method and the number $K$ of MC integration through the keywords `mc_sample` and `K`. 
-    ```julia
-    alg = DeepSplitting(nn, opt = opt, mc_sample = mc_sample, K = 1)
-    ```
-    `mc_sample` can be whether `UniformSampling(a, b)` or ` NormalSampling(σ_sampling, shifted)`.
+!!! tip
+
+
+In practice, if you have a non-local model, you need to provide the sampling method and the number $K$ of MC integration through the keywords `mc_sample` and `K`.
+`julia alg = DeepSplitting(nn, opt = opt, mc_sample = mc_sample, K = 1) `
+`mc_sample` can be whether `UniformSampling(a, b)` or ` NormalSampling(σ_sampling, shifted)`.
 
 ## References
-- Boussange, V., Becker, S., Jentzen, A., Kuckuck, B., Pellissier, L., Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. [arXiv](https://arxiv.org/abs/2205.03672) (2022)
-- Beck, C., Becker, S., Cheridito, P., Jentzen, A., Neufeld, A., Deep splitting method for parabolic PDEs. [arXiv](https://arxiv.org/abs/1907.03452) (2019)
+
+  - Boussange, V., Becker, S., Jentzen, A., Kuckuck, B., Pellissier, L., Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. [arXiv](https://arxiv.org/abs/2205.03672) (2022)
+  - Beck, C., Becker, S., Cheridito, P., Jentzen, A., Neufeld, A., Deep splitting method for parabolic PDEs. [arXiv](https://arxiv.org/abs/1907.03452) (2019)