Diffusion Model

Samuel Isaacson, Chris Rackauckas

using DiffEqBase, Catalyst, DiffEqJump, DiffEqProblemLibrary.JumpProblemLibrary, Plots, Statistics, DataFrames
gr()
fmt = :png
JumpProblemLibrary.importjumpproblems()

Model and example solutions

Here we implement a 1D continuous time random walk approximation of diffusion for $N$ lattice sites on $\left[0,1\right]$, with reflecting boundary conditions at $x=0$ and $x=1$.

N = 256
h = 1 / N
u0 = 10*ones(Int64, N)
tf = .01
methods = (Direct(),DirectFW(),SortingDirect(),NRM(),DirectCR(),RSSA())
shortlabels = [string(leg)[12:end-2] for leg in methods]
jprob   = prob_jump_diffnetwork
prob    = DiscreteProblem(u0, (0.0, tf), [1 / (h*h)])
rn      = jprob.network(N)
ploth   = plot(reuse=false)
for (i,method) in enumerate(methods)
    println("Benchmarking method: ", method)
    jump_prob = JumpProblem(rn, prob, method, save_positions=(false, false))
    sol = solve(jump_prob, SSAStepper(), saveat=tf/1000.)
    plot!(ploth,sol.t,sol[Int(N//2),:],label=shortlabels[i], format=fmt)
end
plot!(ploth, title="Population at middle lattice site", xlabel="time",format=fmt)

Benchmarking method: DiffEqJump.Direct()
Benchmarking method: DiffEqJump.DirectFW()
Benchmarking method: DiffEqJump.SortingDirect()
Benchmarking method: DiffEqJump.NRM()
Benchmarking method: DiffEqJump.DirectCR()
Benchmarking method: DiffEqJump.RSSA()

Benchmarking performance of the methods

function run_benchmark!(t, jump_prob, stepper)
    sol = solve(jump_prob, stepper)
    @inbounds for i in 1:length(t)
        t[i] = @elapsed (sol = solve(jump_prob, stepper))
    end
end

run_benchmark! (generic function with 1 method)

nsims = 50
benchmarks = Vector{Vector{Float64}}()
for method in methods
    jump_prob = JumpProblem(rn, prob, method, save_positions=(false, false))
    stepper = SSAStepper()
    t = Vector{Float64}(undef,nsims)
    run_benchmark!(t, jump_prob, stepper)
    push!(benchmarks, t)
end

medtimes = Vector{Float64}(undef,length(methods))
stdtimes = Vector{Float64}(undef,length(methods))
avgtimes = Vector{Float64}(undef,length(methods))
for i in 1:length(methods)
    medtimes[i] = median(benchmarks[i])
    avgtimes[i] = mean(benchmarks[i])
    stdtimes[i] = std(benchmarks[i])
end

df = DataFrame(names=shortlabels,medtimes=medtimes,relmedtimes=(medtimes/medtimes[1]),
                avgtimes=avgtimes, std=stdtimes, cv=stdtimes./avgtimes)

6 rows × 6 columns

	names	medtimes	relmedtimes	avgtimes	std	cv
	String	Float64	Float64	Float64	Float64	Float64
1	Direct	7.39065	1.0	7.39109	0.00727328	0.00098406
2	DirectFW	7.31383	0.989605	7.3234	0.0218754	0.00298706
3	SortingDirect	1.0762	0.145617	1.07679	0.00375535	0.00348755
4	NRM	0.818555	0.110756	0.819539	0.00338196	0.00412666
5	DirectCR	0.516422	0.069875	0.519925	0.0151209	0.0290829
6	RSSA	1.44363	0.195332	1.44395	0.00256909	0.00177921

Plotting

sa = [string(round(mt,digits=4),"s") for mt in df.medtimes]
bar(df.names,df.relmedtimes,legend=:false, fmt=fmt)
scatter!(df.names, .05 .+ df.relmedtimes, markeralpha=0, series_annotations=sa, fmt=fmt)
ylabel!("median relative to Direct")
title!("256 Site 1D Diffusion CTRW")

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/Jumps","Diffusion_CTRW.jmd")

Computer Information:

Julia Version 1.6.6
Commit b8708f954a (2022-03-28 07:17 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
Environment:
  BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

      Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/Jumps/Project.toml`
  [479239e8] Catalyst v6.13.0
  [a93c6f00] DataFrames v1.1.1
  [2b5f629d] DiffEqBase v6.62.2
  [c894b116] DiffEqJump v6.14.2
  [a077e3f3] DiffEqProblemLibrary v4.13.0
  [961ee093] ModelingToolkit v5.19.1
  [1dea7af3] OrdinaryDiffEq v5.57.0
  [91a5bcdd] Plots v1.16.5
  [31c91b34] SciMLBenchmarks v0.1.0
  [10745b16] Statistics

And the full manifest:

      Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/Jumps/Manifest.toml`
  [c3fe647b] AbstractAlgebra v0.17.1
  [1520ce14] AbstractTrees v0.3.4
  [79e6a3ab] Adapt v3.3.1
  [ec485272] ArnoldiMethod v0.1.0
  [4fba245c] ArrayInterface v3.1.17
  [4c555306] ArrayLayouts v0.7.0
  [aae01518] BandedMatrices v0.16.9
  [8e7c35d0] BlockArrays v0.15.3
  [ffab5731] BlockBandedMatrices v0.10.6
  [00ebfdb7] CSTParser v2.5.0
  [479239e8] Catalyst v6.13.0
  [d360d2e6] ChainRulesCore v0.9.45
  [35d6a980] ColorSchemes v3.12.1
  [3da002f7] ColorTypes v0.11.0
  [5ae59095] Colors v0.12.8
  [861a8166] Combinatorics v1.0.2
  [a80b9123] CommonMark v0.8.1
  [38540f10] CommonSolve v0.2.0
  [bbf7d656] CommonSubexpressions v0.3.0
  [34da2185] Compat v3.30.0
  [b152e2b5] CompositeTypes v0.1.2
  [8f4d0f93] Conda v1.5.2
  [187b0558] ConstructionBase v1.2.1
  [d38c429a] Contour v0.5.7
  [a8cc5b0e] Crayons v4.0.4
  [9a962f9c] DataAPI v1.6.0
  [a93c6f00] DataFrames v1.1.1
  [864edb3b] DataStructures v0.18.9
  [e2d170a0] DataValueInterfaces v1.0.0
  [2b5f629d] DiffEqBase v6.62.2
  [c894b116] DiffEqJump v6.14.2
  [9fdde737] DiffEqOperators v4.28.0
  [a077e3f3] DiffEqProblemLibrary v4.13.0
  [163ba53b] DiffResults v1.0.3
  [b552c78f] DiffRules v1.0.2
  [b4f34e82] Distances v0.10.3
  [31c24e10] Distributions v0.24.18
  [ffbed154] DocStringExtensions v0.8.5
  [e30172f5] Documenter v0.26.3
  [5b8099bc] DomainSets v0.5.2
  [da5c29d0] EllipsisNotation v1.1.0
  [d4d017d3] ExponentialUtilities v1.8.4
  [e2ba6199] ExprTools v0.1.3
  [c87230d0] FFMPEG v0.4.1
  [7034ab61] FastBroadcast v0.1.8
  [9aa1b823] FastClosures v0.3.2
  [1a297f60] FillArrays v0.11.7
  [6a86dc24] FiniteDiff v2.8.0
  [53c48c17] FixedPointNumbers v0.8.4
  [59287772] Formatting v0.4.2
  [f6369f11] ForwardDiff v0.10.18
  [069b7b12] FunctionWrappers v1.1.2
  [28b8d3ca] GR v0.57.5
  [5c1252a2] GeometryBasics v0.3.12
  [d7ba0133] Git v1.2.1
  [42e2da0e] Grisu v1.0.2
  [cd3eb016] HTTP v0.9.10
  [eafb193a] Highlights v0.4.5
  [0e44f5e4] Hwloc v2.0.0
  [7073ff75] IJulia v1.23.2
  [b5f81e59] IOCapture v0.1.1
  [615f187c] IfElse v0.1.0
  [d25df0c9] Inflate v0.1.2
  [83e8ac13] IniFile v0.5.0
  [8197267c] IntervalSets v0.5.3
  [41ab1584] InvertedIndices v1.0.0
  [c8e1da08] IterTools v1.3.0
  [42fd0dbc] IterativeSolvers v0.9.1
  [82899510] IteratorInterfaceExtensions v1.0.0
  [692b3bcd] JLLWrappers v1.3.0
  [682c06a0] JSON v0.21.1
  [98e50ef6] JuliaFormatter v0.13.7
  [b964fa9f] LaTeXStrings v1.2.1
  [2ee39098] LabelledArrays v1.6.1
  [23fbe1c1] Latexify v0.15.6
  [5078a376] LazyArrays v0.21.6
  [d7e5e226] LazyBandedMatrices v0.5.8
  [093fc24a] LightGraphs v1.3.5
  [d3d80556] LineSearches v7.1.1
  [2ab3a3ac] LogExpFunctions v0.2.4
  [bdcacae8] LoopVectorization v0.12.37
  [1914dd2f] MacroTools v0.5.6
  [a3b82374] MatrixFactorizations v0.8.3
  [739be429] MbedTLS v1.0.3
  [442fdcdd] Measures v0.3.1
  [e1d29d7a] Missings v1.0.0
  [961ee093] ModelingToolkit v5.19.1
  [46d2c3a1] MuladdMacro v0.2.2
  [ffc61752] Mustache v1.0.10
  [d41bc354] NLSolversBase v7.8.0
  [2774e3e8] NLsolve v4.5.1
  [872c559c] NNlib v0.7.21
  [77ba4419] NaNMath v0.3.5
  [8913a72c] NonlinearSolve v0.3.8
  [6fe1bfb0] OffsetArrays v1.9.2
  [bac558e1] OrderedCollections v1.4.1
  [1dea7af3] OrdinaryDiffEq v5.57.0
  [90014a1f] PDMats v0.11.1
  [d96e819e] Parameters v0.12.2
  [69de0a69] Parsers v1.1.0
  [ccf2f8ad] PlotThemes v2.0.1
  [995b91a9] PlotUtils v1.0.10
  [91a5bcdd] Plots v1.16.5
  [e409e4f3] PoissonRandom v0.4.0
  [f517fe37] Polyester v0.3.1
  [2dfb63ee] PooledArrays v1.2.1
  [21216c6a] Preferences v1.2.2
  [08abe8d2] PrettyTables v1.1.0
  [1fd47b50] QuadGK v2.4.1
  [fb686558] RandomExtensions v0.4.3
  [e6cf234a] RandomNumbers v1.4.0
  [3cdcf5f2] RecipesBase v1.1.1
  [01d81517] RecipesPipeline v0.3.2
  [731186ca] RecursiveArrayTools v2.11.4
  [f2c3362d] RecursiveFactorization v0.1.12
  [189a3867] Reexport v1.1.0
  [ae029012] Requires v1.1.3
  [79098fc4] Rmath v0.7.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.2
  [476501e8] SLEEFPirates v0.6.21
  [1bc83da4] SafeTestsets v0.0.1
  [0bca4576] SciMLBase v1.13.5
  [31c91b34] SciMLBenchmarks v0.1.0
  [6c6a2e73] Scratch v1.1.0
  [efcf1570] Setfield v0.7.0
  [992d4aef] Showoff v1.0.3
  [699a6c99] SimpleTraits v0.9.3
  [b85f4697] SoftGlobalScope v1.1.0
  [a2af1166] SortingAlgorithms v1.0.0
  [47a9eef4] SparseDiffTools v1.13.2
  [276daf66] SpecialFunctions v1.5.1
  [aedffcd0] Static v0.2.5
  [90137ffa] StaticArrays v1.2.2
  [82ae8749] StatsAPI v1.0.0
  [2913bbd2] StatsBase v0.33.8
  [4c63d2b9] StatsFuns v0.9.8
  [7792a7ef] StrideArraysCore v0.1.13
  [69024149] StringEncodings v0.3.4
  [09ab397b] StructArrays v0.5.1
  [d1185830] SymbolicUtils v0.11.3
  [0c5d862f] Symbolics v0.1.32
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.4.3
  [8290d209] ThreadingUtilities v0.4.4
  [a759f4b9] TimerOutputs v0.5.9
  [0796e94c] Tokenize v0.5.16
  [a2a6695c] TreeViews v0.3.0
  [5c2747f8] URIs v1.3.0
  [3a884ed6] UnPack v1.0.2
  [1986cc42] Unitful v1.8.0
  [3d5dd08c] VectorizationBase v0.20.16
  [81def892] VersionParsing v1.2.0
  [19fa3120] VertexSafeGraphs v0.1.2
  [44d3d7a6] Weave v0.10.9
  [ddb6d928] YAML v0.4.7
  [c2297ded] ZMQ v1.2.1
  [700de1a5] ZygoteRules v0.2.1
  [6e34b625] Bzip2_jll v1.0.6+5
  [83423d85] Cairo_jll v1.16.0+6
  [5ae413db] EarCut_jll v2.1.5+1
  [2e619515] Expat_jll v2.2.10+0
  [b22a6f82] FFMPEG_jll v4.3.1+4
  [a3f928ae] Fontconfig_jll v2.13.1+14
  [d7e528f0] FreeType2_jll v2.10.1+5
  [559328eb] FriBidi_jll v1.0.10+0
  [0656b61e] GLFW_jll v3.3.5+0
  [d2c73de3] GR_jll v0.57.3+0
  [78b55507] Gettext_jll v0.20.1+7
  [f8c6e375] Git_jll v2.31.0+0
  [7746bdde] Glib_jll v2.59.0+4
  [e33a78d0] Hwloc_jll v2.4.1+0
  [aacddb02] JpegTurbo_jll v2.1.0+0
  [c1c5ebd0] LAME_jll v3.100.1+0
  [dd4b983a] LZO_jll v2.10.1+0
  [dd192d2f] LibVPX_jll v1.10.0+0
  [e9f186c6] Libffi_jll v3.2.2+0
  [d4300ac3] Libgcrypt_jll v1.8.7+0
  [7e76a0d4] Libglvnd_jll v1.3.0+3
  [7add5ba3] Libgpg_error_jll v1.42.0+0
  [94ce4f54] Libiconv_jll v1.16.1+0
  [4b2f31a3] Libmount_jll v2.35.0+0
  [89763e89] Libtiff_jll v4.3.0+0
  [38a345b3] Libuuid_jll v2.36.0+0
  [e7412a2a] Ogg_jll v1.3.5+0
  [458c3c95] OpenSSL_jll v1.1.10+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [2f80f16e] PCRE_jll v8.44.0+0
  [30392449] Pixman_jll v0.40.1+0
  [ea2cea3b] Qt5Base_jll v5.15.2+0
  [f50d1b31] Rmath_jll v0.3.0+0
  [a2964d1f] Wayland_jll v1.17.0+4
  [2381bf8a] Wayland_protocols_jll v1.18.0+4
  [02c8fc9c] XML2_jll v2.9.12+0
  [aed1982a] XSLT_jll v1.1.34+0
  [4f6342f7] Xorg_libX11_jll v1.6.9+4
  [0c0b7dd1] Xorg_libXau_jll v1.0.9+4
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.3+4
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
  [c7cfdc94] Xorg_libxcb_jll v1.13.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.0+4
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.2+4
  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
  [c5fb5394] Xorg_xtrans_jll v1.4.0+3
  [8f1865be] ZeroMQ_jll v4.3.2+6
  [3161d3a3] Zstd_jll v1.5.0+0
  [0ac62f75] libass_jll v0.14.0+4
  [f638f0a6] libfdk_aac_jll v0.1.6+4
  [b53b4c65] libpng_jll v1.6.38+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.7+0
  [1270edf5] x264_jll v2020.7.14+2
  [dfaa095f] x265_jll v3.0.0+3
  [d8fb68d0] xkbcommon_jll v0.9.1+5
  [0dad84c5] ArgTools
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [b27032c2] LibCURL
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions
  [44cfe95a] Pkg
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML
  [a4e569a6] Tar
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll
  [deac9b47] LibCURL_jll
  [29816b5a] LibSSH2_jll
  [c8ffd9c3] MbedTLS_jll
  [14a3606d] MozillaCACerts_jll
  [efcefdf7] PCRE2_jll
  [83775a58] Zlib_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll