update docs

Author: yuehhua

docs/make.jl

@@ -15,6 +15,7 @@ makedocs(;
     ),
     pages=[
         "Home" => "index.md",
+        "Introduction" => "introduction.md",
         "APIs" => "apis.md"
     ],
 )

docs/src/apis.md

@@ -1,7 +1,4 @@
-## Index
-
-```@index
-```
+# APIs
 
 ## Layers
 
@@ -40,6 +37,8 @@ OperatorKernel
 
 Reference: [Fourier Neural Operator for Parametric Partial Differential Equations](https://arxiv.org/abs/2010.08895)
 
+---
+
 ### Graph kernel layer
 
 ```math
@@ -55,6 +54,8 @@ GraphKernel
 
 Reference: [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://arxiv.org/abs/2003.03485)
 
+---
+
 ## Models
 
 ### Fourier neural operator

docs/src/index.md

@@ -4,40 +4,13 @@ CurrentModule = NeuralOperators
 
 # NeuralOperators
 
-Documentation for [NeuralOperators](https://github.com/foldfelis/NeuralOperators.jl).
-
-| **Ground Truth** | **Inferenced** |
-|:----------------:|:--------------:|
 | ![](https://github.com/foldfelis/NeuralOperators.jl/blob/master/example/FlowOverCircle/gallery/ans.gif?raw=true) | ![](https://github.com/foldfelis/NeuralOperators.jl/blob/master/example/FlowOverCircle/gallery/inferenced.gif?raw=true) |
+|:----------------:|:--------------:|
+| **Ground Truth** | **Inferenced** |
 
 The demonstration shown above is Navier-Stokes equation learned by the `MarkovNeuralOperator` with only one time step information.
 Example can be found in [`example/FlowOverCircle`](https://github.com/foldfelis/NeuralOperators.jl/tree/master/example/FlowOverCircle).
 
-## Abstract
-
-Neural operator is a novel deep learning architecture.
-It learns a operator, which is a mapping between infinite-dimensional function spaces.
-It can be used to resolve [partial differential equations (PDE)](https://en.wikipedia.org/wiki/Partial_differential_equation).
-Instead of solving by finite element method, a PDE problem can be resolved by training a neural network to learn an operator mapping
-from infinite-dimensional space (u, t) to infinite-dimensional space f(u, t).
-Neural operator learns a continuous function between two continuous function spaces.
-The kernel can be trained on different geometry, which is learned from a graph.
-
-**Fourier neural operator** learns a neural operator with Dirichlet kernel to form a Fourier transformation.
-It performs Fourier transformation across infinite-dimensional function spaces and learns better than neural operator.
-
-**Markov neural operator** learns a neural operator with Fourier operators.
-With only one time step information of learning, it can predict the following few steps with low loss
-by linking the operators into a Markov chain.
-
-**DeepONet operator** (Deep Operator Network) learns a neural operator with the help of two sub-neural net structures described as the branch and the trunk network.
-The branch network is fed the initial conditions data, whereas the trunk is fed with the locations where the target(output) is evaluated from the corresponding initial conditions.
-It is important that the output size of the branch and trunk subnets is same so that a dot product can be performed between them.
-
-Currently, the `OperatorKernel` layer is provided in this work.
-As for model, there are `FourierNeuralOperator` and `MarkovNeuralOperator` provided.
-Please take a glance at them [here](apis.html#Models).
-
 ## Quick start
 
 The package can be installed with the Julia package manager. From the Julia REPL, type `]` to enter the Pkg REPL mode and run:
@@ -80,7 +53,7 @@ model = FourierNeuralOperator(
 And then train as a Flux model.
 
 ```julia
-loss(𝐱, 𝐲) = sum(abs2, 𝐲 .- model(𝐱)) / size(𝐱)[end]
+loss(𝐱, 𝐲) = Flux.Losses.mse(model(𝐱), 𝐲)
 opt = Flux.Optimiser(WeightDecay(1f-4), Flux.ADAM(1f-3))
 Flux.@epochs 50 Flux.train!(loss, params(model), data, opt)
 ```
@@ -112,4 +85,4 @@ opt = ADAM(learning_rate)
 parameters = params(model)
 Flux.@epochs 400 Flux.train!(loss, parameters, [(xtrain, ytrain, grid)], opt, cb=evalcb)
 ```
-A more complete example using DeepONet architecture to solve Burgers' equation can be found in the [examples](../../example/Burgers/src/Burgers_deeponet.jl)
+A more complete example using DeepONet architecture to solve Burgers' equation can be found in the [examples](https://github.com/foldfelis/NeuralOperators.jl/tree/master/example/Burgers/src/Burgers_deeponet.jl).

docs/src/introduction.md

@@ -0,0 +1,30 @@
+# Introduction
+
+Neural operator is a novel deep learning architecture.
+It learns a operator, which is a mapping between infinite-dimensional function spaces.
+It can be used to resolve [partial differential equations (PDE)](https://en.wikipedia.org/wiki/Partial_differential_equation).
+Instead of solving by time-consuming finite element method, a PDE problem can be resolved by training a neural network to learn
+an operator mapping from infinite-dimensional space ``(u, t)`` to infinite-dimensional space ``f(u, t)``.
+Neural operator learns a continuous function between two continuous function spaces.
+The kernel can be trained on different geometry, including regular Euclidean space or a graph topology.
+
+## Fourier Neural Operators
+
+Fourier neural operator (FNO) learns a neural operator with Dirichlet kernel to form a Fourier transformation.
+It performs Fourier transformation across infinite-dimensional function spaces and learns better than neural operator.
+
+## Markov Neural Operators
+
+Markov neural operator (MNO) learns a neural operator with Fourier operators.
+With only one time step information of learning, it can predict the following few steps with low loss
+by linking the operators into a Markov chain.
+
+## Deep Operator Network
+
+Deep operator network (DeepONet) learns a neural operator with the help of two sub-neural network structures described as the branch and the trunk network.
+The branch network is fed the initial conditions data, whereas the trunk is fed with the locations where the target(output) is evaluated from the corresponding initial conditions.
+It is important that the output size of the branch and trunk subnets is same so that a dot product can be performed between them.
+
+Currently, the `OperatorKernel` layer is provided in this work.
+As for model, there are `FourierNeuralOperator` and `MarkovNeuralOperator` provided.
+Please take a glance at [them](apis.html#Models).