mass_spring_intro.jl

### A Pluto.jl notebook ###
# v0.19.26

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

# ╔═╡ 12724474-bc6a-463c-88aa-601f3406d545
using ModelingToolkit, Latexify, OrdinaryDiffEq, Plots, PlutoUI, ForwardDiff, LinearAlgebra, MinimallyDisruptiveCurves

# ╔═╡ e8175462-9bbf-11ec-3a58-0f114bb6cd58
md"""
# Introductory example

### Analysing a forced harmonic oscillator

Here we are going to analyse to death a **really simple** example so you can explore the capabilities of MinimallyDisruptiveCurves.jl.

Points that will be covered in this tutorial:

- Build a barebones, differentiable cost function on the output of a differential equation (the oscillator).
- Evolve a (two-sided) minimally disruptive curve using a cost function and an initial set of parameters.
- Consider options and implications when choosing an initial direction of motion for the minimally disruptive curve.
- Investigate curves that minimise disruption over multiple model conditions (through summation of cost functions)

"""

# ╔═╡ e791cb5f-1d4e-4b4e-987d-fd280cdfb868
md"""
### Background

A harmonic oscillator is the simplest differential equation model on which we can demonstrate how to find a minimally disruptive curve. The differential equation is provided below:

$$m\ddot{x}(t) + c \dot{x}(t) + kx(t) = F(t).$$

Physically, this models a mass bouncing up and down on a spring, while being forced by an input $F(t)$. Here $[m,c,k]$ are the mass, damping-coefficient, and spring constant, respectively. $[x(t), \dot{x}(t), \ddot{x}(t)]$ are the position, velocity, and acceleration at time $t$.

Let's first consider the case where $F(t) = 0$. So we have
$$m\ddot{x}(t) + c \dot{x}(t) + kx(t) = 0.$$

We can immediately see that the equation doesn't change if we divide through by $m$:

$$\ddot{x}(t) + \frac{c}{m} \dot{x}(t) + \frac{k}{m}x(t) = 0.$$

So for the no-input case, we can see that the model is **invariant** to changes that preserve $\frac{c}{m}$ and $\frac{k}{m}$. 

Let's take $\theta$ as the vector of parameters $[m,c,k]$, and $\theta^*$ as a 'nominal' parameter vector. Evolving a minimally disruptive curve equates to generating a parameterised curve $\theta(s)$ that preserves $\frac{c}{m} = \frac{c^*}{m^*}$ and $\frac{k}{m} = \frac{k^*}{m^*}$, since any point on this curve represents a parameter set which gives identical model behaviour as $\theta^*$. 
"""

# ╔═╡ 7ba2b752-c939-4e06-afa0-e0385b121282
md"""
### Building the model
It's below, built using ModelingToolkit.jl
"""

# ╔═╡ e4c7b17f-0399-43c1-9f1a-4a49538280f5
md"""
Mass $$m$$ is $(@bind mval Scrubbable(4.)) 

Damping $$c$$ is $(@bind cval Scrubbable(1.)) 

Spring constant $$k$$ is $(@bind kval Scrubbable(2.)) 

End time is $(@bind tend Scrubbable(100))
"""

# ╔═╡ 14ead89c-7958-4320-98fb-a4ad6a2b29a2
θ₀ = [kval, cval, mval];

# ╔═╡ 6b3fb59a-69ab-4f26-85a4-44c9140c746b
function MassSpringOscillator(input) 
    @variables t
    @parameters k, c, m
    D = Differential(t)
    @variables position(t) velocity(t)

    eqs = [D(position) ~ velocity,
        D(velocity) ~ (-1 / m) * (c * velocity + k * position - input(t))
  ]

    ps = [k,c,m] .=>  θ₀
    ics = [position, velocity] .=> [1.,0.]
    od = ODESystem(eqs, t, first.(ics), first.(ps); name=:mass_spring)
    tspan = (0., tend)
  # prob = ODEProblem(od, ics, tspan, ps)
    return od, ics, tspan, ps
end

# ╔═╡ e2c3c85f-3652-42de-8c3c-8df5290a6180
od,ic,tspan, θs = MassSpringOscillator(t->0.);

# ╔═╡ d393d540-b31c-4e01-a288-b665ef30726a
latexify(od)

# ╔═╡ 48d35dce-fbc7-4eea-9c9c-31dbc0a5bd71
prob = ODEProblem(od,ic,tspan, θs);

# ╔═╡ 756dd82e-bf02-4d09-9ff0-c5ce7a79fefd
nom_sol = solve(prob,Tsit5())

# ╔═╡ 162267ad-e874-4e3a-8cc6-0bd317a07386
plot(nom_sol)

# ╔═╡ 0d5a463f-f621-4460-b385-6550449874e7
md"""
### Building a cost function

First, some background. A cost function is a mapping of the form:

> parameters -> how badly the model behaves

Now you have to choose what your measure of *badly behaved* constitutes. In this example, we will choose something simple: the squared error between the solution and a nominal solution `nom_sol` on a set of timepoints. Something like:

> parameters -> model -> sol -> |sol - nom_sol|^2

Of course, you could use real, noisy data instead of the synthetic `nom_sol`, or an entirely different cost function.

You can make a cost function however you like. You could even code it in a different language, if your model is built in e.g. Python. MinimallyDisruptiveCurves.jl only cares that the cost function has two methods:
```julia
function cost(params)
   ...
    return cost
end
```

```julia
function cost(params, gradient_holder)
   ...
    gradient_holder[:] = ...
    # MUTATE gradient_holder so gradient_holder = d(cost)/d(params
    return cost
end
```

We will just use ForwardDiff to create a barebones version of the second method with the gradient. You can also use DiffEqFlux.jl or DiffEqSensitivity.jl to build your own gradients that might be more efficient or accurate.
"""

# ╔═╡ c4154cc3-c543-4355-b343-a7ec79d03988
sol_cost(sol) = sum(abs2, nom_sol - sol)

# ╔═╡ 9858a01d-3bb7-4735-ae1c-de19c884b4fc
function cost(p)
	_prob = remake(prob, p=p)
	sol = solve(_prob, Tsit5(), saveat = nom_sol.t)
	sol_cost(sol)
end

# ╔═╡ 529aa903-dd50-4b84-ba9b-028d20135c04
function cost(p,g)
	g[:] = ForwardDiff.gradient(cost, p)
	return cost(p)
end

# ╔═╡ 5a4a31cc-5bdf-4610-bdc4-713735f8395b
md"""
Note that the gradient calculation we have coded is **not that accurate**. For instance:

- ```cost(θ₀, g)``` should mutate the gradient to `g=[0., 0., 0.]`

θ₀ is the nominal set of parameters, so the solution at θ₀ is identical to the nominal solution, and thus the cost is at a minimum (it is zero). Gradients at minima are always zero. However, when we try this:

"""

# ╔═╡ 8e3acf8c-f32c-422c-9a01-c7881a6d41a6
begin
	g = [0., 0., 0.] #template
	cost(θ₀ , g)
	g
end

# ╔═╡ 638db2d2-cf26-46f8-a77e-36f76627263c
md"""
...but don't worry! Minimally disruptive curves can evolve accurately even with inaccurate gradient calculations, as we'll see."""

# ╔═╡ b13b68e7-9540-4bb2-b26e-d6efd86608a3
H₀ = ForwardDiff.hessian(cost,θ₀)

# ╔═╡ 348a2244-9ef2-4d3c-8af6-5ab4c1ef427e
md"""
### Initial direction for the MD curve

- We'll consider this more fully in different examples. For now, we'll be lazy and use the least minimally disruptive direction, to infinitesimal order, as our starting direction.
- How? For a small perturbation:
$$C[\theta + \delta \theta] = C[\theta] + \delta \theta^T\nabla_{\theta}C[\theta] + \frac{1}{2} \delta \theta^T\nabla^2_{\theta}C[\theta] \delta \theta,$$
from Taylor's theorem.
- ``\nabla_{\theta} C[\theta] = 0`` at the minimum, which gets rid of one term.
- ``\nabla^2_{\theta}C[\theta] \succeq 0`` (is positive semidefinite).
So we just need to find a $\delta \theta$ corresponding to eigenvector of ``\nabla^2 C[\theta]`` with smallest eigenvalue. 
"""

# ╔═╡ ac4833a0-194e-46f1-97af-7245e6a29f77
δθ₀ = eigen(H₀).vectors[:,1]

# ╔═╡ d0b4f6f5-7048-4579-af33-5aed1e161acc
md"""

### Summary thus far
- *Everything up to this point was **independent** of the MinimallyDisruptiveCurves.jl package*

- We have built a differentiable cost function
- We demonstrated **one way** of finding an initial curve direction in parameter space to which the cost function is insensitive. Feel free to play around with this initial direction.
- Now we want to use the package to generate curves on the parameter space that are minimally disruptive (with respect to this cost function), and starting in our chosen initial direction.

### Evolving a minimally disruptive curve



First let's set the momentum. This is **the one heuristic hyperparameter** of the algorithm.
- Fortunately, the algorithm is (as far as I've seen) pretty robust to the choice of momentum.
- Once $C(\theta) \geq momentum$, the curve stops. So a heuristic is to set the momentum to the highest number you are willing to allow the cost to reach.
- A larger momentum (marginally) speeds up curve evolution, and makes the trajectory (marginally) less curvy

Momentum is $(@bind mom Scrubbable(1.)) 

We also want to set the maximum curve length before automatic termination:

Span is (0. $(@bind endspan Scrubbable(100.)) )

Now we have everything we need to specify an MDCProblem:
"""

# ╔═╡ 1ec6c301-b90c-4030-a853-7db1b1d8dabf
span = (0., endspan);

# ╔═╡ d7b29969-fbb5-4fca-a78e-b3dd4cc058e0
eprob = MDCProblem(cost, θ₀, δθ₀, mom, span);

# ╔═╡ d2440064-c8c1-4627-99ab-b27f6b83c909
md"""
MinimallyDisruptiveCurves.jl has some **custom (optional) callback options** during curve evolution. These can be inserted using the `mdc_callback` keyword argument. To demonstrate, let's (**arbitrarily**):
- Constrain parameters 1 and 3 to lie in the interval $[-1000, 1000]$
- Provide online output of distance as the curve evolves, on the range 0.1:1:10
- Show the numerical residual on the derivative of the momentum wrt to the curve direction (which should be zero), on the range 2.3:4:10
You can see the full list of solving options by looking at src/evolve_options.jl in the source code.
"""

# ╔═╡ bc36827b-b968-41fc-8302-58a3663fbf3a
cb = [
    Verbose([CurveDistance(0.1:1:10), HamiltonianResidual(2.3:4:10)]),
    ParameterBounds([1,3], [-1000.,-1000.], [1000.,1000.])
    ]

# ╔═╡ 44503314-f126-4903-9ef5-010e2a70f053
md"""
We also have a special keyword argument: `momentum_tol = 1e-3`. This is the numerical residual at which momentum of the particle tracing the curve is recalculated. **Don't use this keyword argument in general, I've just put it here for demonstration**. The exception is if the curve is sawtoothing back on itself, in which case set `momentum_tol = NaN`.

You can also use **any callback compatible with DifferentialEquations.jl and DiffEqCallbacks.jl**. These can be inserted using the `callback` keyword argument.

Finally, the `solve` function that evolves MD curves inherits all keyword arguments compatible with `DifferentialEquations.solve`. For instance, we have specified `dtmin` below.
"""

# ╔═╡ 4430dcd3-474c-42c6-a4c9-9b41b45fd17a
@time mdc = evolve(eprob, Tsit5; mdc_callback = cb, callback=nothing, dtmin=0.01, momentum_tol = NaN);

# ╔═╡ 2dfebbff-3d71-48e0-883b-020e13f38c14
md"""
### MD curve analysis

In different tutorials we will put more emphasis on post-curve analysis (both conceptually and by demonstrating helper functions in the codebase)

For now, let's just plot the curve:
"""

# ╔═╡ 332dec4f-c942-403a-b86e-d7829e6810ae
cc = cost_trajectory(mdc, distances(mdc));

# ╔═╡ ae4b44cb-5ce0-49cc-9e4b-3067fd03afae
p1= plot(mdc; idxs=[1,2,3], pnames = first.(θs), what = :trajectory);

# ╔═╡ 2ada1933-4cb2-43fc-8569-dc33a53340c0
p2 = plot(distances(mdc), log.(cc), ylabel = "log(cost)", xlabel = "distance", title = "cost over MD curve");

# ╔═╡ a8dc1660-8977-4ad9-88e0-ba29d01eec0b
p3 = plot(mdc; idxs=[1,2,3], pnames = first.(θs), what = :final_changes);

# ╔═╡ 48f8da36-97f1-498f-b585-fe6a32f801cb
p = plot(p1,p2,p3, layout = (3,1))

# ╔═╡ 3e99fb45-cbac-4a4d-a608-ecc54605e410
md"""
Notice that the cost stays at numerical zero ($\leq exp(-18)$). As we predicted, the ratios $\frac{c}{m}$ and $\frac{k}{m}$ are preserved over the curve. 

Notice the numerical accuracy isn't shabby, given the degree of numerical error in calculations of the cost gradient!:
"""

# ╔═╡ bc1fcf93-7877-433c-86d7-0e21d2925b51
ratios = θ -> [θ[2]/θ[1] ,θ[3]/θ[1]]

# ╔═╡ 865fe978-0f4e-4fca-b9e4-382032996f4b
ratios(θ₀)

# ╔═╡ 806c8f47-d66d-4755-960a-0201e15cd940
ratios(mdc(100)[:states])

# ╔═╡ d8bb862b-548b-4454-bc23-c2aaf5d5368e
md"""
### Summing different costs

Recall from the preliminary maths that the above minimally disruptive curve only holds in the unforced case, when $F(t) = 0$.

Let's now try to find a minimally disruptive curve that holds under **two** conditions:
- the unforced case
- a sinusoidal forcing term
So points on the minimally disruptive curve will correspond to parameters that minimally disrupt the collective behaviours of the nominal system in both of these cases.

What do you imagine will happen? For nonzero $F(t)$ it looks clear that all parameters in

$$m\ddot{x}(t) + c \dot{x}(t) + kx(t) = F(t).$$
will affect behaviour.

So any curve on parameter space will be disruptive, to some extent. What is the least disruptive curve? This is no longer an analytically tractable problem like before, when $F(t) = 0$ held.

1. Let's build and plot the forced system:
"""

# ╔═╡ 29aa55b2-8d11-4bf0-afbe-8c381fcbc291
begin
	od2,_ , _ , _ = MassSpringOscillator(sin);
	prob2 = ODEProblem(od2,ic,tspan,θs)
	nom_sol2 = solve(prob2,Tsit5())
	plot(nom_sol2)
end

# ╔═╡ 2de1b351-cd94-4554-80ca-50f2516d2be2
md"""
2. Let's build a cost function that reflects discrepancy from the nominal trajectory above, for the forced system.
"""

# ╔═╡ bc17b9c0-c837-4bb2-8191-963b00feb32e
sol_cost2(sol) = sum(abs2, nom_sol2 - sol)

# ╔═╡ 6b672051-6192-443e-a2f7-e886a03b4861
function cost2(p)
	_prob2 = remake(prob2, p=p)
	sol2 = solve(_prob2, Tsit5(), saveat = nom_sol2.t)
	sol_cost2(sol2)
end

# ╔═╡ 7e7d2cb1-f58d-45ee-b650-0a62c0e70689
function cost2(p,g)
	g[:] = ForwardDiff.gradient(cost2, p)
	return cost2(p)
end

# ╔═╡ 77732289-c2bd-490f-9313-32c4c2b6e298
md"""
... and sum the costs!
"""

# ╔═╡ bfc1381e-17cd-4f89-bd35-a72e8c69948e
summed_cost = sum_losses([cost, cost2], θ₀);

# ╔═╡ 4dee52c6-e683-45bd-964d-f43f265601f1
md"""
- Note that above we summed the two cost functions, using `sum_losses`. This works for **any** compatible cost functions. Doesn't matter how they are generated individually.

- If you restart julia with multiple threads, than evaluating each subcomponent of the summed cost runs on a separate thread, increasing speed.

Again, let's find the direction of minimal sensitivity for the initial curve direction, and run the curve:
"""

# ╔═╡ 4c31745b-6e98-4a56-8925-2fc8c0379dab
begin
	H₁ = ForwardDiff.hessian(summed_cost,θ₀)
	δθ₁ = eigen(H₁).vectors[:,1];
end

# ╔═╡ 36b6da5e-4576-4704-8cd8-5a960c367b8d
begin
	mom2 = 10.
	span2 = (-5.,15.)
	eprob2 = MDCProblem(summed_cost, θ₀, δθ₁, mom2, span2);
	cb2 = [      Verbose([CurveDistance(0.1:10:99)]),
	            ParameterBounds([1,2,3], [0.,0.,0.], [1000.,1000.,1000.])
	        ]
end

# ╔═╡ 2540e729-9a04-45db-8570-ac2f310ac869
@time mdc2 = evolve(eprob2, Tsit5; mdc_callback=cb2);

# ╔═╡ 29c0211b-965f-4dc8-8310-5e618a22219d
begin
	plot(mdc2; idxs=[1,2,3], pnames = first.(θs), what = :trajectory)
end

# ╔═╡ fc7deb80-bfe6-4d3c-8eb5-92cf116668a1
md"""
The curve is looking for directions that keep the cost low, but it cant find them so it keeps doubling back on itself. Note that the cost at the point it doubles back (see below) is just below the momentum level.
"""

# ╔═╡ 8c7ae698-b517-4978-a109-188be87ee90d
begin
	summed_cost(mdc2(11.0)[:states])
end

# ╔═╡ dd76e380-f5eb-4d6c-8fb8-64f7d2ccf919
begin
	pp = mdc2(11.)[:states]
	mprob = remake(prob2, p = pp)
	msol = solve(mprob, Tsit5(), saveat=nom_sol2.t)
	plot(msol, linewidth=3)
	plot!(nom_sol2)
end

# ╔═╡ 180828e8-60b7-4e4d-bf06-f6ba04c06358
md"""
## Interpretation

There are no directions, mathematically that give the same response to a sinusoidal forcing term. So the MDC finds a direction that maintains the phase and frequency, while decreasing the amplitude from increased mass. 
"""

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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.ZygoteRules]]
deps = ["ChainRulesCore", "MacroTools"]
git-tree-sha1 = "977aed5d006b840e2e40c0b48984f7463109046d"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.3"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+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 = "5.7.0+0"

[[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", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+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 = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[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"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

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