Multistate Work-Precision Diagrams

The following benchmark is of 9 ODEs with 18 terms that describe a chemical reaction network. This multistate model was used as a benchmark model in Gupta et al.. We use ReactionNetworkImporters to load the BioNetGen model files as a Catalyst model, and then use ModelingToolkit to convert the Catalyst network model to ODEs.

using DiffEqBase, OrdinaryDiffEq, Catalyst, ReactionNetworkImporters,
      Sundials, Plots, DiffEqDevTools, ODEInterface, ODEInterfaceDiffEq,
      LSODA, TimerOutputs, LinearAlgebra, ModelingToolkit, BenchmarkTools

gr()
const to = TimerOutput()
tf       = 20.0

# generate ModelingToolkit ODEs
@timeit to "Parse Network" prnbng = loadrxnetwork(BNGNetwork(), joinpath(@__DIR__, "Models/multistate.net"))
show(to) 
rn    = prnbng.rn
obs = [eq.lhs for eq in observed(rn)]

@timeit to "Create ODESys" osys = convert(ODESystem, rn)
show(to) 

tspan = (0.,tf)
@timeit to "ODEProb No Jac" oprob = ODEProblem{true, SciMLBase.FullSpecialize}(osys, Float64[], tspan, Float64[])
show(to);

Parsing parameters...done
Creating parameters...done
Parsing species...done
Creating species...done
Creating species and parameters for evaluating expressions...done
Parsing and adding reactions...done
Parsing groups...done
 ──────────────────────────────────────────────────────────────────────────
                                  Time                    Allocations      
                         ───────────────────────   ────────────────────────
    Tot / % measured:        6.83ms /  80.5%            493KiB /  95.9%    

 Section         ncalls     time    %tot     avg     alloc    %tot      avg
 ──────────────────────────────────────────────────────────────────────────
 Parse Network        1   5.50ms  100.0%  5.50ms    473KiB  100.0%   473KiB
 ──────────────────────────────────────────────────────────────────────────
 ──────────────────────────────────────────────────────────────────────────
                                  Time                    Allocations      
                         ───────────────────────   ────────────────────────
    Tot / % measured:        88.8ms /  10.3%           3.58MiB /  29.1%    

 Section         ncalls     time    %tot     avg     alloc    %tot      avg
 ──────────────────────────────────────────────────────────────────────────
 Parse Network        1   5.50ms   60.3%  5.50ms    473KiB   44.3%   473KiB
 Create ODESys        1   3.62ms   39.7%  3.62ms    594KiB   55.7%   594KiB
 ──────────────────────────────────────────────────────────────────────────
 ──────────────────────────────────────────────────────────────────────────
─
                                   Time                    Allocations     
 
                          ───────────────────────   ───────────────────────
─
     Tot / % measured:         465ms /  82.4%           25.8MiB /  90.0%   
 

 Section          ncalls     time    %tot     avg     alloc    %tot      av
g
 ──────────────────────────────────────────────────────────────────────────
─
 ODEProb No Jac        1    374ms   97.6%   374ms   22.2MiB   95.5%  22.2Mi
B
 Parse Network         1   5.50ms    1.4%  5.50ms    473KiB    2.0%   473Ki
B
 Create ODESys         1   3.62ms    0.9%  3.62ms    594KiB    2.5%   594Ki
B
 ──────────────────────────────────────────────────────────────────────────
─

@timeit to "ODEProb SparseJac" sparsejacprob = ODEProblem{true, SciMLBase.FullSpecialize}(osys, Float64[], tspan, Float64[], jac=true, sparse=true)
show(to)

──────────────────────────────────────────────────────────────────────────
────
                                      Time                    Allocations  
    
                             ───────────────────────   ────────────────────
────
      Tot / % measured:           747ms /  88.6%           44.6MiB /  94.1%
    

 Section             ncalls     time    %tot     avg     alloc    %tot     
 avg
 ──────────────────────────────────────────────────────────────────────────
────
 ODEProb No Jac           1    374ms   56.5%   374ms   22.2MiB   52.9%  22.
2MiB
 ODEProb SparseJac        1    279ms   42.1%   279ms   18.7MiB   44.6%  18.
7MiB
 Parse Network            1   5.50ms    0.8%  5.50ms    473KiB    1.1%   47
3KiB
 Create ODESys            1   3.62ms    0.5%  3.62ms    594KiB    1.4%   59
4KiB
 ──────────────────────────────────────────────────────────────────────────
────

@show numspecies(rn) # Number of ODEs
@show numreactions(rn) # Apprx. number of terms in the ODE
@show length(parameters(rn)); # Number of Parameters

numspecies(rn) = 9
numreactions(rn) = 18
length(parameters(rn)) = 9

Time ODE derivative function compilation

As compiling the ODE derivative functions has in the past taken longer than running a simulation, we first force compilation by evaluating these functions one time.

u  = ModelingToolkit.varmap_to_vars(nothing, species(rn); defaults=ModelingToolkit.defaults(rn))
du = copy(u)
p  = ModelingToolkit.varmap_to_vars(nothing, parameters(rn); defaults=ModelingToolkit.defaults(rn))
@timeit to "ODE rhs Eval1" oprob.f(du,u,p,0.)
@timeit to "ODE rhs Eval2" oprob.f(du,u,p,0.)
sparsejacprob.f(du,u,p,0.)

We also time the ODE rhs function with BenchmarkTools as it is more accurate given how fast evaluating f is:

@btime oprob.f($du,$u,$p,0.)

279.737 ns (3 allocations: 1.12 KiB)

Picture of the solution

sol = solve(oprob, CVODE_BDF(), saveat=tf/1000., reltol=1e-5, abstol=1e-5)
plot(sol; idxs=obs, legend=false, fmt=:png)

For these benchmarks we will be using the time-series error with these saving points.

Generate Test Solution

@time sol = solve(oprob, CVODE_BDF(), reltol=1e-15, abstol=1e-15)
test_sol  = TestSolution(sol);

0.782015 seconds (752.22 k allocations: 41.591 MiB, 96.48% compilation ti
me)

Setups

Sets plotting defaults

default(legendfontsize=7,framestyle=:box,gridalpha=0.3,gridlinewidth=2.5)

Declares a plotting helper function

function plot_settings(wp)
    times = vcat(map(wp -> wp.times, wp.wps)...)
    errors = vcat(map(wp -> wp.errors, wp.wps)...)
    xlimit = 10 .^ (floor(log10(minimum(errors))), ceil(log10(maximum(errors))))
    ylimit = 10 .^ (floor(log10(minimum(times))), ceil(log10(maximum(times))))
    return xlimit,ylimit
end

plot_settings (generic function with 1 method)

Sets tolerances

abstols = 1.0 ./ 10.0 .^ (6:10)
reltols = 1.0 ./ 10.0 .^ (6:10);

Work-Precision Diagram

We start by trying lsoda and CVODE solvers.

Declare solvers

We designate the solvers (and options) we wish to use.

setups = [
          Dict(:alg=>lsoda()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>CVODE_BDF(linear_solver=:LapackDense)),
          Dict(:alg=>CVODE_BDF(linear_solver=:GMRES))
          ];

Plot Work-Precision Diagram

Finally, we generate a work-precision diagram for the selection of solvers.

wp = WorkPrecisionSet(oprob,abstols,reltols,setups;error_estimate=:l2,
                      saveat=tf/10000.,appxsol=test_sol,maxiters=Int(1e9),numruns=200)

names = ["lsoda" "CVODE_BDF" "CVODE_BDF (LapackDense)" "CVODE_BDF (GMRES)"]
xlimit,ylimit = plot_settings(wp)
plot(wp;label=names,xlimit=xlimit,ylimit=ylimit)

Implicit Work-Precision Diagram

Next, we try a couple of implicit Julia solvers.

Declare solvers

We designate the solvers we wish to use.

setups = [
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>QNDF()),
          Dict(:alg=>FBDF()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Rodas4()),
          Dict(:alg=>Rodas5P())
          ];

Plot Work-Precision Diagram

Finally, we generate a work-precision diagram for the selection of solvers.

wp = WorkPrecisionSet(oprob,abstols,reltols,setups;error_estimate=:l2,
                      saveat=tf/10000.,appxsol=test_sol,maxiters=Int(1e12),dtmin=1e-18,numruns=200)

names = ["TRBDF2" "QNDF" "FBDF" "KenCarp4" "Rosenbrock23" "Rodas4" "Rodas5P"]
xlimit,ylimit = plot_settings(wp)
plot(wp;label=names,xlimit=xlimit,ylimit=ylimit)

Implicit methods doing poorly suggests it's non-stiff.

Explicit Work-Precision Diagram

Benchmarks for explicit solvers.

Declare solvers

We designate the solvers we wish to use, this also includes lsoda and CVODE.

setups = [
          Dict(:alg=>lsoda()), 
          Dict(:alg=>Tsit5()), 
          Dict(:alg=>BS5()), 
          Dict(:alg=>VCABM()),          
          Dict(:alg=>Vern6()), 
          Dict(:alg=>Vern7()), 
          Dict(:alg=>Vern8()), 
          Dict(:alg=>Vern9()),
          Dict(:alg=>ROCK4())
          ];

Plot Work-Precision Diagram

wp = WorkPrecisionSet(oprob,abstols,reltols,setups;error_estimate=:l2,
                      saveat=tf/10000.,appxsol=test_sol,maxiters=Int(1e9),numruns=200)

names = ["lsoda" "Tsit5" "BS5" "VCABM" "Vern6" "Vern7" "Vern8" "Vern9" "ROCK4"]
xlimit,ylimit = plot_settings(wp)
plot(wp;label=names,xlimit=xlimit,ylimit=ylimit)

Additional explicit solvers

Two additional explicit solvers, CVODE_Adams and ROCK2, perform noticeably worse as compared to the other ones.

setups = [Dict(:alg=>CVODE_Adams()), Dict(:alg=>ROCK2())];
wp = WorkPrecisionSet(oprob,abstols,reltols,setups;error_estimate=:l2,
                      saveat=tf/10000.,appxsol=test_sol,maxiters=Int(1e9),numruns=200)
names = ["CVODE_Adams" "ROCK2"]
xlimit,ylimit = plot_settings(wp)
plot(wp;label=names,xlimit=xlimit,ylimit=ylimit)

Summary of results

Finally, we compute a single diagram comparing the various solvers used.

Declare solvers

We designate the solvers we wish to compare.

setups = [
          Dict(:alg=>lsoda()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>QNDF()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>Rodas5P()),
          Dict(:alg=>Tsit5()), 
          Dict(:alg=>BS5()), 
          Dict(:alg=>VCABM()),   
          Dict(:alg=>Vern7())
          ];

Plot Work-Precision Diagram

For these, we generate a work-precision diagram for the selection of solvers.

wp = WorkPrecisionSet(oprob,abstols,reltols,setups;error_estimate=:l2,
                      saveat=tf/10000.,appxsol=test_sol,maxiters=Int(1e9),numruns=200)

names = ["lsoda" "CVODE_BDF" "QNDF" "KenCarp4" "Rodas5P" "Tsit5" "BS5" "VCABM" "Vern7"]
colors = [:seagreen1 :chartreuse1 :deepskyblue1 :lightskyblue :blue :orchid2 :thistle2 :lightsteelblue2 :mediumpurple1]
markershapes = [:star4 :circle :hexagon :star5 :heptagon :ltriangle :star8 :heptagon :star6]
xlimit,ylimit = plot_settings(wp)
xlimit = xlimit .* [0.95,1/0.95]; ylimit = ylimit .* [0.95,1/0.95]; 
plot(wp;label=names,left_margin=10Plots.mm,right_margin=10Plots.mm,xlimit=xlimit,ylimit=ylimit,xticks=[1e-12,1e-11,1e-10,1e-9,1e-8,1e-7,1e-6,1e-5,1e-4,1e-3,1e-2],yticks=[1e-3,1e-2],color=colors,markershape=markershapes,legendfontsize=15,tickfontsize=15,guidefontsize=15, legend=:topright, lw=20, la=0.8, markersize=20,markerstrokealpha=1.0, markerstrokewidth=1.5, gridalpha=0.3, gridlinewidth=7.5,size=(1100,1000))

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/Bio","multistate.jmd")

Computer Information:

Julia Version 1.8.5
Commit 17cfb8e65ea (2023-01-08 06:45 UTC)
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-13.0.1 (ORCJIT, znver2)
  Threads: 128 on 128 virtual cores
Environment:
  JULIA_CPU_THREADS = 128
  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/Bio/Project.toml`
  [6e4b80f9] BenchmarkTools v1.3.2
⌅ [479239e8] Catalyst v12.3.1
⌃ [2b5f629d] DiffEqBase v6.114.0
  [f3b72e0c] DiffEqDevTools v2.35.0
  [40713840] IncompleteLU v0.2.1
⌃ [033835bb] JLD2 v0.4.29
  [7f56f5a3] LSODA v0.7.3
⌃ [7ed4a6bd] LinearSolve v1.33.0
⌃ [961ee093] ModelingToolkit v8.41.0
  [54ca160b] ODEInterface v0.5.0
  [09606e27] ODEInterfaceDiffEq v3.12.0
⌃ [1dea7af3] OrdinaryDiffEq v6.38.0
  [91a5bcdd] Plots v1.38.7
⌅ [b4db0fb7] ReactionNetworkImporters v0.13.5
  [31c91b34] SciMLBenchmarks v0.1.1
⌃ [c3572dad] Sundials v4.12.0
  [a759f4b9] TimerOutputs v0.5.22
Info Packages marked with ⌃ and ⌅ have new versions available, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`

And the full manifest:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/Bio/Manifest.toml`
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  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.1.0
⌃ [727e6d20] SimpleNonlinearSolve v0.1.12
  [699a6c99] SimpleTraits v0.9.4
  [66db9d55] SnoopPrecompile v1.0.3
  [b85f4697] SoftGlobalScope v1.1.0
  [a2af1166] SortingAlgorithms v1.1.0
⌃ [47a9eef4] SparseDiffTools v1.30.0
  [e56a9233] Sparspak v0.3.9
  [276daf66] SpecialFunctions v2.2.0
  [171d559e] SplittablesBase v0.1.15
  [aedffcd0] Static v0.8.4
  [90137ffa] StaticArrays v1.5.16
  [1e83bf80] StaticArraysCore v1.4.0
  [82ae8749] StatsAPI v1.5.0
  [2913bbd2] StatsBase v0.33.21
  [4c63d2b9] StatsFuns v1.3.0
⌃ [7792a7ef] StrideArraysCore v0.4.7
  [69024149] StringEncodings v0.3.6
⌃ [c3572dad] Sundials v4.12.0
  [2efcf032] SymbolicIndexingInterface v0.2.2
⌅ [d1185830] SymbolicUtils v0.19.11
⌅ [0c5d862f] Symbolics v4.14.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.10.0
  [62fd8b95] TensorCore v0.1.1
⌅ [8ea1fca8] TermInterface v0.2.3
  [8290d209] ThreadingUtilities v0.5.1
  [ac1d9e8a] ThreadsX v0.1.11
  [a759f4b9] TimerOutputs v0.5.22
  [0796e94c] Tokenize v0.5.25
  [3bb67fe8] TranscodingStreams v0.9.11
  [28d57a85] Transducers v0.4.75
  [a2a6695c] TreeViews v0.3.0
  [d5829a12] TriangularSolve v0.1.19
  [410a4b4d] Tricks v0.1.6
  [781d530d] TruncatedStacktraces v1.1.0
  [5c2747f8] URIs v1.4.2
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.12.4
  [41fe7b60] Unzip v0.2.0
⌃ [3d5dd08c] VectorizationBase v0.21.58
  [81def892] VersionParsing v1.3.0
  [19fa3120] VertexSafeGraphs v0.2.0
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.8
  [c2297ded] ZMQ v1.2.2
  [700de1a5] ZygoteRules v0.2.2
  [6e34b625] Bzip2_jll v1.0.8+0
  [83423d85] Cairo_jll v1.16.1+1
  [2e619515] Expat_jll v2.4.8+0
  [b22a6f82] FFMPEG_jll v4.4.2+2
  [a3f928ae] Fontconfig_jll v2.13.93+0
  [d7e528f0] FreeType2_jll v2.10.4+0
  [559328eb] FriBidi_jll v1.0.10+0
  [0656b61e] GLFW_jll v3.3.8+0
  [d2c73de3] GR_jll v0.71.7+0
  [78b55507] Gettext_jll v0.21.0+0
  [f8c6e375] Git_jll v2.36.1+2
  [7746bdde] Glib_jll v2.74.0+2
  [3b182d85] Graphite2_jll v1.3.14+0
  [2e76f6c2] HarfBuzz_jll v2.8.1+1
  [aacddb02] JpegTurbo_jll v2.1.91+0
  [c1c5ebd0] LAME_jll v3.100.1+0
  [88015f11] LERC_jll v3.0.0+1
  [aae0fff6] LSODA_jll v0.1.1+0
  [dd4b983a] LZO_jll v2.10.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
  [d4300ac3] Libgcrypt_jll v1.8.7+0
  [7e76a0d4] Libglvnd_jll v1.6.0+0
  [7add5ba3] Libgpg_error_jll v1.42.0+0
  [94ce4f54] Libiconv_jll v1.16.1+2
  [4b2f31a3] Libmount_jll v2.35.0+0
  [89763e89] Libtiff_jll v4.4.0+0
  [38a345b3] Libuuid_jll v2.36.0+0
  [c771fb93] ODEInterface_jll v0.0.1+0
  [e7412a2a] Ogg_jll v1.3.5+1
  [458c3c95] OpenSSL_jll v1.1.20+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [30392449] Pixman_jll v0.40.1+0
  [ea2cea3b] Qt5Base_jll v5.15.3+2
  [f50d1b31] Rmath_jll v0.4.0+0
  [fb77eaff] Sundials_jll v5.2.1+0
  [a2964d1f] Wayland_jll v1.21.0+0
  [2381bf8a] Wayland_protocols_jll v1.25.0+0
  [02c8fc9c] XML2_jll v2.10.3+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 v4.3.4+0
  [3161d3a3] Zstd_jll v1.5.4+0
⌅ [214eeab7] fzf_jll v0.29.0+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [b53b4c65] libpng_jll v1.6.38+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.8.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.0
  [a4e569a6] Tar v1.10.1
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.0.1+0
  [deac9b47] LibCURL_jll v7.84.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.0+0
  [14a3606d] MozillaCACerts_jll v2022.2.1
  [4536629a] OpenBLAS_jll v0.3.20+0
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.40.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+0
  [83775a58] Zlib_jll v1.2.12+3
  [8e850b90] libblastrampoline_jll v5.1.1+0
  [8e850ede] nghttp2_jll v1.48.0+0
  [3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`