Introduction.jl

### A Pluto.jl notebook ###
# v0.19.42

#> [frontmatter]
#> title = "01 -- Introduction to Julia"
#> date = "2024-06-11"
#> license = "MIT"
#> tags = ["Differentiable Programming in Julia with Enzyme"]
#> 
#>     [[frontmatter.author]]
#>     name = "Valentin Churavy"
#>     affiliation = "Massachussetts Institute of Technology"
#>     url = "https://vchuravy.dev"

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ c4c4ca62-2d63-4876-9027-9ca7f0d6cb37
using PlutoUI

# ╔═╡ f6dcabfb-9f97-4be7-a729-eacd9402d4bf
begin 
	using CairoMakie
	# using WGLMakie
	# WGLMakie.activate!()
end

# ╔═╡ ac379b14-de26-43b0-bf29-df8764430683
using Optim

# ╔═╡ 05267988-ff9d-4e60-9a10-55864390fd22
using Enzyme

# ╔═╡ 1195dec8-2d9b-4bae-a2b6-c0835ad6e27b
begin
	using BenchmarkTools
	using Unitful
end

# ╔═╡ fcc7571b-5e57-4ec8-87f1-0639f78bab8d
md"""
# MPE24 -- Differentiable programming with Julia and Enzyme

### Agenda:
1. Introduction
2. Enzyme
3. Burgers equation and Checkpointing

Material: 
- https://github.com/dj4earth/mpe24
- https://dj4earth.github.io/MPE24/

### How to follow along
Go to https://julialang.org/downloads/

```
curl -fsSL https://install.julialang.org | sh
```

#### Using Git

```
git clone https://github.com/dj4earth/mpe24
cd mpe24
julia --project=.

julia> ]
pkg> up
julia> using Pluto
julia> Pluto.run()
```

## Using Pluto

```
julia
julia> ]
pkg> activate --temp
pkg> add Pluto
julia> using Pluto
julia> Pluto.run()
```

and then download the notebooks from https://dj4earth.github.io/MPE24/

"""

# ╔═╡ 9b96d09c-9c41-42ab-b2cf-a70669647bc8
md"""
## What's Julia? 🟢 🟣 🔴

Julia is a modern, dynamic, general-purpose, compiled programming language.
It's interactive ("like Python"), can be used in a REPL or notebooks, like Jupyter (it's the "Ju") or Pluto (this one🎈).
Julia has a runtime which includes a just-in-time (JIT) compiler and a garbage collector (GC), for automatic memory management.

Julia is mainly used for technical computing, and addresses a gap in the programming language landscape for numerical computing.


Main paradigm of Julia is multiple dispatch, what functions do depend on type and number of _all_ arguments.
"""

# ╔═╡ 9b770ce2-2285-42e5-9f5e-dcdbcf387768
md"""
## Why Julia? 😍

From "[My Target Audience](https://scientificcoder.com/my-target-audience)" by Matthijs Cox:

$(Resource("https://cdn.hashnode.com/res/hashnode/image/upload/v1681735971356/91b6e886-7ce1-41a3-9d9f-29b7b096e7f2.png"))
$(Resource("https://cdn.hashnode.com/res/hashnode/image/upload/v1681735992315/62fdd58f-4630-4120-8eb4-5238740543e8.png"))

* Explorable & Understandable
* Composability thanks to multiple dispatch (ask me more about this at the end!)
* User-defined types are as fast and compact as built-ins
* Code that is close to the mathematics
* No need to switch languages for performance...
* ...but you can still call C-like shared libraries with simple Foreign Function Interface (FFI) if you want to
* MIT licensed: free and open source
"""

# ╔═╡ 3902e31a-b2bf-4c7c-b44b-d5a5340ca574
md"""
## What is the 2 language problem?

You start out proto-typing in one language (high-level, dynamic), but performance forces you to switch to a different one (low-level, static).

- For convinience use a scripting language (Python, R, Matlab, ...)
- but do all the hard stuff in a systems language (C, C++, Fortran)

Pragmatic for many applications, but has drawbacks

- aren't the hard parts exactly where you need an easier language
- creates a social barrier -- a wall between users and developers
- "sandwich problem" -- layering of system & user code is expensive
- prohibits full stack optimisations
"""

# ╔═╡ 73f61e66-5217-4fac-9607-cf2dc3889a39
md"""
## A brief introduction to Julia
"""

# ╔═╡ 0e8168af-baad-444f-9f5e-e0c8559fc81b
function mandel(z)
	c = z
	maxiter = 80
	for n = 1:maxiter
	  if abs(z) > 2
		  return n-1
	  end
	  z = z^2 + c
	end
	return maxiter
end

# ╔═╡ 5cc72a26-0469-40cd-95c0-8b44a747197b
mandel(complex(.3, -.6))

# ╔═╡ 920c775c-be47-4302-8481-87cd3b873807
md"""

Real component: $(@bind x_real PlutoUI.Slider(-2:0.01:1)) 

Imaginary component: $(@bind x_img PlutoUI.Slider(-1:0.01:1))
"""

# ╔═╡ a83228cc-6281-4a08-9784-c40f35ebac3e
md"""

Resolution: $(@bind resolution PlutoUI.Slider([0.1, 0.01, 0.001], show_value=true))

Real shift: $(@bind real_shift PlutoUI.Slider(-3:resolution:3, show_value=true, default=0))

Img shift: $(@bind img_shift PlutoUI.Slider(-3:resolution:3, show_value=true, default=0))

Zoom: $(@bind zoom PlutoUI.Slider(1:100, show_value=true, default=1))
"""

# ╔═╡ 8cc83acf-dddf-43c0-b9dc-5e39e541583e
# evaluate the mandelbrot set over a complex plane
begin
	reals = ((-2:resolution:1) .+ real_shift) ./ zoom
	imgs = ((-1:resolution:1) .+ img_shift) ./ zoom

	plane = [complex(re, img) for (re, img) in Iterators.product(reals, imgs)]
end;

# ╔═╡ 3893e73e-d335-4c8b-96e2-6a8496623a53
heatmap(mandel.(plane))

# ╔═╡ 22fb5605-04fd-4f0a-b41a-6c9d65cd9425
md"""
### Rosenbrock function 🗻

Everyone's favourite example function for optimisation problems.

```math
f(x, y) = (1 - x) ^ 2 + 100 (y - x ^ 2) ^ 2
```

```julia
rosenbrock(x, y) = (1 - x) ^ 2 + 100 * (y - x ^ 2) ^ 2
```
"""

# ╔═╡ 9536e1ca-7d5f-47bb-8ecc-55f38e7df1a2
rosenbrock(x, y) = (1 - x) ^ 2 + 100 * (y - x ^ 2) ^ 2;

# ╔═╡ 92b71ff8-0dc5-481c-9d05-19d3db73cbcc
begin
	xs = -6:0.04:6
	ys = -2:0.04:5
	zs = [log10(rosenbrock(x, y)) for x in xs, y in ys]
end;

# ╔═╡ 81bd99c3-3e38-4fff-83af-37faa014429f
surface(xs, ys, zs, axis=(type=Axis3,))

# ╔═╡ efd6eb23-be46-437e-9c87-a6f770dc3613
contourf(xs, ys, zs)

# ╔═╡ 44b65bb0-cf1b-41c0-b792-01f5e5e9022d
let
	f(x) = rosenbrock(x...)
	sol = optimize(f, [0.0, 0.0])
	@show Optim.minimizer(sol)
	sol
end

# ╔═╡ 12d80cd2-542b-4844-8957-9337640a84c2
md"""
## Automatic differentiation 📈

In mathematics, contrary to integration, computing the derivative of an expression is a mechanical process.
Wouldn't it be nice to automatically compute the exact derivative of mathematical functions in your code?

Automatic differentiation (autodiff in short) is a set of techniques to automatically compute the derivative of a function in a computer program.

[Enzyme](https://enzyme.mit.edu/) is a source code transformation autodiff engine operating at the LLVM level: it analysis the LLVM IR of the source code and takes the derivative of all the instructions, based on the built-in differentiation rules.
The fact that it works at the LLVM level means that it's _mostly_ front-end-language-agnostic and can be used to take derivative of source code combining multiple languages or using parallelisation frameworks (["Scalable automatic differentiation of multiple parallel paradigms through compiler augmentation"](https://dl.acm.org/doi/abs/10.5555/3571885.3571964).
"""

# ╔═╡ 7553f962-90fc-4ec2-892f-f73b300e0850
md"""
Finding the valley is trivial, but converging to the global minimum is hard.

The Rosenbrock function is a polynomial function, easy to compute its gradient:

```math
\nabla{}f(x, y) =
\begin{bmatrix}
    400x^3 - 400xy + 2x - 2\\
    200 (y -x^2)
\end{bmatrix}
```

We can ask Enzyme to compute the partial derivative on the second argument:
"""

# ╔═╡ 6f87e98a-3767-4c5e-a24f-f9cb0698346b
∂f_∂x(f, x, y) = first(autodiff(Enzyme.Reverse, f, Active(x), Const(y)));

# ╔═╡ 8bde3cd7-32c5-4d21-ba13-34d63bc143b5
∂f_∂y(f, x, y) = first(autodiff(Enzyme.Reverse, f, Const(x), Active(y)));

# ╔═╡ 415df7a4-a127-4cb6-a7fc-132a83883474
∂f(f, x, y) = first(autodiff(Enzyme.Reverse, f, Active(x), Active(y)));

# ╔═╡ b85ec6d8-735f-49ab-b9d5-dccf76754335
∂f_∂x(rosenbrock, 1.0, 0.0)

# ╔═╡ 0676ecd3-41ff-43f5-865c-1681591d5fcf
∂f_∂x(rosenbrock, 0.0, 1.0)

# ╔═╡ b1c40c2a-6c6c-457a-ad27-a73b8bdffa78
∂f_∂y(rosenbrock, 0.0, 1.0)

# ╔═╡ 9d9c6b7f-a73b-4e39-812f-00d30ab3c201
∂f(rosenbrock, 0.0, 1.0)

# ╔═╡ fafcf585-4f4e-4f2d-89a1-8f410a683754
contourf(xs, ys, ((x,y)->∂f_∂x(rosenbrock, x, y)[1]).(xs, ys'))

# ╔═╡ 99f8f148-5c6c-436e-a2e7-a69ad3c1bdbe
contourf(xs, ys, ((x,y)->∂f_∂y(rosenbrock, x, y)[2]).(xs, ys'))

# ╔═╡ 8f9294c6-172e-41d9-83b8-93991a7a989a
with_terminal() do
Enzyme.Compiler.enzyme_code_llvm( 
	rosenbrock, Active, Tuple{Const{Float64}, Active{Float64}},
	debuginfo=:none)
end

# ╔═╡ ecc9581f-4efc-408c-832a-ac33b2392d55
let
	f(x) = rosenbrock(x[1], x[2])
	function g!(dx, x)
		fill!(dx, 0)
		autodiff(Enzyme.Reverse, f, Duplicated(x, dx))
		nothing
	end
	sol = optimize(f, g!, [0.0, 0.0])
	@show Optim.minimizer(sol)
	sol
end

# ╔═╡ 0b26e518-9fd5-46db-9a80-78c0a3fe0f62
md"""
## On Performance
"""

# ╔═╡ 37ba14ea-caf3-4d7f-81a3-a46e13c5042c
function sum(X::Vector{Float64})
    acc = 0::Int64
    for x in X
        (acc += x)::Float64
    end
    acc::Union{Int64, Float64}
end
    

# ╔═╡ b4e65db8-3256-440a-9723-b5360f054a7d
md"""
## Figuring out what is happening
The stages of the compiler

- `@code_lowered`
- `@code_typed` & `@code_warntype`
- `@code_llvm`
- `@code_native`

Where is a function defined `@which` & `@edit`
"""

# ╔═╡ 175a1bb0-926c-4aac-9da1-0c09ec1cce53

@bind N PlutoUI.Slider(1:9)

# ╔═╡ 61dbaace-d43a-4469-97f1-2760d9b5e059
data = rand(10^N);

# ╔═╡ 49c85664-208b-4c46-8407-44a5119888c7
@code_lowered sum(data)

# ╔═╡ 7a2b56c5-cf55-411f-bf38-409c3a507be5
md"""
## A simple example: counting
"""

# ╔═╡ 1dcfd086-7922-47be-b993-d22368decd68
function f(N)
    acc = 0
    for i in 1:N
        acc += 1
    end
    return acc
end

# ╔═╡ ba85941f-86dd-4c8f-b509-2fe79e4f4c2f
K = 100_000_000

# ╔═╡ 8e208f2d-69dd-4336-9a19-0914dd0182ee
result = @benchmark f($K)

# ╔═╡ 80135e68-6da5-4f97-a678-8d0589d25a9d
begin
	t = time(minimum(result)) * u"ns" # in ns
	pFreq = round(typeof(1u"PHz"), K/t)
end;

# ╔═╡ 135013ce-ee5a-46bd-8659-79e1fedcd702
@benchmark f($(Ref(10*K))[])

# ╔═╡ cc9732f4-fee6-4c31-be16-c4c74ef046bb
md"""
Let's explore what code we are **actually** running.

Using Julia's reflection macros we can see all of the stages of code-generation.
"""

# ╔═╡ e11d83d9-5f78-4f56-b78e-5a5fff8606d1
@code_lowered f(K)

# ╔═╡ 992a2eb2-ef18-49f1-993f-69c987859e5f
@code_typed optimize=false f(K)

# ╔═╡ e768df55-b2ea-4bbd-988b-08c403403882
@code_typed optimize=true f(K)

# ╔═╡ ac177383-f9a0-42ad-89bf-2bd1ec64145f
with_terminal() do
	@code_llvm optimize=false f(K)
end

# ╔═╡ 32c79d3b-1745-4f88-b479-f4b74f6a8249
with_terminal() do
	@code_llvm optimize=true f(K)
end

# ╔═╡ f3840a59-c0d3-47ff-ba77-371c7a9299ad
with_terminal() do
	@code_native f(K)
end

# ╔═╡ 2b0cf1c9-d00c-4fdf-b649-549a7e92d208
md"""
### Conclusion
LLVM realised that our loop:
```julia
for i in 1:N
  acc += 1
end
```

Just ended up being $acc = 1 * N$
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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Unitful = "1986cc42-f94f-5a68-af5c-568840ba703d"

[compat]
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CairoMakie = "~0.12.2"
Enzyme = "~0.12.0"
Optim = "~1.9.4"
PlutoUI = "~0.7.58"
Unitful = "~1.20.0"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6035850dcc70518ca32f012e46015b9beeda49d8"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+0"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "34d526d318358a859d7de23da945578e8e8727b7"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "d2d1a5c49fae4ba39983f63de6afcbea47194e85"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.6+0"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "47e45cd78224c53109495b3e324df0c37bb61fbe"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.11+0"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.15.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.isoband_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51b5eeb3f98367157a7a12a1fb0aa5328946c03c"
uuid = "9a68df92-36a6-505f-a73e-abb412b6bfb4"
version = "0.2.3+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1827acba325fdcdf1d2647fc8d5301dd9ba43a9d"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.9.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+1"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "d7015d2e18a5fd9a4f47de711837e980519781a4"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.43+1"

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"]
git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.3+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[deps.oneTBB_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "7d0ea0f4895ef2f5cb83645fa689e52cb55cf493"
uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e"
version = "2021.12.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"
"""

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