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Fceri_gamma2 Work-Precision Diagrams

The following benchmark is of 356 ODEs with 3749 terms that describe a stiff chemical reaction network. This fceri_gamma2 model was used as a benchmark model in Gupta et al.. It describes high-affinity human IgE receptor signalling Faeder 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,
      LinearSolve, RecursiveFactorization

gr()
const to = TimerOutput()
tf = 150.0

# generate ModelingToolkit ODEs
@timeit to "Parse Network" prnbng = loadrxnetwork(BNGNetwork(), joinpath(@__DIR__, "Models/fceri_gamma2.net"))
show(to)
rn = complete(prnbng.rn)
obs = [eq.lhs for eq in observed(rn)]

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

tspan = (0.0, tf)
@timeit to "ODEProb SparseJac" sparsejacprob = ODEProblem{true, SciMLBase.FullSpecialize}(
    osys, Float64[], tspan, Float64[], jac = true, sparse = true)
show(to)
@timeit to "ODEProb No Jac" oprob = ODEProblem{true, SciMLBase.FullSpecialize}(
    osys, Float64[], tspan, Float64[])
show(to)
oprob_sparse = ODEProblem{true, SciMLBase.FullSpecialize}(
    osys, Float64[], tspan, Float64[]; sparse = true);
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:         10.1s /  97.7%           2.47GiB /  99.5%    

Section         ncalls     time    %tot     avg     alloc    %tot      avg
──────────────────────────────────────────────────────────────────────────
Parse Network        1    9.86s  100.0%   9.86s   2.46GiB  100.0%  2.46GiB
───────────────────────────────────────────────────────────────────────────
─────────────────────────────────────────────────────────────────────────
                                 Time                    Allocations      
                        ───────────────────────   ────────────────────────
   Tot / % measured:         28.4s /  95.0%           11.4GiB /  98.3%    

Section         ncalls     time    %tot     avg     alloc    %tot      avg
──────────────────────────────────────────────────────────────────────────
Create ODESys        1    17.1s   63.4%   17.1s   8.79GiB   78.1%  8.79GiB
Parse Network        1    9.86s   36.6%   9.86s   2.46GiB   21.9%  2.46GiB
───────────────────────────────────────────────────────────────────────────
───────────────────────────────────────────────────────────────────────────
──
                                     Time                    Allocations   
   
                            ───────────────────────   ─────────────────────
───
     Tot / % measured:           1.34h / 100.0%           8.73TiB / 100.0% 
   

Section             ncalls     time    %tot     avg     alloc    %tot      
avg
───────────────────────────────────────────────────────────────────────────
───
ODEProb SparseJac        1    1.33h   99.4%   1.33h   8.72TiB   99.9%  8.72
TiB
Create ODESys            1    17.1s    0.4%   17.1s   8.79GiB    0.1%  8.79
GiB
Parse Network            1    9.86s    0.2%   9.86s   2.46GiB    0.0%  2.46
GiB
───────────────────────────────────────────────────────────────────────────
───────────────────────────────────────────────────────────────────────────
──────
                                     Time                    Allocations   
   
                            ───────────────────────   ─────────────────────
───
     Tot / % measured:           1.35h / 100.0%           8.74TiB / 100.0% 
   

Section             ncalls     time    %tot     avg     alloc    %tot      
avg
───────────────────────────────────────────────────────────────────────────
───
ODEProb SparseJac        1    1.33h   98.8%   1.33h   8.72TiB   99.8%  8.72
TiB
ODEProb No Jac           1    30.0s    0.6%   30.0s   8.80GiB    0.1%  8.80
GiB
Create ODESys            1    17.1s    0.4%   17.1s   8.79GiB    0.1%  8.79
GiB
Parse Network            1    9.86s    0.2%   9.86s   2.46GiB    0.0%  2.46
GiB
───────────────────────────────────────────────────────────────────────────
───
@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) = 3744
numreactions(rn) = 58276
length(parameters(rn)) = 26

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 = oprob.u0
du = copy(u)
p = oprob.p
@timeit to "ODE rhs Eval1" sparsejacprob.f(du, u, p, 0.0)
sparsejacprob.f(du, u, p, 0.0)

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

@btime sparsejacprob.f($du, $u, $p, 0.0)
81.020 μs (2 allocations: 832 bytes)

Picture of the solution

sol = solve(oprob, CVODE_BDF(), saveat = tf/1000.0, 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(sparsejacprob, CVODE_BDF(linear_solver = :GMRES), reltol = 1e-15, abstol = 1e-15)
test_sol = TestSolution(sol);
45.983801 seconds (10.56 M allocations: 1.337 GiB, 42.43% gc time, 61.53% 
compilation time)

Setups

Sets plotting defaults

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

Declare pre-conditioners

using IncompleteLU, LinearAlgebra

jaccache = sparsejacprob.f.jac(oprob.u0, oprob.p, 0.0)
W = I - 1.0*jaccache
prectmp = ilu(W, τ = 50.0)
preccache = Ref(prectmp)

const τ1 = 5
function psetupilu(p, t, u, du, jok, jcurPtr, gamma)
    if !jok
        sparsejacprob.f.jac(jaccache, u, p, t)
        jcurPtr[] = true

        # W = I - gamma*J
        @. W = -gamma*jaccache
        idxs = diagind(W)
        @. @view(W[idxs]) = @view(W[idxs]) + 1

        # Build preconditioner on W
        preccache[] = ilu(W, τ = τ1)
    end
end
function precilu(z, r, p, t, y, fy, gamma, delta, lr)
    ldiv!(z, preccache[], r)
end

const τ2 = 5
function incompletelu(W, du, u, p, t, newW, Plprev, Prprev, solverdata)
    if newW === nothing || newW
        Pl = ilu(convert(AbstractMatrix, W), τ = τ2)
    else
        Pl = Plprev
    end
    Pl, nothing
end;

Sets tolerances

abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (5:8);

Work-Precision Diagrams (CVODE and lsoda solvers)

Declare solvers.

setups = [
    Dict(:alg=>lsoda(), :prob_choice => 1),
    Dict(:alg=>CVODE_BDF(), :prob_choice => 1),
    Dict(:alg=>CVODE_BDF(linear_solver = :LapackDense), :prob_choice => 1),
    Dict(:alg=>CVODE_BDF(linear_solver = :GMRES), :prob_choice => 1),
    Dict(
        :alg=>CVODE_BDF(linear_solver = :GMRES, prec = precilu, psetup = psetupilu, prec_side = 1),
        :prob_choice => 2),
    Dict(:alg=>CVODE_BDF(linear_solver = :KLU), :prob_choice => 3)
];

Plot Work-Precision Diagram.

wp = WorkPrecisionSet(
    [oprob, oprob_sparse, sparsejacprob], abstols, reltols, setups; error_estimate = :l2,
    saveat = tf/10000.0, appxsol = [test_sol, test_sol, test_sol], maxiters = Int(1e9), numruns = 10)

names = ["lsoda" "CVODE_BDF" "CVODE_BDF (LapackDense)" "CVODE_BDF (GMRES)" "CVODE_BDF (GMRES, iLU)" "CVODE_BDF (KLU, sparse jac)"]
plot(wp; label = names)

Work-Precision Diagrams (various Julia solvers)

Declare solvers (using default linear solver).

setups = [
    Dict(:alg=>TRBDF2(autodiff = false)),
    Dict(:alg=>QNDF(autodiff = false)),
    Dict(:alg=>FBDF(autodiff = false)),
    Dict(:alg=>KenCarp4(autodiff = false))
];

Plot Work-Precision Diagram (using default linear solver).

wp = WorkPrecisionSet(oprob, abstols, reltols, setups; error_estimate = :l2,
    saveat = tf/10000.0, appxsol = test_sol, maxiters = Int(1e12), dtmin = 1e-18, numruns = 10)

names = ["TRBDF2" "QNDF" "FBDF" "KenCarp4"]
plot(wp; label = names)

Declare solvers (using GMRES linear solver).

setups = [
    Dict(:alg=>TRBDF2(linsolve = KrylovJL_GMRES(), autodiff = false)),
    Dict(:alg=>QNDF(linsolve = KrylovJL_GMRES(), autodiff = false)),
    Dict(:alg=>FBDF(linsolve = KrylovJL_GMRES(), autodiff = false)),
    Dict(:alg=>KenCarp4(linsolve = KrylovJL_GMRES(), autodiff = false))
];

Plot Work-Precision Diagram (using GMRES linear solver).

wp = WorkPrecisionSet(oprob, abstols, reltols, setups; error_estimate = :l2,
    saveat = tf/10000.0, appxsol = test_sol, maxiters = Int(1e12), dtmin = 1e-18, numruns = 10)

names = ["TRBDF2 (GMRES)" "QNDF (GMRES)" "FBDF (GMRES)" "KenCarp4 (GMRES)"]
plot(wp; label = names)

Declare solvers (using GMRES linear solver, with pre-conditioner).

setups = [
    Dict(:alg=>TRBDF2(
        linsolve = KrylovJL_GMRES(), autodiff = false, precs = incompletelu, concrete_jac = true)),
    Dict(:alg=>QNDF(linsolve = KrylovJL_GMRES(), autodiff = false, precs = incompletelu, concrete_jac = true)),
    Dict(:alg=>FBDF(linsolve = KrylovJL_GMRES(), autodiff = false, precs = incompletelu, concrete_jac = true)),
    Dict(:alg=>KenCarp4(
        linsolve = KrylovJL_GMRES(), autodiff = false, precs = incompletelu, concrete_jac = true))
];

Plot Work-Precision Diagram (using GMRES linear solver, with pre-conditioner).

wp = WorkPrecisionSet(sparsejacprob, abstols, reltols, setups; error_estimate = :l2,
    saveat = tf/10000.0, appxsol = test_sol, maxiters = Int(1e12), dtmin = 1e-18, numruns = 10)

names = ["TRBDF2 (GMRES, iLU)" "QNDF (GMRES, iLU)" "FBDF (GMRES, iLU)" "KenCarp4 (GMRES, iLU)"]
plot(wp; label = names)

Declare solvers (using sparse jacobian)

We designate the solvers we wish to use.

setups = [
    Dict(:alg=>TRBDF2(linsolve = KLUFactorization(), autodiff = false)),
    Dict(:alg=>QNDF(linsolve = KLUFactorization(), autodiff = false)),
    Dict(:alg=>FBDF(linsolve = KLUFactorization(), autodiff = false)),
    Dict(:alg=>KenCarp4(linsolve = KLUFactorization(), autodiff = false))
];

Plot Work-Precision Diagram (using sparse jacobian)

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

wp = WorkPrecisionSet(sparsejacprob, abstols, reltols, setups; error_estimate = :l2,
    saveat = tf/10000.0, appxsol = test_sol, maxiters = Int(1e12), dtmin = 1e-18, numruns = 10)

names = ["TRBDF2 (KLU, sparse jac)" "QNDF (KLU, sparse jac)" "FBDF (KLU, sparse jac)" "KenCarp4 (KLU, sparse jac)"]
plot(wp; label = names)

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=>CVODE_Adams()),
    Dict(:alg=>Tsit5()),
    Dict(:alg=>BS5()),
    Dict(:alg=>VCABM()),
    Dict(:alg=>Vern6()),
    Dict(:alg=>Vern7()),
    Dict(:alg=>Vern8()),
    Dict(:alg=>Vern9())
];

Plot Work-Precision Diagram

wp = WorkPrecisionSet(oprob, abstols, reltols, setups; error_estimate = :l2,
    saveat = tf/10000.0, appxsol = test_sol, maxiters = Int(1e9), numruns = 10)

names = ["CVODE_Adams" "Tsit5" "BS5" "Vern6" "Vern7" "Vern8" "Vern9"]
plot(wp; label = names)

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=>CVODE_BDF(linear_solver = :GMRES), :prob_choice => 1),
    Dict(
        :alg=>CVODE_BDF(linear_solver = :GMRES, prec = precilu, psetup = psetupilu, prec_side = 1),
        :prob_choice => 2),
    Dict(
        :alg=>QNDF(linsolve = KrylovJL_GMRES(), autodiff = false, precs = incompletelu, concrete_jac = true),
        :prob_choice => 3),
    Dict(
        :alg=>FBDF(linsolve = KrylovJL_GMRES(), autodiff = false, precs = incompletelu, concrete_jac = true),
        :prob_choice => 3),
    Dict(:alg=>Tsit5())
];

Plot Work-Precision Diagram

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

wp = WorkPrecisionSet(
    [oprob, oprob_sparse, sparsejacprob], abstols, reltols, setups; error_estimate = :l2,
    saveat = tf/10000.0, appxsol = [test_sol, test_sol, test_sol], maxiters = Int(1e9), numruns = 200)

names = ["CVODE_BDF (GMRES)" "CVODE_BDF (GMRES, iLU)" "QNDF (GMRES, iLU)" "FBDF (GMRES, iLU)" "Tsit5"]
colors = [:darkgreen :green :deepskyblue1 :dodgerblue2 :orchid2]
markershapes = [:rect :octagon :hexagon :rtriangle :ltriangle]
plot(wp; label = names, left_margin = 10Plots.mm, right_margin = 10Plots.mm,
    xticks = [1e-9, 1e-8, 1e-7, 1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1],
    yticks = [1e-1, 1e0, 1e1, 1e2], 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))

echo = false
using SciMLBenchmarks
SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder], WEAVE_ARGS[:file])

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","fceri_gamma2.jmd")

Computer Information:

Julia Version 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/Bio/Project.toml`
⌃ [6e4b80f9] BenchmarkTools v1.5.0
⌅ [479239e8] Catalyst v14.4.1
⌃ [2b5f629d] DiffEqBase v6.160.0
⌃ [f3b72e0c] DiffEqDevTools v2.45.1
  [40713840] IncompleteLU v0.2.1
⌃ [033835bb] JLD2 v0.5.8
  [7f56f5a3] LSODA v0.7.5
⌅ [7ed4a6bd] LinearSolve v2.37.0
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⌃ [1dea7af3] OrdinaryDiffEq v6.90.1
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⌃ [c3572dad] Sundials v4.26.1
⌃ [a759f4b9] TimerOutputs v0.5.25
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.

And the full manifest:

Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/Bio/Manifest.toml`
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.