notebook.jl

### A Pluto.jl notebook ###
# v0.15.0

using Markdown
using InteractiveUtils

# ╔═╡ 1344246e-87ab-11eb-0d86-d523bcdf699a
begin
	using DiffEqFlux
	using Flux
	using Zygote
	using SparseArrays
	using LinearAlgebra
end

# ╔═╡ 07cd8bcd-803f-4ea2-acc1-70d2d58d2be0
md"# Standard example"

# ╔═╡ 7b738e20-2712-4349-93b7-f9b5c1ba5265
md"Take a small neural network, `NN`,"

# ╔═╡ 7955b232-0cc7-491d-8f3e-900f9030dcab
NN = FastChain(FastDense(2, 16, tanh), FastDense(16, 2, x->x^2)) # a NN

# ╔═╡ ad58b8aa-e8f1-4a64-b146-11a2393bab80
md"Assign some parameters `p` and an input vector randomly,"

# ╔═╡ c041026b-9043-4851-9b81-d4b75c117f80
p, u = Float64.(initial_params(NN)), rand(2)

# ╔═╡ e9a8b319-39c7-493f-aa74-edd29122047a
md"And we can easily get its derivatives with Zygote"

# ╔═╡ b481e279-fb5f-474a-aca5-45b1792e19b4
Zygote.jacobian(NN, u, p)

# ╔═╡ bece2b73-699a-4492-962a-006d1ab89e17
md"""
# With extra *prescribed* input variables

Now let's make this a bit more complicated.
Let's assume that we want a 3-element input `vars = [x, y, z]`, and 1-element output.
But for the input, we will keep `y` and `z` constant.
So we make `NN2` with a 3-element input and 1-element output:
"""

# ╔═╡ 8b67da12-d1bc-4c3c-88af-cd0dec448abd
NN2 = FastChain(FastDense(3, 16, tanh), FastDense(16, 1, x->x^2))

# ╔═╡ 6150b3c7-7b7b-425b-a5d1-bb62eee8c835
md"""
So we write a function `f` with single argument `x` by "wrapping" `NN2([x, y, z])`:
"""

# ╔═╡ 1cab1c36-494d-4b69-9aef-9ccb5c026e4e
begin
	y = 15.9
	z = 22.3
	f(x, p) = NN2([x, y, z], p)[1] # note I extract the first element here
end

# ╔═╡ 84450eb1-28da-4aaa-b876-f0f59b7d3936
md"""
And we can get its derivatives with `jacobian`:
"""

# ╔═╡ 2d858321-61f2-4390-ae7b-8ae43ba5f866
begin
	x = 0.24
	Zygote.jacobian(f, x, p)
end

# ╔═╡ 92ab5dec-2764-4a19-a997-39a1f37f96e7
md"""
# Multiple inputs from different locations

Now let's assume that we have values of `x` at many different locations, so we want to use the same function `f`, but with many different `x` values, provided as a vector `X`.
And then we will want to compute its Jacobian with respect to `X`.
"""

# ╔═╡ 652bb26e-a53c-48eb-9721-39cf29987345
F(X, p) = [f(x, p) for x in X] # almost like a broadcast, F(X) = f.(X)

# ╔═╡ d144ed58-0012-43ac-b2d7-193dd393388a
md"""
We can get the derivatives with the same approach:
"""

# ╔═╡ ff34fe18-be1f-4a51-9e7e-83a783e9cb21
begin
	X = rand(1000)
	Zygote.jacobian(F, X, p)
end

# ╔═╡ 251d673c-f6ff-42ee-bcdb-8d12dd5ef362
md"""
Notice that now the jacobian with respect to `x1vec` is a diagonal. 
This is simply the consequence of the structure here.
Indeed, `NN2funvec` is "local" in `x1vec`, i.e., its $i$th output only depends on the $i$th entry of `x1vec`.
Maybe we can exploit that.
Let's define some functions to test different approaches.
"""

# ╔═╡ 06bcda0d-cee8-4f2b-81a6-8a3850dec56c
begin
	A(X, p) = Zygote.jacobian(F, X, p)[1]
	B(X, p) = spdiagm([Zygote.jacobian(f, x, p)[1][1] for x in X])
end

# ╔═╡ 9bc37149-6f74-4e5d-af97-9d2bbe303549
A(X, p) # ~10s

# ╔═╡ ac95ea06-4adf-4b12-b8a4-a999421c1677
B(X, p) # ~0.15s

# ╔═╡ 57f0795a-d3d6-4534-b98c-45451d84168b
A(X, p) == Matrix(B(X, p)) #Check that they are the same

# ╔═╡ 6774973f-1369-4d2e-a7a4-80f6700186e7
md"""
So it seems getting the jacobian of `F` through `B` is much faster.
That's because we did not let Zygote try all the combinations of off diagonal indices that we already know would give `0`.
Instead, we dug up the underlying "scalar" function `f`, applied Zygote's derivatives element by element and constructed the diagonal Jacobian "by hand".

The **question is, is that the best way to go about this?**
Should I have constructed `F` differently?
Can I use this under the hood with some solver of DiffEqFlux and SciML in general?
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
DiffEqFlux = "aae7a2af-3d4f-5e19-a356-7da93b79d9d0"
Flux = "587475ba-b771-5e3f-ad9e-33799f191a9c"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"

[compat]
DiffEqFlux = "^1.34.1"
Flux = "^0.11.6"
Zygote = "^0.6.4"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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[[StableRNGs]]
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uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

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[[TOML]]
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[[Tar]]
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[[TerminalLoggers]]
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[[ZygoteRules]]
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[[p7zip_jll]]
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"""

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