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

namesmedtimesrelmedtimesavgtimesstdcv
StringFloat64Float64Float64Float64Float64
1Direct7.390651.07.391090.007273280.00098406
2DirectFW7.313830.9896057.32340.02187540.00298706
3SortingDirect1.07620.1456171.076790.003755350.00348755
4NRM0.8185550.1107560.8195390.003381960.00412666
5DirectCR0.5164220.0698750.5199250.01512090.0290829
6RSSA1.443630.1953321.443950.002569090.00177921

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
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  [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:

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