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November 8, 2021 14:32
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SDE Demo
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| ### A Pluto.jl notebook ### | |
| # v0.16.1 | |
| 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 el = $(esc(element)) | |
| global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing | |
| el | |
| end | |
| end | |
| # ╔═╡ 8370af88-403c-11ec-3411-d7f78b38c809 | |
| using Plots | |
| # ╔═╡ c1fe5bef-707c-4b43-a4b2-ba42036778a5 | |
| using Distributions | |
| # ╔═╡ 460d695a-c1b2-4a02-bb79-5b85c26ebef2 | |
| using DifferentialEquations | |
| # ╔═╡ b55cb9bb-3fef-4267-b2dc-b3f953a2915f | |
| using PlutoUI | |
| # ╔═╡ 314f6a4d-3ffa-4721-8aad-b33400561bbc | |
| using NearestNeighbors | |
| # ╔═╡ 1726d0ec-d9c5-4a1a-88b4-67cb791938dd | |
| using LinearAlgebra | |
| # ╔═╡ 79fc6597-4a69-4c39-bd50-99b90c8e113d | |
| using ReverseDiff | |
| # ╔═╡ 6dd7ca98-a75a-4c29-a72d-d7b2861dcff8 | |
| using Statistics | |
| # ╔═╡ 7b9439f2-ef9e-431c-8bbd-d30552314323 | |
| using Base.Iterators: flatten, product | |
| # ╔═╡ a52ed8bf-629b-4a70-b686-04dbbb95f321 | |
| id(x) = x | |
| # ╔═╡ ad82a0f2-8796-456c-a65d-0a90f1849635 | |
| md" | |
| # Swiss-Roll Distribution | |
| " | |
| # ╔═╡ 9a698ae0-5636-486f-8adf-ef8db52ce375 | |
| N = 300 | |
| # ╔═╡ 767d5721-6564-494d-b4d5-f79f8031e196 | |
| swissify(x) = [x .* sin.(x) x .* cos.(x)] | |
| # ╔═╡ 0626a286-d713-489a-9b4a-366e542a2499 | |
| z = rand(Uniform(5, 15), N) | |
| # ╔═╡ 2c054894-f642-4537-9236-6b2db6b23755 | |
| x = swissify(z) | |
| # ╔═╡ 3aab31f4-6f9d-473f-a498-d864590b2c71 | |
| swissplt(x) = scatter(x[:, 1], x[:, 2], zcolor=z, legend = false, lims=[-20, 20], size=(600, 600)) | |
| # ╔═╡ 1598cb2f-a90d-40ed-8d16-e8a5ec5eb754 | |
| swissplt(x) | |
| # ╔═╡ a5db8ef0-8fcb-46fe-b86d-b52b2b4149ee | |
| md" | |
| # SDEs in the Wild | |
| " | |
| # ╔═╡ e05b69d3-db65-4f58-aa30-51bc569f8213 | |
| md" | |
| ## Brownian SDE | |
| $dx = dw$ | |
| " | |
| # ╔═╡ c53717cb-a1a4-41bc-8e3e-70effaca96b9 | |
| T = 20. | |
| # ╔═╡ dec9a17b-69dc-4b08-b1b3-0715fcd2bd53 | |
| function brownian_sde() | |
| f(x,p,t) = zero(x) | |
| g(x,p,t) = fill(1, size(x)) | |
| SDEProblem(f, g, x, (0.0, T)), id | |
| end | |
| # ╔═╡ 8fd26590-7b80-4dd3-b949-b13edaffd1f2 | |
| md" | |
| ## Variance-Preserving SDE | |
| $dx = -\frac{1}{2} \beta_t x \, dt + \sqrt{\beta_t} \, dw$ | |
| " | |
| # ╔═╡ 2e24a242-fcc4-4010-ac57-fc55a043bfe0 | |
| function vp_sde() | |
| β₀= .1 | |
| β₁ = 20 | |
| β(t) = β₀ + t * (β₁ - β₀) | |
| f(x,p,t) = -0.5 * β(t) * x | |
| g(x,p,t) = sqrt(β(t)) | |
| SDEProblem(f, g, x ./ 4, (0.0, 1.0)), x -> x .* 4 | |
| end | |
| # ╔═╡ e586040e-de82-4786-9086-3ef45623d2c4 | |
| md" | |
| ## Variance-Exploding SDE | |
| $dx = \sigma_0 \left(\frac{\sigma_1}{\sigma_0}\right)^t \sqrt{2 \log \frac{\sigma_1}{\sigma_0}} \, dw$ | |
| " | |
| # ╔═╡ 83144131-b88c-4356-b8fb-d481350d00bc | |
| function ve_sde() | |
| σ₀= .05 | |
| σ₁ = 20 | |
| f(x,p,t) = zero(x) | |
| g(x,p,t) = σ₀ * (σ₁/σ₀)^t * sqrt(2 * log(σ₁/σ₀)) | |
| SDEProblem(f, g, x, (0.0, 1.0)), id | |
| end | |
| # ╔═╡ 50a9d2da-6f80-4811-9425-5b3aa7533c05 | |
| md" | |
| # Solving an SDE | |
| The Euler-Maruyama method can be used to sample solutions to SDEs of the form | |
| $dx = f(x, t) \, dt + g(x, t) \, dw$ | |
| " | |
| # ╔═╡ b5d81f94-c82d-4b5f-acf7-1c4c0ea4d969 | |
| sdes = Dict(["vp" => vp_sde, "ve" => ve_sde, "brownian" => brownian_sde]) | |
| # ╔═╡ cd2a7836-9e60-44ac-92df-3b682c3f4a1b | |
| dt = 1//2^5 | |
| # ╔═╡ c531667e-7b86-40b3-ba6a-a95c502c706b | |
| @bind sde_name Select(["brownian", "ve", "vp"]) | |
| # ╔═╡ 6e0dc15d-9d55-4c07-b7a1-a90377456b0a | |
| demo_sde, map_from = sdes[sde_name](); | |
| # ╔═╡ e6962b93-6327-4215-b02b-5853c4f35272 | |
| demo_xs = solve(demo_sde, EM(), dt=dt); | |
| # ╔═╡ 5c60b950-191d-480d-80b7-1d5edc57c6e4 | |
| @bind t Slider(demo_sde.tspan[1]:dt:demo_sde.tspan[2]) | |
| # ╔═╡ 7b344acf-55ed-4ad0-899d-b8660714a95a | |
| swissplt(map_from(demo_xs(t))) | |
| # ╔═╡ f09981d6-0c5e-40b0-bbd8-45d97f359b87 | |
| md" | |
| # Reverse SDEs | |
| $dx = \left( f(x, t) - g^2(t) \nabla_x \log p_t(x) \right) \, dt + g(t) \, dw$ | |
| " | |
| # ╔═╡ 5bc246a6-474d-4b4a-a509-390e8eba3ca6 | |
| md" | |
| ## Reverse Brownian SDE | |
| $dx = - \nabla_x \log p_t(x) \, dt + dw$ | |
| " | |
| # ╔═╡ 65ffa262-0100-454e-8c86-9b0f6b99e11a | |
| md" | |
| ## Calculating $p(x)$ and $\nabla_x \log p(x)$ | |
| " | |
| # ╔═╡ c48b8a99-1ac1-4a22-bc8d-68c733ce7966 | |
| xs = solve(brownian_sde()[1], EM(), dt=dt); | |
| # ╔═╡ b1234b1a-a150-4efe-8147-989b5e7d8dc6 | |
| begin | |
| nns = KDTree(x') | |
| nearest(v) = x[nn(nns, v)[1], :] | |
| end | |
| # ╔═╡ 1ee68e6c-015d-47e5-8d6a-7d056c2842d1 | |
| function p(x, t) | |
| x̂ = nearest(ReverseDiff.value(x)) | |
| t = max(t, 0.01) | |
| pdf(MvNormal([0, 0], [t 0; 0 t]), x .- x̂) | |
| end | |
| # ╔═╡ 8e58f0a8-2a23-4699-bb22-a7063ed5300a | |
| score(x, t) = ReverseDiff.gradient(x -> log(p(x, t)), x) | |
| # ╔═╡ 15203992-224b-4e82-ba84-cd53667f8757 | |
| @bind p_t Slider(0.2:dt:T) | |
| # ╔═╡ d64b9ad6-cde1-4847-82c4-dae9dbe770e2 | |
| begin | |
| contour(-20:20, -20:20, (a, b) -> p([a, b], p_t), size=(600, 600), lims=(-20, 20)) | |
| grid = product(-20:1.5:20, -20:1.5:20) | |
| x_coords = reshape(map(a -> a[1], grid), :) | |
| y_coords = reshape(map(a -> a[2], grid), :) | |
| arrows = reshape(collect(flatten(map(a -> score([a...], p_t), grid))), (2, :)) | |
| quiver!(x_coords, y_coords, quiver=(arrows[1, :], arrows[2, :]), c=:teal) | |
| end | |
| # ╔═╡ 23ca0b4b-d23d-486a-98a3-0be7fb556427 | |
| md" | |
| ## Solving the Reverse SDE | |
| " | |
| # ╔═╡ 8c09dabe-ab60-484e-9ef7-170825173c95 | |
| function reverse_brownian_sde() | |
| x₀ = randn(300, 2) .* std(xs(T)) | |
| f(x,p,t) = mapslices(x -> -score(x, t), x, dims=[2]) | |
| g(x,p,t) = fill(1, size(x)) | |
| SDEProblem(f, g, x₀, (T, 0.0)) | |
| end | |
| # ╔═╡ 7c87b677-215a-46aa-9039-c713d1401270 | |
| x̂s = solve(reverse_brownian_sde(), EM(), dt=-dt); | |
| # ╔═╡ 2e27d753-9885-4ada-8119-3a6aba8833fc | |
| @bind sample_t Slider(0:dt:T) | |
| # ╔═╡ 1f54b672-684c-4c21-878b-3807de6ca264 | |
| swissplt(x̂s(T - sample_t)) | |
| # ╔═╡ 00000000-0000-0000-0000-000000000001 | |
| PLUTO_PROJECT_TOML_CONTENTS = """ | |
| [deps] | |
| DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa" | |
| Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" | |
| LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" | |
| NearestNeighbors = "b8a86587-4115-5ab1-83bc-aa920d37bbce" | |
| Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" | |
| PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" | |
| ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267" | |
| Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" | |
| [compat] | |
| DifferentialEquations = "~6.19.0" | |
| Distributions = "~0.25.24" | |
| NearestNeighbors = "~0.4.9" | |
| Plots = "~1.23.5" | |
| PlutoUI = "~0.7.18" | |
| ReverseDiff = "~1.10.0" | |
| """ | |
| # ╔═╡ 00000000-0000-0000-0000-000000000002 | |
| PLUTO_MANIFEST_TOML_CONTENTS = """ | |
| # This file is machine-generated - editing it directly is not advised | |
| [[AbstractPlutoDingetjes]] | |
| deps = ["Pkg"] | |
| git-tree-sha1 = "0ec322186e078db08ea3e7da5b8b2885c099b393" | |
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| version = "1.1.0" | |
| [[AbstractTrees]] | |
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| [[Adapt]] | |
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| [[ArgTools]] | |
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| [[ArnoldiMethod]] | |
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| git-tree-sha1 = "f87e559f87a45bece9c9ed97458d3afe98b1ebb9" | |
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| [[ArrayInterface]] | |
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| [[ArrayLayouts]] | |
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| [[Artifacts]] | |
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| [[BandedMatrices]] | |
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| [[Base64]] | |
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| [[BitTwiddlingConvenienceFunctions]] | |
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| [[BoundaryValueDiffEq]] | |
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| [[Bzip2_jll]] | |
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| [[CSTParser]] | |
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| [[Colors]] | |
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| [[ConstructionBase]] | |
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| [[Contour]] | |
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| uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e" | |
| version = "1.3.0" | |
| [[IterativeSolvers]] | |
| deps = ["LinearAlgebra", "Printf", "Random", "RecipesBase", "SparseArrays"] | |
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| uuid = "42fd0dbc-a981-5370-80f2-aaf504508153" | |
| version = "0.9.2" | |
| [[IteratorInterfaceExtensions]] | |
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| uuid = "82899510-4779-5014-852e-03e436cf321d" | |
| version = "1.0.0" | |
| [[JLLWrappers]] | |
| deps = ["Preferences"] | |
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| uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210" | |
| version = "1.3.0" | |
| [[JSON]] | |
| deps = ["Dates", "Mmap", "Parsers", "Unicode"] | |
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| uuid = "682c06a0-de6a-54ab-a142-c8b1cf79cde6" | |
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| [[JpegTurbo_jll]] | |
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| uuid = "aacddb02-875f-59d6-b918-886e6ef4fbf8" | |
| version = "2.1.0+0" | |
| [[JuliaFormatter]] | |
| deps = ["CSTParser", "CommonMark", "DataStructures", "Pkg", "Tokenize"] | |
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| [[LAME_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "c1c5ebd0-6772-5130-a774-d5fcae4a789d" | |
| version = "3.100.1+0" | |
| [[LZO_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac" | |
| version = "2.10.1+0" | |
| [[LaTeXStrings]] | |
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| [[LabelledArrays]] | |
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| [[Latexify]] | |
| deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "Printf", "Requires"] | |
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| [[LayoutPointers]] | |
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| [[LibCURL]] | |
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| [[LibCURL_jll]] | |
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| uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0" | |
| [[LibGit2]] | |
| deps = ["Base64", "NetworkOptions", "Printf", "SHA"] | |
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| [[LibSSH2_jll]] | |
| deps = ["Artifacts", "Libdl", "MbedTLS_jll"] | |
| uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8" | |
| [[Libdl]] | |
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| [[Libffi_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| version = "3.2.2+0" | |
| [[Libgcrypt_jll]] | |
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| [[Libglvnd_jll]] | |
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| [[Libgpg_error_jll]] | |
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| [[Libiconv_jll]] | |
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| [[Libmount_jll]] | |
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| version = "2.35.0+0" | |
| [[Libtiff_jll]] | |
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| uuid = "89763e89-9b03-5906-acba-b20f662cd828" | |
| version = "4.3.0+0" | |
| [[Libuuid_jll]] | |
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| [[LightGraphs]] | |
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| [[LineSearches]] | |
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| [[LinearAlgebra]] | |
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| [[LogExpFunctions]] | |
| deps = ["ChainRulesCore", "DocStringExtensions", "InverseFunctions", "IrrationalConstants", "LinearAlgebra"] | |
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| [[Logging]] | |
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| [[LoopVectorization]] | |
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| git-tree-sha1 = "caaa2d3518fe6312327819cdd485a4258e52ece0" | |
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| [[MacroTools]] | |
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| [[ManualMemory]] | |
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| [[Markdown]] | |
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| [[MbedTLS]] | |
| deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"] | |
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| [[MbedTLS_jll]] | |
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| [[Measures]] | |
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| [[Missings]] | |
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| [[Mmap]] | |
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| [[ModelingToolkit]] | |
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| [[MultivariatePolynomials]] | |
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| [[MutableArithmetics]] | |
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| [[NLSolversBase]] | |
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| [[NLsolve]] | |
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| [[NaNMath]] | |
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| [[NearestNeighbors]] | |
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| [[NetworkOptions]] | |
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| [[NonlinearSolve]] | |
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| version = "1.10.8" | |
| [[Ogg_jll]] | |
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| [[OpenBLAS_jll]] | |
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| [[OpenLibm_jll]] | |
| deps = ["Artifacts", "Libdl"] | |
| uuid = "05823500-19ac-5b8b-9628-191a04bc5112" | |
| [[OpenSSL_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| version = "1.1.10+0" | |
| [[OpenSpecFun_jll]] | |
| deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"] | |
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| [[Optim]] | |
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| [[Opus_jll]] | |
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| [[OrderedCollections]] | |
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| [[OrdinaryDiffEq]] | |
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| [[PCRE_jll]] | |
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| uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc" | |
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| [[PDMats]] | |
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| [[ParameterizedFunctions]] | |
| deps = ["DataStructures", "DiffEqBase", "DocStringExtensions", "Latexify", "LinearAlgebra", "ModelingToolkit", "Reexport", "SciMLBase"] | |
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| uuid = "65888b18-ceab-5e60-b2b9-181511a3b968" | |
| version = "5.12.1" | |
| [[Parameters]] | |
| deps = ["OrderedCollections", "UnPack"] | |
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| uuid = "d96e819e-fc66-5662-9728-84c9c7592b0a" | |
| version = "0.12.3" | |
| [[Parsers]] | |
| deps = ["Dates"] | |
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| uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0" | |
| version = "2.1.2" | |
| [[Pixman_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
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| uuid = "30392449-352a-5448-841d-b1acce4e97dc" | |
| version = "0.40.1+0" | |
| [[Pkg]] | |
| deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"] | |
| uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f" | |
| [[PlotThemes]] | |
| deps = ["PlotUtils", "Requires", "Statistics"] | |
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| [[PlotUtils]] | |
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| deps = ["CloseOpenIntervals", "IfElse", "LayoutPointers", "LinearAlgebra", "LoopVectorization", "Polyester", "Static", "VectorizationBase"] | |
| git-tree-sha1 = "ec9a310324dd2c546c07f33a599ded9c1d00a420" | |
| uuid = "d5829a12-d9aa-46ab-831f-fb7c9ab06edf" | |
| version = "0.1.8" | |
| [[URIs]] | |
| git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355" | |
| uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4" | |
| version = "1.3.0" | |
| [[UUIDs]] | |
| deps = ["Random", "SHA"] | |
| uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4" | |
| [[UnPack]] | |
| git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b" | |
| uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed" | |
| version = "1.0.2" | |
| [[Unicode]] | |
| uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5" | |
| [[UnicodeFun]] | |
| deps = ["REPL"] | |
| git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf" | |
| uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1" | |
| version = "0.4.1" | |
| [[Unitful]] | |
| deps = ["ConstructionBase", "Dates", "LinearAlgebra", "Random"] | |
| git-tree-sha1 = "880f77d2cd4c6948e6bd55425b7b52f34dcd7f4b" | |
| uuid = "1986cc42-f94f-5a68-af5c-568840ba703d" | |
| version = "1.9.1" | |
| [[VectorizationBase]] | |
| deps = ["ArrayInterface", "CPUSummary", "HostCPUFeatures", "Hwloc", "IfElse", "LayoutPointers", "Libdl", "LinearAlgebra", "SIMDTypes", "Static"] | |
| git-tree-sha1 = "5239606cf3552aff43d79ecc75b1af1ce4625109" | |
| uuid = "3d5dd08c-fd9d-11e8-17fa-ed2836048c2f" | |
| version = "0.21.21" | |
| [[VertexSafeGraphs]] | |
| deps = ["LightGraphs"] | |
| git-tree-sha1 = "b9b450c99a3ca1cc1c6836f560d8d887bcbe356e" | |
| uuid = "19fa3120-7c27-5ec5-8db8-b0b0aa330d6f" | |
| version = "0.1.2" | |
| [[Wayland_jll]] | |
| deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"] | |
| git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23" | |
| uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89" | |
| version = "1.19.0+0" | |
| [[Wayland_protocols_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"] | |
| git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670" | |
| uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91" | |
| version = "1.18.0+4" | |
| [[XML2_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"] | |
| git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a" | |
| uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a" | |
| version = "2.9.12+0" | |
| [[XSLT_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"] | |
| git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a" | |
| uuid = "aed1982a-8fda-507f-9586-7b0439959a61" | |
| version = "1.1.34+0" | |
| [[Xorg_libX11_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"] | |
| git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527" | |
| uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc" | |
| version = "1.6.9+4" | |
| [[Xorg_libXau_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e" | |
| uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec" | |
| version = "1.0.9+4" | |
| [[Xorg_libXcursor_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"] | |
| git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd" | |
| uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724" | |
| version = "1.2.0+4" | |
| [[Xorg_libXdmcp_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4" | |
| uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05" | |
| version = "1.1.3+4" | |
| [[Xorg_libXext_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
| git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3" | |
| uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3" | |
| version = "1.3.4+4" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[Xorg_libpthread_stubs_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb" | |
| uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74" | |
| version = "0.1.0+3" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[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" | |
| [[Xorg_xtrans_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845" | |
| uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" | |
| version = "1.4.0+3" | |
| [[Zlib_jll]] | |
| deps = ["Libdl"] | |
| uuid = "83775a58-1f1d-513f-b197-d71354ab007a" | |
| [[Zstd_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6" | |
| uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" | |
| version = "1.5.0+0" | |
| [[ZygoteRules]] | |
| deps = ["MacroTools"] | |
| git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0" | |
| uuid = "700de1a5-db45-46bc-99cf-38207098b444" | |
| version = "0.2.2" | |
| [[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" | |
| [[libfdk_aac_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" | |
| uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" | |
| version = "2.0.2+0" | |
| [[libpng_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] | |
| git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" | |
| uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" | |
| version = "1.6.38+0" | |
| [[libvorbis_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] | |
| git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb" | |
| uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" | |
| version = "1.3.7+0" | |
| [[nghttp2_jll]] | |
| deps = ["Artifacts", "Libdl"] | |
| uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" | |
| [[p7zip_jll]] | |
| deps = ["Artifacts", "Libdl"] | |
| uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" | |
| [[x264_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" | |
| uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" | |
| version = "2021.5.5+0" | |
| [[x265_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" | |
| uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" | |
| version = "3.5.0+0" | |
| [[xkbcommon_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] | |
| git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6" | |
| uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" | |
| version = "0.9.1+5" | |
| """ | |
| # ╔═╡ Cell order: | |
| # ╠═8370af88-403c-11ec-3411-d7f78b38c809 | |
| # ╠═c1fe5bef-707c-4b43-a4b2-ba42036778a5 | |
| # ╠═460d695a-c1b2-4a02-bb79-5b85c26ebef2 | |
| # ╠═b55cb9bb-3fef-4267-b2dc-b3f953a2915f | |
| # ╠═314f6a4d-3ffa-4721-8aad-b33400561bbc | |
| # ╠═1726d0ec-d9c5-4a1a-88b4-67cb791938dd | |
| # ╠═79fc6597-4a69-4c39-bd50-99b90c8e113d | |
| # ╠═6dd7ca98-a75a-4c29-a72d-d7b2861dcff8 | |
| # ╠═7b9439f2-ef9e-431c-8bbd-d30552314323 | |
| # ╠═a52ed8bf-629b-4a70-b686-04dbbb95f321 | |
| # ╟─ad82a0f2-8796-456c-a65d-0a90f1849635 | |
| # ╠═9a698ae0-5636-486f-8adf-ef8db52ce375 | |
| # ╠═767d5721-6564-494d-b4d5-f79f8031e196 | |
| # ╠═0626a286-d713-489a-9b4a-366e542a2499 | |
| # ╠═2c054894-f642-4537-9236-6b2db6b23755 | |
| # ╠═3aab31f4-6f9d-473f-a498-d864590b2c71 | |
| # ╠═1598cb2f-a90d-40ed-8d16-e8a5ec5eb754 | |
| # ╟─a5db8ef0-8fcb-46fe-b86d-b52b2b4149ee | |
| # ╟─e05b69d3-db65-4f58-aa30-51bc569f8213 | |
| # ╠═c53717cb-a1a4-41bc-8e3e-70effaca96b9 | |
| # ╠═dec9a17b-69dc-4b08-b1b3-0715fcd2bd53 | |
| # ╟─8fd26590-7b80-4dd3-b949-b13edaffd1f2 | |
| # ╠═2e24a242-fcc4-4010-ac57-fc55a043bfe0 | |
| # ╟─e586040e-de82-4786-9086-3ef45623d2c4 | |
| # ╠═83144131-b88c-4356-b8fb-d481350d00bc | |
| # ╟─50a9d2da-6f80-4811-9425-5b3aa7533c05 | |
| # ╠═b5d81f94-c82d-4b5f-acf7-1c4c0ea4d969 | |
| # ╠═cd2a7836-9e60-44ac-92df-3b682c3f4a1b | |
| # ╠═c531667e-7b86-40b3-ba6a-a95c502c706b | |
| # ╠═6e0dc15d-9d55-4c07-b7a1-a90377456b0a | |
| # ╠═e6962b93-6327-4215-b02b-5853c4f35272 | |
| # ╟─5c60b950-191d-480d-80b7-1d5edc57c6e4 | |
| # ╟─7b344acf-55ed-4ad0-899d-b8660714a95a | |
| # ╠═f09981d6-0c5e-40b0-bbd8-45d97f359b87 | |
| # ╟─5bc246a6-474d-4b4a-a509-390e8eba3ca6 | |
| # ╟─65ffa262-0100-454e-8c86-9b0f6b99e11a | |
| # ╠═c48b8a99-1ac1-4a22-bc8d-68c733ce7966 | |
| # ╠═b1234b1a-a150-4efe-8147-989b5e7d8dc6 | |
| # ╠═1ee68e6c-015d-47e5-8d6a-7d056c2842d1 | |
| # ╠═8e58f0a8-2a23-4699-bb22-a7063ed5300a | |
| # ╟─15203992-224b-4e82-ba84-cd53667f8757 | |
| # ╟─d64b9ad6-cde1-4847-82c4-dae9dbe770e2 | |
| # ╟─23ca0b4b-d23d-486a-98a3-0be7fb556427 | |
| # ╠═8c09dabe-ab60-484e-9ef7-170825173c95 | |
| # ╠═7c87b677-215a-46aa-9039-c713d1401270 | |
| # ╟─2e27d753-9885-4ada-8119-3a6aba8833fc | |
| # ╟─1f54b672-684c-4c21-878b-3807de6ca264 | |
| # ╟─00000000-0000-0000-0000-000000000001 | |
| # ╟─00000000-0000-0000-0000-000000000002 |
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