From 853e826dd17912c8ebb253de8ee1523b5bb17655 Mon Sep 17 00:00:00 2001
From: Sathvik Bhagavan <sathvik.bhagavan@juliahub.com>
Date: Thu, 11 Apr 2024 09:04:07 +0000
Subject: [PATCH 1/8] build(docs): add deps

---
 docs/Project.toml | 21 +++++++++++++++++++++
 1 file changed, 21 insertions(+)

diff --git a/docs/Project.toml b/docs/Project.toml
index 79f849e..d886980 100644
--- a/docs/Project.toml
+++ b/docs/Project.toml
@@ -1,6 +1,27 @@
 [deps]
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
+Lux = "b2108857-7c20-44ae-9111-449ecde12c47"
+ModelingToolkit = "961ee093-0014-501f-94e3-6117800e7a78"
+ModelingToolkitStandardLibrary = "16a59e39-deab-5bd0-87e4-056b12336739"
+Optimization = "7f7a1694-90dd-40f0-9382-eb1efda571ba"
+OptimizationOptimisers = "42dfb2eb-d2b4-4451-abcd-913932933ac1"
+OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+SciMLStructures = "53ae85a6-f571-4167-b2af-e1d143709226"
+StableRNGs = "860ef19b-820b-49d6-a774-d7a799459cd3"
+SymbolicIndexingInterface = "2efcf032-c050-4f8e-a9bb-153293bab1f5"
 ModelingToolkitNeuralNets = "f162e290-f571-43a6-83d9-22ecc16da15f"
 
 [compat]
 Documenter = "1.3"
+Lux = "0.5.32"
+ModelingToolkit = "9.9"
+ModelingToolkitStandardLibrary = "2.6"
+Optimization = "3.24"
+OptimizationOptimisers = "0.2.1"
+OrdinaryDiffEq = "6.74"
+Plots = "1"
+SciMLStructures = "1.1.0"
+StableRNGs = "1"
+SymbolicIndexingInterface = "0.3.15"
+ModelingToolkitNeuralNets = "1"

From 0eeee0d9ed7fe74f3a9859e0b5d5a6a827a70b6a Mon Sep 17 00:00:00 2001
From: Sathvik Bhagavan <sathvik.bhagavan@juliahub.com>
Date: Thu, 11 Apr 2024 09:10:42 +0000
Subject: [PATCH 2/8] docs: add friction tutorial

---
 docs/make.jl                |  19 ++--
 docs/src/api.md             |   5 +
 docs/src/assets/favicon.ico | Bin 0 -> 1394 bytes
 docs/src/assets/logo.png    | Bin 0 -> 26575 bytes
 docs/src/friction.md        | 178 ++++++++++++++++++++++++++++++++++++
 5 files changed, 195 insertions(+), 7 deletions(-)
 create mode 100644 docs/src/api.md
 create mode 100644 docs/src/assets/favicon.ico
 create mode 100644 docs/src/assets/logo.png
 create mode 100644 docs/src/friction.md

diff --git a/docs/make.jl b/docs/make.jl
index 54367ee..2393190 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -4,19 +4,24 @@ using Documenter
 cp("./docs/Manifest.toml", "./docs/src/assets/Manifest.toml", force = true)
 cp("./docs/Project.toml", "./docs/src/assets/Project.toml", force = true)
 
-DocMeta.setdocmeta!(ModelingToolkitNeuralNets, :DocTestSetup, :(using ModelingToolkitNeuralNets); recursive = true)
+DocMeta.setdocmeta!(ModelingToolkitNeuralNets, :DocTestSetup,
+    :(using ModelingToolkitNeuralNets); recursive = true)
 
 makedocs(;
     modules = [ModelingToolkitNeuralNets],
     authors = "Sebastian Micluța-Câmpeanu <sebastian.mc95@proton.me> and contributors",
     sitename = "ModelingToolkitNeuralNets.jl",
-    format = Documenter.HTML(;
-        canonical = "https://SciML.github.io/ModelingToolkitNeuralNets.jl",
-        edit_link = "main",
-        assets = String[]
-    ),
+    format = Documenter.HTML(assets = ["assets/favicon.ico"],
+        canonical = "https://docs.sciml.ai/ModelingToolkitNeuralNets.jl/stable/"),
+    clean = true,
+    doctest = false,
+    linkcheck = true,
     pages = [
-        "Home" => "index.md"
+        "Home" => "index.md",
+        "Tutorials" => [
+            "Friction Model" => "friction.md"
+        ],
+        "API" => "api.md"
     ]
 )
 
diff --git a/docs/src/api.md b/docs/src/api.md
new file mode 100644
index 0000000..28b49d5
--- /dev/null
+++ b/docs/src/api.md
@@ -0,0 +1,5 @@
+# API
+
+```@docs
+NeuralNetworkBlock
+```
diff --git a/docs/src/assets/favicon.ico b/docs/src/assets/favicon.ico
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diff --git a/docs/src/friction.md b/docs/src/friction.md
new file mode 100644
index 0000000..d05ca9a
--- /dev/null
+++ b/docs/src/friction.md
@@ -0,0 +1,178 @@
+# Modeling Non Linear Friction Model using UDEs
+
+Friction between moving bodies is not trivial to model. There have been idealised linear models which are not always useful in complicated systems. There have been many theories and non linear models which we can use, but they are not perfect. The aim of this tutorial to use Universal Differential Equations to showcase how we can embed a neural network to learn an unknown non linear friction model.
+
+## Julia Environment
+
+First, lets import the required packages.
+
+```@example friction
+using ModelingToolkitNeuralNets
+using ModelingToolkit
+import ModelingToolkit.t_nounits as t
+import ModelingToolkit.D_nounits as Dt
+using ModelingToolkitStandardLibrary.Blocks
+using OrdinaryDiffEq
+using Optimization
+using OptimizationOptimisers: Adam
+using SciMLStructures
+using SciMLStructures: Tunable
+using SymbolicIndexingInterface
+using StableRNGs
+using Lux
+using Plots
+```
+
+## Problem Setup
+
+Let's use the friction model presented in https://www.mathworks.com/help/simscape/ref/translationalfriction.html for generating data.
+
+```@example friction
+Fbrk = 100.0
+vbrk = 10.0
+Fc = 80.0
+vst = vbrk / 10
+vcol = vbrk * sqrt(2)
+function friction(v)
+    sqrt(2 * MathConstants.e) * (Fbrk - Fc) * exp(-(v / vst)^2) * (v / vst) +
+    Fc * tanh(v / vcol)
+end
+```
+
+Next, we define the model - an object sliding in 1D plane with a constant force `Fu` acting on it and friction force opposing the motion.
+
+```@example friction
+function friction_true()
+    @variables y(t) = 0.0
+    @constants Fu = 120.0
+    eqs = [
+        Dt(y) ~ Fu - friction(y)
+    ]
+    return ODESystem(eqs, t, name = :friction_true)
+end
+```
+
+Now that we have defined the model, we will simulate it from 0 to 0.1 seconds.
+
+```@example friction
+model_true = structural_simplify(friction_true())
+prob_true = ODEProblem(model_true, [], (0, 0.1), [])
+sol_ref = solve(prob_true, Rodas4(); saveat = 0.001)
+```
+
+Let's plot it.
+
+```@example friction
+scatter(sol_ref, label = "velocity")
+```
+
+That was the velocity. Let's also plot the friction force acting on the object throughout the simulation.
+
+```@example friction
+scatter(sol_ref.t, friction.(first.(sol_ref.u)), label = "friction force")
+```
+
+## Model Setup
+
+Now, we will try to learn the same friction model using a neural network. We will use [`NeuralNetworkBlock`](@ref) to define neural network as a component. The input of the neural network is the velocity and the output is the friction force. We connect the neural network with the model using `RealInputArray` and `RealOutputArray` blocks.
+
+```@example friction
+function friction_ude(Fu)
+    @variables y(t) = 0.0
+    @constants Fu = Fu
+    @named nn_in = RealInputArray(nin = 1)
+    @named nn_out = RealOutputArray(nout = 1)
+    eqs = [Dt(y) ~ Fu - nn_in.u[1]
+           y ~ nn_out.u[1]]
+    return ODESystem(eqs, t, name = :friction, systems = [nn_in, nn_out])
+end
+
+Fu = 120.0
+model = friction_ude(Fu)
+
+chain = Lux.Chain(
+    Lux.Dense(1 => 10, Lux.mish, use_bias = false),
+    Lux.Dense(10 => 10, Lux.mish, use_bias = false),
+    Lux.Dense(10 => 1, use_bias = false)
+)
+nn = NeuralNetworkBlock(1, 1; chain = chain, rng = StableRNG(1111))
+
+eqs = [connect(model.nn_in, nn.output)
+       connect(model.nn_out, nn.input)]
+
+ude_sys = complete(ODESystem(eqs, t, systems = [model, nn], name = :ude_sys))
+sys = structural_simplify(ude_sys)
+```
+
+## Optimization Setup
+
+We now setup the loss function and the optimization loop.
+
+```@example friction
+function loss(x, (prob, sol_ref, get_vars, get_refs))
+    new_p = SciMLStructures.replace(Tunable(), prob.p, x)
+    new_prob = remake(prob, p = new_p, u0 = eltype(x).(prob.u0))
+    ts = sol_ref.t
+    new_sol = solve(new_prob, Rodas4(), saveat = ts, abstol = 1e-8, reltol = 1e-8)
+    loss = zero(eltype(x))
+    for i in eachindex(new_sol.u)
+        loss += sum(abs2.(get_vars(new_sol, i) .- get_refs(sol_ref, i)))
+    end
+    if SciMLBase.successful_retcode(new_sol)
+        loss
+    else
+        Inf
+    end
+end
+
+of = OptimizationFunction{true}(loss, AutoForwardDiff())
+
+prob = ODEProblem(sys, [], (0, 0.1), [])
+get_vars = getu(sys, [sys.friction.y])
+get_refs = getu(model_true, [model_true.y])
+x0 = reduce(vcat, getindex.((default_values(sys),), tunable_parameters(sys)))
+
+cb = (opt_state, loss) -> begin
+    @info "step $(opt_state.iter), loss: $loss"
+    return false
+end
+
+op = OptimizationProblem(of, x0, (prob, sol_ref, get_vars, get_refs))
+res = solve(op, Adam(5e-3); maxiters = 10000, callback = cb)
+```
+
+## Visualization of results
+
+We now have a trained neural network! We can check whether running the simulation of the model embedded with the neural network matches the data or not.
+
+```@example friction
+res_p = SciMLStructures.replace(Tunable(), prob.p, res)
+res_prob = remake(prob, p = res_p)
+res_sol = solve(res_prob, Rodas4(), saveat = sol_ref.t)
+@test first.(sol_ref.u)≈first.(res_sol.u) rtol=1e-3 #hide
+@test friction.(first.(sol_ref.u))≈(Fu .- first.(res_sol(res_sol.t, Val{1}).u)) rtol=1e-1 #hide
+```
+
+Also, it would be interesting to check the simulation before the training to get an idea of the starting point of the network.
+
+```@example friction
+initial_sol = solve(prob, Rodas4(), saveat = sol_ref.t)
+```
+
+Now we plot it.
+
+```@example friction
+scatter(sol_ref, idxs = [model_true.y], label = "ground truth velocity")
+plot!(res_sol, idxs = [sys.friction.y], label = "velocity after training")
+plot!(initial_sol, idxs = [sys.friction.y], label = "velocity before training")
+```
+
+It matches the data well! Let's also check the predictions for the friction force and whether the network learnt the friction model or not.
+
+```@example friction
+scatter(sol_ref.t, friction.(first.(sol_ref.u)), label = "ground truth friction")
+plot!(res_sol.t, Fu .- first.(res_sol(res_sol.t, Val{1}).u),
+    label = "friction from neural network")
+```
+
+It learns the friction model well!

From 02041f1f9b50d52518fd0adc4f560a9746982da1 Mon Sep 17 00:00:00 2001
From: Sathvik Bhagavan <sathvik.bhagavan@juliahub.com>
Date: Thu, 11 Apr 2024 09:03:42 +0000
Subject: [PATCH 3/8] refactor: [temp] add RealInput/RealOutput which works for
 size equal to 1

---
 src/ModelingToolkitNeuralNets.jl | 25 ++++++++++++++++++++++---
 1 file changed, 22 insertions(+), 3 deletions(-)

diff --git a/src/ModelingToolkitNeuralNets.jl b/src/ModelingToolkitNeuralNets.jl
index b9346ba..3f03c26 100644
--- a/src/ModelingToolkitNeuralNets.jl
+++ b/src/ModelingToolkitNeuralNets.jl
@@ -1,6 +1,7 @@
 module ModelingToolkitNeuralNets
 
-using ModelingToolkit: @parameters, @named, ODESystem, t_nounits
+using ModelingToolkit: @parameters, @named, ODESystem, t_nounits, @connector, @variables,
+                       Equation
 using ModelingToolkitStandardLibrary.Blocks: RealInput, RealOutput
 using Symbolics: Symbolics, @register_array_symbolic, @wrapped
 using LuxCore: stateless_apply
@@ -12,6 +13,24 @@ export NeuralNetworkBlock, multi_layer_feed_forward
 
 include("utils.jl")
 
+@connector function RealInput2(; name, nin = 1, u_start = zeros(nin))
+    @variables u(t_nounits)[1:nin]=u_start [
+        input = true,
+        description = "Inner variable in RealInput $name"
+    ]
+    u = collect(u)
+    ODESystem(Equation[], t_nounits, [u...], []; name = name)
+end
+
+@connector function RealOutput2(; name, nout = 1, u_start = zeros(nout))
+    @variables u(t_nounits)[1:nout]=u_start [
+        output = true,
+        description = "Inner variable in RealOutput $name"
+    ]
+    u = collect(u)
+    ODESystem(Equation[], t_nounits, [u...], []; name = name)
+end
+
 """
     NeuralNetworkBlock(n_input = 1, n_output = 1;
         chain = multi_layer_feed_forward(n_input, n_output),
@@ -32,8 +51,8 @@ function NeuralNetworkBlock(n_input = 1,
     @parameters p[1:length(ca)] = Vector(ca)
     @parameters T::typeof(typeof(p))=typeof(p) [tunable = false]
 
-    @named input = RealInput(nin = n_input)
-    @named output = RealOutput(nout = n_output)
+    @named input = RealInput2(nin = n_input)
+    @named output = RealOutput2(nout = n_output)
 
     out = stateless_apply(chain, input.u, lazyconvert(typeof(ca), p))
 

From 578a56121f91f668bfe67ce01cc580476af62508 Mon Sep 17 00:00:00 2001
From: Sathvik Bhagavan <sathvik.bhagavan@juliahub.com>
Date: Fri, 19 Apr 2024 07:32:11 +0000
Subject: [PATCH 4/8] refactor: use RealInputArray and RealOutputArray

---
 src/ModelingToolkitNeuralNets.jl | 31 ++++++-------------------------
 1 file changed, 6 insertions(+), 25 deletions(-)

diff --git a/src/ModelingToolkitNeuralNets.jl b/src/ModelingToolkitNeuralNets.jl
index 3f03c26..fd55d09 100644
--- a/src/ModelingToolkitNeuralNets.jl
+++ b/src/ModelingToolkitNeuralNets.jl
@@ -1,8 +1,7 @@
 module ModelingToolkitNeuralNets
 
-using ModelingToolkit: @parameters, @named, ODESystem, t_nounits, @connector, @variables,
-                       Equation
-using ModelingToolkitStandardLibrary.Blocks: RealInput, RealOutput
+using ModelingToolkit: @parameters, @named, ODESystem, t_nounits
+using ModelingToolkitStandardLibrary.Blocks: RealInputArray, RealOutputArray
 using Symbolics: Symbolics, @register_array_symbolic, @wrapped
 using LuxCore: stateless_apply
 using Lux: Lux
@@ -13,24 +12,6 @@ export NeuralNetworkBlock, multi_layer_feed_forward
 
 include("utils.jl")
 
-@connector function RealInput2(; name, nin = 1, u_start = zeros(nin))
-    @variables u(t_nounits)[1:nin]=u_start [
-        input = true,
-        description = "Inner variable in RealInput $name"
-    ]
-    u = collect(u)
-    ODESystem(Equation[], t_nounits, [u...], []; name = name)
-end
-
-@connector function RealOutput2(; name, nout = 1, u_start = zeros(nout))
-    @variables u(t_nounits)[1:nout]=u_start [
-        output = true,
-        description = "Inner variable in RealOutput $name"
-    ]
-    u = collect(u)
-    ODESystem(Equation[], t_nounits, [u...], []; name = name)
-end
-
 """
     NeuralNetworkBlock(n_input = 1, n_output = 1;
         chain = multi_layer_feed_forward(n_input, n_output),
@@ -49,12 +30,12 @@ function NeuralNetworkBlock(n_input = 1,
     ca = ComponentArray{eltype}(init_params)
 
     @parameters p[1:length(ca)] = Vector(ca)
-    @parameters T::typeof(typeof(p))=typeof(p) [tunable = false]
+    @parameters T::typeof(typeof(ca))=typeof(ca) [tunable = false]
 
-    @named input = RealInput2(nin = n_input)
-    @named output = RealOutput2(nout = n_output)
+    @named input = RealInputArray(nin = n_input)
+    @named output = RealOutputArray(nout = n_output)
 
-    out = stateless_apply(chain, input.u, lazyconvert(typeof(ca), p))
+    out = stateless_apply(chain, input.u, lazyconvert(T, p))
 
     eqs = [output.u ~ out]
 

From aa625f86c010205a7d30ba3e6cec24abd8d5c56e Mon Sep 17 00:00:00 2001
From: Sathvik Bhagavan <sathvik.bhagavan@juliahub.com>
Date: Fri, 19 Apr 2024 07:32:58 +0000
Subject: [PATCH 5/8] test: update lv tutorial

---
 test/lotka_volterra.jl | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/test/lotka_volterra.jl b/test/lotka_volterra.jl
index 3879c64..6b6f7a9 100644
--- a/test/lotka_volterra.jl
+++ b/test/lotka_volterra.jl
@@ -16,8 +16,8 @@ function lotka_ude()
     @variables t x(t)=3.1 y(t)=1.5
     @parameters α=1.3 [tunable = false] δ=1.8 [tunable = false]
     Dt = ModelingToolkit.D_nounits
-    @named nn_in = RealInput(nin = 2)
-    @named nn_out = RealOutput(nout = 2)
+    @named nn_in = RealInputArray(nin = 2)
+    @named nn_out = RealOutputArray(nout = 2)
 
     eqs = [
         Dt(x) ~ α * x + nn_in.u[1],
@@ -50,7 +50,8 @@ eqs = [connect(model.nn_in, nn.output)
        connect(model.nn_out, nn.input)]
 
 ude_sys = complete(ODESystem(
-    eqs, ModelingToolkit.t_nounits, systems = [model, nn], name = :ude_sys))
+    eqs, ModelingToolkit.t_nounits, systems = [model, nn],
+    name = :ude_sys, defaults = [nn.input.u => [0.0, 0.0]]))
 
 sys = structural_simplify(ude_sys)
 

From e7e702e1eb2a4fcd4389b4fbde03d7700b8c7da4 Mon Sep 17 00:00:00 2001
From: Sathvik Bhagavan <sathvik.bhagavan@juliahub.com>
Date: Fri, 19 Apr 2024 08:09:35 +0000
Subject: [PATCH 6/8] docs: import Test

---
 docs/make.jl         | 2 ++
 docs/src/friction.md | 1 +
 2 files changed, 3 insertions(+)

diff --git a/docs/make.jl b/docs/make.jl
index 2393190..8575c56 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -1,5 +1,7 @@
 using ModelingToolkitNeuralNets
 using Documenter
+ENV["GKSwstype"] = "100"
+ENV["JULIA_DEBUG"] = "Documenter"
 
 cp("./docs/Manifest.toml", "./docs/src/assets/Manifest.toml", force = true)
 cp("./docs/Project.toml", "./docs/src/assets/Project.toml", force = true)
diff --git a/docs/src/friction.md b/docs/src/friction.md
index d05ca9a..842d6b7 100644
--- a/docs/src/friction.md
+++ b/docs/src/friction.md
@@ -21,6 +21,7 @@ using SymbolicIndexingInterface
 using StableRNGs
 using Lux
 using Plots
+using Test #hide
 ```
 
 ## Problem Setup

From 4bac4dcbd68c9ac8fbfd625ad42c319b40c4e3a1 Mon Sep 17 00:00:00 2001
From: Sathvik Bhagavan <sathvik.bhagavan@juliahub.com>
Date: Fri, 19 Apr 2024 08:09:57 +0000
Subject: [PATCH 7/8] build: bump MTKStdLib compat

---
 Project.toml      | 2 +-
 docs/Project.toml | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/Project.toml b/Project.toml
index 599404d..09c07f0 100644
--- a/Project.toml
+++ b/Project.toml
@@ -20,7 +20,7 @@ JET = "0.8"
 Lux = "0.5.32"
 LuxCore = "0.1.14"
 ModelingToolkit = "9.9"
-ModelingToolkitStandardLibrary = "2.6"
+ModelingToolkitStandardLibrary = "2.7"
 Optimization = "3.24"
 OptimizationOptimisers = "0.2.1"
 OrdinaryDiffEq = "6.74"
diff --git a/docs/Project.toml b/docs/Project.toml
index d886980..92889db 100644
--- a/docs/Project.toml
+++ b/docs/Project.toml
@@ -16,7 +16,7 @@ ModelingToolkitNeuralNets = "f162e290-f571-43a6-83d9-22ecc16da15f"
 Documenter = "1.3"
 Lux = "0.5.32"
 ModelingToolkit = "9.9"
-ModelingToolkitStandardLibrary = "2.6"
+ModelingToolkitStandardLibrary = "2.7"
 Optimization = "3.24"
 OptimizationOptimisers = "0.2.1"
 OrdinaryDiffEq = "6.74"

From b70f2cca81a24241b8d0d71c070f879e1f416c43 Mon Sep 17 00:00:00 2001
From: Sathvik Bhagavan <sathvik.bhagavan@juliahub.com>
Date: Fri, 19 Apr 2024 11:57:25 +0000
Subject: [PATCH 8/8] docs: fix link in index.md

---
 docs/src/index.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/docs/src/index.md b/docs/src/index.md
index f2945cc..a9ed1cf 100644
--- a/docs/src/index.md
+++ b/docs/src/index.md
@@ -4,7 +4,7 @@ CurrentModule = ModelingToolkitNeuralNets
 
 # ModelingToolkitNeuralNets
 
-Documentation for [ModelingToolkitNeuralNets](https://github.com/SebastianM-C/ModelingToolkitNeuralNets.jl).
+Documentation for [ModelingToolkitNeuralNets](https://github.com/SciML/ModelingToolkitNeuralNets.jl).
 
 ```@index
 ```