BCR Work-Precision Diagrams
The following benchmark is of 1122 ODEs with 24388 terms that describe a stiff chemical reaction network modeling the BCR signaling network from Barua 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
using OrdinaryDiffEqBDF, OrdinaryDiffEqSDIRK
gr()
datadir = joinpath(dirname(pathof(ReactionNetworkImporters)), "../data/bcr")
const to = TimerOutput()
tf = 100000.0
# generate ModelingToolkit ODEs
@timeit to "Parse Network" prnbng = loadrxnetwork(BNGNetwork(), joinpath(datadir, "bcr.net"))
show(to)
rn = complete(prnbng)
obs = [eq.lhs for eq in observed(rn)]
@timeit to "Create ODESys" osys = complete(Catalyst.ode_model(rn))
show(to)
tspan = (0.0, tf)
@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);Scanning blocks...done
Parsing parameters...done
Creating parameters...done
Parsing species...done
Creating variables...done
Setting up expression bindings...done
Parsing groups...done
Parsing functions...done
Parsing and adding reactions...done
──────────────────────────────────────────────────────────────────────────
Time Allocations
─────────────────────── ────────────────────────
Tot / % measured: 6.56s / 99.6% 525MiB / 100.0%
Section ncalls time %tot avg alloc %tot avg
──────────────────────────────────────────────────────────────────────────
Parse Network 1 6.53s 100.0% 6.53s 525MiB 100.0% 525MiB
───────────────────────────────────────────────────────────────────────────
─────────────────────────────────────────────────────────────────────────
Time Allocations
─────────────────────── ────────────────────────
Tot / % measured: 23.8s / 80.7% 2.18GiB / 76.0%
Section ncalls time %tot avg alloc %tot avg
──────────────────────────────────────────────────────────────────────────
Create ODESys 1 12.7s 66.0% 12.7s 1.14GiB 69.0% 1.14GiB
Parse Network 1 6.53s 34.0% 6.53s 525MiB 31.0% 525MiB
───────────────────────────────────────────────────────────────────────────
──────────────────────────────────────────────────────────────────────────
Time Allocations
─────────────────────── ────────────────────────
Tot / % measured: 64.6s / 92.9% 4.63GiB / 88.7%
Section ncalls time %tot avg alloc %tot avg
───────────────────────────────────────────────────────────────────────────
ODEProb No Jac 1 40.8s 68.0% 40.8s 2.45GiB 59.7% 2.45GiB
Create ODESys 1 12.7s 21.2% 12.7s 1.14GiB 27.8% 1.14GiB
Parse Network 1 6.53s 10.9% 6.53s 525MiB 12.5% 525MiB
───────────────────────────────────────────────────────────────────────────@timeit to "ODEProb SparseJac" sparsejacprob = ODEProblem{true, SciMLBase.FullSpecialize}(
osys, Float64[], tspan, Float64[], jac = true, sparse = true)
show(to)───────────────────────────────────────────────────────────────────────────
───
Time Allocations
─────────────────────── ─────────────────────
───
Tot / % measured: 139s / 91.2% 11.8GiB / 86.9%
Section ncalls time %tot avg alloc %tot
avg
───────────────────────────────────────────────────────────────────────────
───
ODEProb SparseJac 1 67.2s 52.8% 67.2s 6.17GiB 60.0% 6.17
GiB
ODEProb No Jac 1 40.8s 32.1% 40.8s 2.45GiB 23.9% 2.45
GiB
Create ODESys 1 12.7s 10.0% 12.7s 1.14GiB 11.1% 1.14
GiB
Parse Network 1 6.53s 5.1% 6.53s 525MiB 5.0% 525
MiB
───────────────────────────────────────────────────────────────────────────
───@show numspecies(rn) # Number of ODEs
@show numreactions(rn) # Approx. number of terms in the ODE
@show length(parameters(rn)); # Number of Parametersnumspecies(rn) = 1122
numreactions(rn) = 24388
length(parameters(rn)) = 128Time 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" oprob.f(du, u, p, 0.0)
@timeit to "ODE rhs spjac Eval1" sparsejacprob.f(du, u, p, 0.0)
show(to)───────────────────────────────────────────────────────────────────────────
─────
Time Allocations
─────────────────────── ───────────────────
─────
Tot / % measured: 239s / 94.9% 13.5GiB / 88.5
%
Section ncalls time %tot avg alloc %tot
avg
───────────────────────────────────────────────────────────────────────────
─────
ODE rhs Eval1 1 99.2s 43.8% 99.2s 1.64GiB 13.7% 1.
64GiB
ODEProb SparseJac 1 67.2s 29.7% 67.2s 6.17GiB 51.8% 6.
17GiB
ODEProb No Jac 1 40.8s 18.0% 40.8s 2.45GiB 20.6% 2.
45GiB
Create ODESys 1 12.7s 5.6% 12.7s 1.14GiB 9.6% 1.
14GiB
Parse Network 1 6.53s 2.9% 6.53s 525MiB 4.3% 5
25MiB
ODE rhs spjac Eval1 1 13.2ms 0.0% 13.2ms 123KiB 0.0% 1
23KiB
───────────────────────────────────────────────────────────────────────────
─────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, 0.0)55.530 μs (2 allocations: 336 bytes)
1122-element Vector{Float64}:
-61.028082045662714
5.941959152600332e-5
-0.00017585525801146217
1.0927353287115693e-5
-2.375821949685643e-10
0.0
-0.021580601559854376
-1.674730187857901e-11
-2.0210128871802965e-9
-0.021503434567899174
⋮
-1.3438399696192487e-31
-1.821222651639453e-23
-2.0469554560482396e-23
-1.5706899341477363e-14
-3.706198991389375e-21
-3.422528102939304e-13
-7.529912023768182e-13
-5.53409527730824e-23
-1.6608734902441886e-17Js = similar(sparsejacprob.f.jac_prototype)
@timeit to "SparseJac Eval1" sparsejacprob.f.jac(Js, u, p, 0.0)
@timeit to "SparseJac Eval2" sparsejacprob.f.jac(Js, u, p, 0.0)
show(to)───────────────────────────────────────────────────────────────────────────
─────
Time Allocations
─────────────────────── ───────────────────
─────
Tot / % measured: 355s / 94.2% 14.2GiB / 88.6
%
Section ncalls time %tot avg alloc %tot
avg
───────────────────────────────────────────────────────────────────────────
─────
SparseJac Eval1 1 108s 32.3% 108s 675MiB 5.2% 6
75MiB
ODE rhs Eval1 1 99.2s 29.6% 99.2s 1.64GiB 13.0% 1.
64GiB
ODEProb SparseJac 1 67.2s 20.1% 67.2s 6.17GiB 49.1% 6.
17GiB
ODEProb No Jac 1 40.8s 12.2% 40.8s 2.45GiB 19.5% 2.
45GiB
Create ODESys 1 12.7s 3.8% 12.7s 1.14GiB 9.1% 1.
14GiB
Parse Network 1 6.53s 2.0% 6.53s 525MiB 4.1% 5
25MiB
ODE rhs spjac Eval1 1 13.2ms 0.0% 13.2ms 123KiB 0.0% 1
23KiB
SparseJac Eval2 1 108μs 0.0% 108μs 912B 0.0%
912B
───────────────────────────────────────────────────────────────────────────
─────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)
Generate Test Solution
@time sol = solve(oprob, CVODE_BDF(), abstol = 1/10^12, reltol = 1/10^12)
test_sol = TestSolution(sol);842.433575 seconds (6.70 M allocations: 2.130 GiB, 0.19% gc time, 0.09% com
pilation time)Setups
Sets plotting defaults
default(legendfontsize = 7, framestyle = :box, gridalpha = 0.3, gridlinewidth = 2.5)Declare pre-conditioners
using IncompleteLU, LinearAlgebra
const τ = 1e2
const τ2 = 1e2
jaccache = sparsejacprob.f.jac(oprob.u0, oprob.p, 0.0)
W = I - 1.0*jaccache
prectmp = ilu(W, τ = τ)
preccache = Ref(prectmp)
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 = τ)
end
end
function precilu(z, r, p, t, y, fy, gamma, delta, lr)
ldiv!(z, preccache[], r)
end
function incompletelu(A, p)
Pl = ilu(convert(AbstractMatrix, A); τ = τ2)
return Pl, I
end;Sets tolerances
abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (5:8);Work-Precision Diagrams (competitive solvers)
Main suites: methods that remain competitive on this large sparse stiff system. Everything else is timed once, in isolation, in the loser section below.
lsoda and the default dense CVODE_BDF used to be swept across this whole tolerance grid. Measured on CI (2026-07-27 build, per point): lsoda 500-1200 s and CVODE_BDF 210-370 s, against 9-22 s for CVODE_BDF with GMRES + incomplete LU. That one panel alone cost 2 h 44 min of the document's 7 h 57 min, so both were moved to the isolation section. TRBDF2 (880 s / 975 s summed over the grid for the GMRES+iLU and KLU variants) and KenCarp4 with GMRES+iLU (870 s) were removed from the Julia panels for the same reason - roughly 7x QNDF/FBDF in the same configuration.
GMRES + incomplete LU
setups = [
Dict(
:alg=>CVODE_BDF(linear_solver = :GMRES, prec = precilu, psetup = psetupilu, prec_side = 1),
:prob_choice => 2),
Dict(:alg=>QNDF(linsolve = KrylovJL_GMRES(; precs = incompletelu), autodiff = AutoFiniteDiff(), concrete_jac = true),
:prob_choice => 3),
Dict(:alg=>FBDF(linsolve = KrylovJL_GMRES(; precs = incompletelu), autodiff = AutoFiniteDiff(), concrete_jac = true),
:prob_choice => 3),
Dict(:alg=>NordsieckBDF(linsolve = KrylovJL_GMRES(; precs = incompletelu), autodiff = AutoFiniteDiff(), concrete_jac = true),
:prob_choice => 3)
];wp = WorkPrecisionSet(
[oprob, oprob_sparse, sparsejacprob], abstols, reltols, setups; error_estimate = :l2,
saveat = tf/1000.0, appxsol = [test_sol, test_sol, test_sol], maxiters = Int(1e6), numruns = 1)
names = ["CVODE_BDF (GMRES, iLU)" "QNDF (GMRES, iLU)" "FBDF (GMRES, iLU)" "NordsieckBDF (GMRES, iLU)"]
plot(wp; label = names)
Sparse Jacobian + KLU
setups = [
Dict(:alg=>CVODE_BDF(linear_solver = :KLU)),
Dict(:alg=>QNDF(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff())),
Dict(:alg=>FBDF(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff())),
Dict(:alg=>NordsieckBDF(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff())),
Dict(:alg=>KenCarp4(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff()))
];wp = WorkPrecisionSet(sparsejacprob, abstols, reltols, setups; error_estimate = :l2,
saveat = tf/1000.0, appxsol = test_sol, maxiters = Int(1e6), numruns = 1)
names = ["CVODE_BDF (KLU, sparse jac)" "QNDF (KLU, sparse jac)" "FBDF (KLU, sparse jac)" "NordsieckBDF (KLU, sparse jac)" "KenCarp4 (KLU, sparse jac)"]
plot(wp; label = names)
Loser methods (large cost in isolation)
On this ~1122-ODE sparse chemistry system the following are not competitive with the sparse/preconditioned setups above: lsoda, dense direct CVODE (default and Lapack), default dense Julia Newton/linear solves, GMRES without a preconditioner, and TRBDF2/KenCarp4 even with the good linear solvers. We do not fold them into the main work-precision suites. Instead each is timed once, in isolation, at a fixed tolerance, next to a competitive sparse reference so the wall-time gap is visible.
The size of that gap varies a lot between methods. In the latest run, relative to the FBDF + KLU reference: lsoda took roughly 10x as long; dense CVODE_BDF, default dense TRBDF2 and every unpreconditioned-GMRES entry hit the wall-clock cap; KenCarp4 (default dense), TRBDF2/KenCarp4 with GMRES + iLU and TRBDF2 with KLU took roughly 2-4x as long; while CVODE_BDF with LapackDense and QNDF/FBDF with their default dense linear solver were within about 1.4x of the reference at this single tolerance.
Each isolated solve is capped by wall clock. In the 2026-07-27 CI build the uncapped versions of the four unpreconditioned-GMRES entries alone measured 2110 s, 3763 s, 4707 s and 7267 s - about 4.9 h for four data points, i.e. this "cheap isolation" section had itself become one of the most expensive parts of the document. A solve that hits the cap is reported as a lower bound (>cap), which is all the diagram needs in order to show the gap.
const _loser_tol = 1e-6
const _loser_maxiters = Int(1e6)
const _loser_cap = 180.0 # seconds of wall clock per isolated solve
loser_labels = String[]
loser_elapsed = Float64[]
# LSODA.jl does not support callbacks, so `lsoda` is the one entry that has to
# run to completion; everything else is stopped by the wall-clock callback.
function _time_loser!(label, prob, alg; cap = true)
println("--- $label ---")
tstart = time()
kw = if cap
capcb = DiscreteCallback(
(u, t, integrator) -> time() - tstart > _loser_cap,
integrator -> terminate!(integrator); save_positions = (false, false))
(; callback = capcb)
else
(;)
end
t = @elapsed sol = solve(prob, alg; abstol = _loser_tol, reltol = _loser_tol,
maxiters = _loser_maxiters, save_everystep = false, kw...)
hit_cap = cap && t >= _loser_cap
@show sol.retcode
println("elapsed = ", t, " s", hit_cap ? " (hit the $(_loser_cap) s cap)" : "")
push!(loser_labels, hit_cap ? label * " (>cap)" : label)
push!(loser_elapsed, t)
return sol
end
# Competitive reference (sparse KLU)
_time_loser!("FBDF + KLU (reference)", sparsejacprob,
FBDF(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff()))
# Multistep dense direct solvers
_time_loser!("lsoda", oprob, lsoda(); cap = false)
_time_loser!("CVODE_BDF (dense)", oprob, CVODE_BDF())
_time_loser!("CVODE_BDF LapackDense", oprob, CVODE_BDF(linear_solver = :LapackDense))
# Bare CVODE GMRES (no preconditioner)
_time_loser!("CVODE_BDF GMRES (no prec)", oprob, CVODE_BDF(linear_solver = :GMRES))
# Default dense Julia factorizations on the non-sparse problem
_time_loser!("TRBDF2 (default dense)", oprob, TRBDF2(autodiff = AutoFiniteDiff()))
_time_loser!("QNDF (default dense)", oprob, QNDF(autodiff = AutoFiniteDiff()))
_time_loser!("FBDF (default dense)", oprob, FBDF(autodiff = AutoFiniteDiff()))
_time_loser!("KenCarp4 (default dense)", oprob, KenCarp4(autodiff = AutoFiniteDiff()))
# Unpreconditioned GMRES on the dense residual problem
_time_loser!("TRBDF2 GMRES (no prec)", oprob,
TRBDF2(linsolve = KrylovJL_GMRES(), autodiff = AutoFiniteDiff()))
_time_loser!("QNDF GMRES (no prec)", oprob,
QNDF(linsolve = KrylovJL_GMRES(), autodiff = AutoFiniteDiff()))
_time_loser!("FBDF GMRES (no prec)", oprob,
FBDF(linsolve = KrylovJL_GMRES(), autodiff = AutoFiniteDiff()))
_time_loser!("KenCarp4 GMRES (no prec)", oprob,
KenCarp4(linsolve = KrylovJL_GMRES(), autodiff = AutoFiniteDiff()))
# Slow methods with the *good* linear solvers, dropped from the panels above
_time_loser!("TRBDF2 (GMRES, iLU)", sparsejacprob,
TRBDF2(linsolve = KrylovJL_GMRES(; precs = incompletelu), autodiff = AutoFiniteDiff(),
concrete_jac = true))
_time_loser!("KenCarp4 (GMRES, iLU)", sparsejacprob,
KenCarp4(linsolve = KrylovJL_GMRES(; precs = incompletelu), autodiff = AutoFiniteDiff(),
concrete_jac = true))
_time_loser!("TRBDF2 (KLU, sparse jac)", sparsejacprob,
TRBDF2(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff()))--- FBDF + KLU (reference) ---
sol.retcode = SciMLBase.ReturnCode.Success
elapsed = 88.411921846 s
--- lsoda ---
sol.retcode = SciMLBase.ReturnCode.Success
elapsed = 612.852325055 s
--- CVODE_BDF (dense) ---
sol.retcode = SciMLBase.ReturnCode.Terminated
elapsed = 180.11362113 s (hit the 180.0 s cap)
--- CVODE_BDF LapackDense ---
sol.retcode = SciMLBase.ReturnCode.Success
elapsed = 69.813875676 s
--- CVODE_BDF GMRES (no prec) ---
sol.retcode = SciMLBase.ReturnCode.Terminated
elapsed = 180.000300542 s (hit the 180.0 s cap)
--- TRBDF2 (default dense) ---
sol.retcode = SciMLBase.ReturnCode.Terminated
elapsed = 180.017240579 s (hit the 180.0 s cap)
--- QNDF (default dense) ---
sol.retcode = SciMLBase.ReturnCode.Success
elapsed = 67.963230758 s
--- FBDF (default dense) ---
sol.retcode = SciMLBase.ReturnCode.Success
elapsed = 83.894787238 s
--- KenCarp4 (default dense) ---
sol.retcode = SciMLBase.ReturnCode.Success
elapsed = 131.954577134 s
--- TRBDF2 GMRES (no prec) ---
sol.retcode = SciMLBase.ReturnCode.Terminated
elapsed = 180.067492574 s (hit the 180.0 s cap)
--- QNDF GMRES (no prec) ---
sol.retcode = SciMLBase.ReturnCode.Terminated
elapsed = 180.10812495 s (hit the 180.0 s cap)
--- FBDF GMRES (no prec) ---
sol.retcode = SciMLBase.ReturnCode.Terminated
elapsed = 180.294773986 s (hit the 180.0 s cap)
--- KenCarp4 GMRES (no prec) ---
sol.retcode = SciMLBase.ReturnCode.Terminated
elapsed = 180.129939635 s (hit the 180.0 s cap)
--- TRBDF2 (GMRES, iLU) ---
sol.retcode = SciMLBase.ReturnCode.Success
elapsed = 141.712205903 s
--- KenCarp4 (GMRES, iLU) ---
sol.retcode = SciMLBase.ReturnCode.Success
elapsed = 159.864261982 s
--- TRBDF2 (KLU, sparse jac) ---
sol.retcode = SciMLBase.ReturnCode.Terminated
elapsed = 180.058387272 s (hit the 180.0 s cap)
retcode: Terminated
Interpolation: 1st order linear
t: 2-element Vector{Float64}:
0.0
84399.59836499131
u: 2-element Vector{Vector{Float64}}:
[299717.8348854, 47149.15480798, 46979.01102231, 290771.2428252, 299980.73
96749, 300000.0, 141.3151575495, 0.1256496403614, 0.4048783555301, 140.8052
338618 … 5.279974499715e-11, 1.005585387399e-24, 6.724953378237e-17, 3.39
5560698281e-16, 1.787990228838e-5, 8.761844379939e-13, 0.0002517949074779,
0.0005539124513976, 2.281251822741e-14, 1.78232055967e-8]
[299999.06489081174, 118.92753626531007, 118.92753891743403, 291731.860806
9263, 291779.4664725298, 299999.99999997875, 0.3576170509290214, 7.97718681
3255676e-7, 2.4468444059256953e-6, 0.3576170590425217 … 7.809348291941137
e-18, 5.808696030007352e-31, -6.197005263515671e-26, -1.1508763549970694e-2
5, 1.5026176928674508e-12, 1.3483982121473098e-23, 6.993403462903959e-12, 1
.5241483830366825e-11, 4.24399476747901e-24, 1.5025076845753974e-15]# Relative cost vs the sparse KLU reference (first entry)
ref_t = loser_elapsed[1]
bar(loser_labels, loser_elapsed ./ ref_t; xrotation = 45, legend = false,
ylabel = "wall time / (FBDF+KLU reference)",
title = "BCR loser isolation (tol=$_loser_tol, one capped solve each)",
size = (900, 500), left_margin = 5Plots.mm, bottom_margin = 15Plots.mm)
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, prec = precilu, psetup = psetupilu, prec_side = 1),
:prob_choice => 2),
Dict(
:alg=>QNDF(linsolve = KrylovJL_GMRES(; precs = incompletelu), autodiff = AutoFiniteDiff(), concrete_jac = true),
:prob_choice => 3),
Dict(
:alg=>FBDF(linsolve = KrylovJL_GMRES(; precs = incompletelu), autodiff = AutoFiniteDiff(), concrete_jac = true),
:prob_choice => 3),
Dict(
:alg=>NordsieckBDF(linsolve = KrylovJL_GMRES(; precs = incompletelu), autodiff = AutoFiniteDiff(), concrete_jac = true),
:prob_choice => 3),
Dict(:alg=>CVODE_BDF(linear_solver = :KLU), :prob_choice => 3),
Dict(:alg=>QNDF(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff()), :prob_choice => 3),
Dict(:alg=>FBDF(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff()), :prob_choice => 3),
Dict(:alg=>NordsieckBDF(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff()), :prob_choice => 3),
Dict(:alg=>KenCarp4(linsolve = KLUFactorization(), autodiff = AutoFiniteDiff()), :prob_choice => 3)
];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/1000.0, appxsol = [test_sol, test_sol, test_sol], maxiters = Int(1e9), numruns = 200)
names = ["CVODE_BDF (GMRES, iLU)" "QNDF (GMRES, iLU)" "FBDF (GMRES, iLU)" "NordsieckBDF (GMRES, iLU)" "CVODE_BDF (KLU, sparse jac)" "QNDF (KLU, sparse jac)" "FBDF (KLU, sparse jac)" "NordsieckBDF (KLU, sparse jac)" "KenCarp4 (KLU, sparse jac)"]
colors = [:green :deepskyblue1 :dodgerblue2 :mediumorchid :seagreen :royalblue2 :slateblue3 :orchid :lightskyblue]
markershapes = [:octagon :hexagon :rtriangle :diamond :circle :pentagon :ltriangle :dtriangle :star5]
plot(wp; label = names, left_margin = 10Plots.mm, right_margin = 10Plots.mm,
xticks = [1e-3, 1e-2, 1e-1, 1e0, 1e1, 1e2, 1e3], yticks = [1e0, 1e1, 1e2, 1e3],
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","BCR.jmd")Computer Information:
Julia Version 1.10.12
Commit d93beab124c (2026-08-15 10:29 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: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
JULIA_NUM_THREADS = auto
Package Information:
Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Bio/Project.toml`
[47edcb42] ADTypes v1.24.0
[6e4b80f9] BenchmarkTools v1.8.0
[479239e8] Catalyst v16.4.3
[d360d2e6] ChainRulesCore v1.26.1
⌃ [2b5f629d] DiffEqBase v7.21.1
[f3b72e0c] DiffEqDevTools v3.6.3
[40713840] IncompleteLU v0.2.1
⌃ [033835bb] JLD2 v0.6.6
[7f56f5a3] LSODA v1.2.0
⌃ [7ed4a6bd] LinearSolve v5.17.3
⌃ [961ee093] ModelingToolkit v11.43.1
[54ca160b] ODEInterface v0.5.2
⌅ [09606e27] ODEInterfaceDiffEq v4.1.0
[1dea7af3] OrdinaryDiffEq v7.8.1
[89bda076] OrdinaryDiffEqAdamsBashforthMoulton v2.2.0
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.9
⌃ [bbf590c4] OrdinaryDiffEqCore v4.17.2
[becaefa8] OrdinaryDiffEqExtrapolation v2.6.3
[1344f307] OrdinaryDiffEqLowOrderRK v2.2.5
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.7.3
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.3
⌃ [358294b1] OrdinaryDiffEqStabilizedRK v2.7.0
[79d7bb75] OrdinaryDiffEqVerner v2.4.1
[91a5bcdd] Plots v1.41.7
[b4db0fb7] ReactionNetworkImporters v1.5.0
[f2c3362d] RecursiveFactorization v0.2.30
[31c91b34] SciMLBenchmarks v0.2.1
[c3572dad] Sundials v6.7.1
⌅ [a759f4b9] TimerOutputs v0.5.29
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`And the full manifest:
Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Bio/Manifest.toml`
[47edcb42] ADTypes v1.24.0
[14f7f29c] AMD v0.5.4
[6e696c72] AbstractPlutoDingetjes v1.4.1
[1520ce14] AbstractTrees v0.4.5
[7d9f7c33] Accessors v0.1.45
⌃ [79e6a3ab] Adapt v4.7.0
[66dad0bd] AliasTables v1.1.3
[ec485272] ArnoldiMethod v0.4.0
⌃ [4fba245c] ArrayInterface v7.30.1
⌃ [4c555306] ArrayLayouts v1.12.2
⌃ [aae01518] BandedMatrices v1.12.0
[6e4b80f9] BenchmarkTools v1.8.0
[e2ed5e7c] Bijections v0.2.2
[b2a6c25c] BinaryHeaps v1.1.0
[caf10ac8] BipartiteGraphs v0.1.14
[62783981] BitTwiddlingConvenienceFunctions v0.1.6
[8e7c35d0] BlockArrays v1.10.0
[70df07ce] BracketingNonlinearSolve v1.12.7
[fa961155] CEnum v0.5.0
[2a0fbf3d] CPUSummary v0.2.7
[479239e8] Catalyst v16.4.3
[d360d2e6] ChainRulesCore v1.26.1
[0b6fb165] ChunkCodecCore v1.0.2
[4c0bbee4] ChunkCodecLibZlib v1.1.0
[55437552] ChunkCodecLibZstd v1.0.0
[fb6a15b2] CloseOpenIntervals v0.1.13
[35d6a980] ColorSchemes v3.31.0
⌃ [3da002f7] ColorTypes v0.12.1
[c3611d14] ColorVectorSpace v0.11.0
⌃ [5ae59095] Colors v0.13.1
⌅ [861a8166] Combinatorics v1.0.2
[38540f10] CommonSolve v0.2.14
[bbf7d656] CommonSubexpressions v0.3.1
[f70d9fcc] CommonWorldInvalidations v1.2.2
[34da2185] Compat v4.18.1
[b152e2b5] CompositeTypes v0.1.4
[a33af91c] CompositionsBase v0.1.2
[2569d6c7] ConcreteStructs v0.2.8
[187b0558] ConstructionBase v1.6.0
[d38c429a] Contour v0.6.3
[adafc99b] CpuId v0.3.1
[9a962f9c] DataAPI v1.16.0
[864edb3b] DataStructures v0.19.6
[e2d170a0] DataValueInterfaces v1.0.0
[8bb1440f] DelimitedFiles v1.9.1
⌃ [2b5f629d] DiffEqBase v7.21.1
[459566f4] DiffEqCallbacks v4.19.4
[f3b72e0c] DiffEqDevTools v3.6.3
⌃ [77a26b50] DiffEqNoiseProcess v5.36.3
[163ba53b] DiffResults v1.1.0
[b552c78f] DiffRules v1.16.0
[a0c0ee7d] DifferentiationInterface v0.7.21
⌃ [8d63f2c5] DispatchDoctor v0.4.28
[31c24e10] Distributions v0.25.131
[ffbed154] DocStringExtensions v0.9.5
⌃ [5b8099bc] DomainSets v0.8.1
[7c1d4256] DynamicPolynomials v0.6.8
[06fc5a27] DynamicQuantities v1.13.0
[4e289a0a] EnumX v1.0.7
[f151be2c] EnzymeCore v0.8.21
[e2ba6199] ExprTools v0.1.11
[55351af7] ExproniconLite v0.10.14
[c87230d0] FFMPEG v0.4.5
[7034ab61] FastBroadcast v1.4.0
[9aa1b823] FastClosures v0.3.2
[a4df4552] FastPower v1.5.0
[5789e2e9] FileIO v1.20.0
⌃ [1a297f60] FillArrays v1.17.0
⌃ [64ca27bc] FindFirstFunctions v3.2.1
[6a86dc24] FiniteDiff v2.33.0
⌅ [53c48c17] FixedPointNumbers v0.8.6
[1fa38f19] Format v1.3.7
[f6369f11] ForwardDiff v1.4.6
[a85aefff] FunctionMaps v0.1.2
[069b7b12] FunctionWrappers v1.1.3
[77dc65aa] FunctionWrappersWrappers v1.13.0
⌃ [46192b85] GPUArraysCore v0.2.0
[28b8d3ca] GR v0.73.27
[a0844989] Gamma v1.2.0
[86223c79] Graphs v1.15.0
[076d061b] HashArrayMappedTries v0.2.0
⌅ [eafb193a] Highlights v0.5.3
[3e5b6fbb] HostCPUFeatures v0.1.18
[34004b35] HypergeometricFunctions v0.3.30
[615f187c] IfElse v0.1.1
[3263718b] ImplicitDiscreteSolve v2.3.0
[40713840] IncompleteLU v0.2.1
[d25df0c9] Inflate v0.1.5
[18e54dd8] IntegerMathUtils v0.1.4
[8197267c] IntervalSets v0.7.14
[3587e190] InverseFunctions v0.1.17
[92d709cd] IrrationalConstants v0.2.6
[82899510] IteratorInterfaceExtensions v1.0.0
⌃ [033835bb] JLD2 v0.6.6
[1019f520] JLFzf v0.1.11
[692b3bcd] JLLWrappers v1.8.0
⌅ [682c06a0] JSON v0.21.4
[ae98c720] Jieko v0.2.1
⌃ [ccbc3e58] JumpProcesses v9.32.3
[ba0b0d4f] Krylov v0.10.10
[2faa5264] LHLFactorization v2.2.2
[7f56f5a3] LSODA v1.2.0
[b964fa9f] LaTeXStrings v1.4.1
[23fbe1c1] Latexify v0.16.12
[10f19ff3] LayoutPointers v0.1.17
[87fe0de2] LineSearch v0.1.18
⌃ [7ed4a6bd] LinearSolve v5.17.3
⌃ [2ab3a3ac] LogExpFunctions v1.0.1
[e6f89c97] LoggingExtras v1.2.0
[bdcacae8] LoopVectorization v0.12.174
[1914dd2f] MacroTools v0.5.16
[d125e4d3] ManualMemory v0.1.8
[bb5d69b7] MaybeInplace v0.1.8
[442fdcdd] Measures v0.3.3
[e1d29d7a] Missings v1.2.0
⌃ [961ee093] ModelingToolkit v11.43.1
⌃ [7771a370] ModelingToolkitBase v1.70.0
[6bb917b9] ModelingToolkitTearing v1.20.6
⌅ [2e0e35c7] Moshi v0.3.9
[46d2c3a1] MuladdMacro v0.2.7
⌃ [102ac46a] MultivariatePolynomials v0.5.19
⌃ [ffc61752] Mustache v1.0.21
⌃ [d8a4904e] MutableArithmetics v1.8.0
[77ba4419] NaNMath v1.1.4
⌃ [8913a72c] NonlinearSolve v4.30.0
⌃ [be0214bd] NonlinearSolveBase v2.48.0
⌃ [5959db7a] NonlinearSolveFirstOrder v2.6.1
[9a2c21bd] NonlinearSolveQuasiNewton v1.15.3
[26075421] NonlinearSolveSpectralMethods v1.8.3
[54ca160b] ODEInterface v0.5.2
⌅ [09606e27] ODEInterfaceDiffEq v4.1.0
[6fe1bfb0] OffsetArrays v1.17.0
⌅ [bac558e1] OrderedCollections v1.8.2
[1dea7af3] OrdinaryDiffEq v7.8.1
[89bda076] OrdinaryDiffEqAdamsBashforthMoulton v2.2.0
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.9
⌃ [bbf590c4] OrdinaryDiffEqCore v4.17.2
[50262376] OrdinaryDiffEqDefault v2.6.2
⌃ [4302a76b] OrdinaryDiffEqDifferentiation v3.12.0
[becaefa8] OrdinaryDiffEqExtrapolation v2.6.3
[1344f307] OrdinaryDiffEqLowOrderRK v2.2.5
[127b3ac7] OrdinaryDiffEqNonlinearSolve v2.9.8
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.7.3
[b4bd8bb3] OrdinaryDiffEqRosenbrockTableaus v2.4.2
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.3
⌃ [358294b1] OrdinaryDiffEqStabilizedRK v2.7.0
[b1df2697] OrdinaryDiffEqTsit5 v2.1.4
[79d7bb75] OrdinaryDiffEqVerner v2.4.1
[90014a1f] PDMats v0.11.41
⌅ [d96e819e] Parameters v0.12.3
⌅ [69de0a69] Parsers v2.8.8
[ccf2f8ad] PlotThemes v3.3.0
⌃ [995b91a9] PlotUtils v1.4.4
[91a5bcdd] Plots v1.41.7
[e409e4f3] PoissonRandom v0.4.13
[f517fe37] Polyester v0.7.19
[1d0040c9] PolyesterWeave v0.2.2
[d236fae5] PreallocationTools v1.7.1
⌅ [aea7be01] PrecompileTools v1.2.1
[21216c6a] Preferences v1.6.0
[27ebfcd6] Primes v0.5.7
[43287f4e] PtrArrays v1.4.0
⌃ [0c0d3e7f] PureKLU v1.5.0
[1fd47b50] QuadGK v2.11.3
[b4db0fb7] ReactionNetworkImporters v1.5.0
[988b38a3] ReadOnlyArrays v0.2.0
[795d4caa] ReadOnlyDicts v1.0.1
[3cdcf5f2] RecipesBase v1.3.4
[01d81517] RecipesPipeline v0.6.12
⌃ [731186ca] RecursiveArrayTools v4.5.1
[f2c3362d] RecursiveFactorization v0.2.30
[189a3867] Reexport v1.2.2
[05181044] RelocatableFolders v1.0.1
[ae029012] Requires v1.3.1
[ae5879a3] ResettableStacks v1.4.0
[9fe22ead] RespecializeParams v1.3.0
[79098fc4] Rmath v0.9.0
[47965b36] RootedTrees v2.27.0
[f2b01f46] Roots v3.0.8
⌃ [7e49a35a] RuntimeGeneratedFunctions v0.5.26
⌃ [9dfe8606] SCCNonlinearSolve v1.15.3
[94e857df] SIMDTypes v0.1.0
[476501e8] SLEEFPirates v0.6.46
⌃ [0bca4576] SciMLBase v3.54.0
[31c91b34] SciMLBenchmarks v0.2.1
[19f34311] SciMLJacobianOperators v0.1.19
[a6db7da4] SciMLLogging v2.1.0
⌃ [c0aeaf25] SciMLOperators v1.30.0
[431bcebd] SciMLPublic v1.3.0
[53ae85a6] SciMLStructures v1.10.5
[7e506255] ScopedValues v1.6.2
[6c6a2e73] Scratch v1.3.0
[efcf1570] Setfield v1.1.2
[992d4aef] Showoff v1.1.1
[727e6d20] SimpleNonlinearSolve v2.14.5
[699a6c99] SimpleTraits v0.9.6
[a2af1166] SortingAlgorithms v1.2.3
[bd59d7e1] SparseBandedMatrices v1.4.0
[a57abbd0] SparseColumnPivotedQR v2.1.8
[0a514795] SparseMatrixColorings v0.4.28
[276daf66] SpecialFunctions v2.9.0
[860ef19b] StableRNGs v1.0.4
[0c0c59c1] StarAlgebras v0.3.0
[64909d44] StateSelection v1.11.1
[aedffcd0] Static v1.4.6
[0d7ed370] StaticArrayInterface v1.10.0
⌃ [90137ffa] StaticArrays v1.9.20
[1e83bf80] StaticArraysCore v1.4.4
[82ae8749] StatsAPI v1.8.0
[2913bbd2] StatsBase v0.34.13
[4c63d2b9] StatsFuns v2.2.1
[7792a7ef] StrideArraysCore v0.5.9
[69024149] StringEncodings v0.3.7
[09ab397b] StructArrays v0.7.3
[c3572dad] Sundials v6.7.1
[2efcf032] SymbolicIndexingInterface v0.3.55
[19f23fe9] SymbolicLimits v1.2.1
⌃ [d1185830] SymbolicUtils v4.46.6
⌃ [0c5d862f] Symbolics v7.39.2
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.14.0
[ed4db957] TaskLocalValues v0.1.3
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v2.0.0
[1c621080] TestItems v1.1.0
[8290d209] ThreadingUtilities v0.5.6
⌅ [a759f4b9] TimerOutputs v0.5.29
[d5829a12] TriangularSolve v0.2.6
[410a4b4d] Tricks v0.1.13
[781d530d] TruncatedStacktraces v1.4.0
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
[41fe7b60] Unzip v0.2.0
[3d5dd08c] VectorizationBase v0.21.74
[33b4df10] VectorizedRNG v0.2.26
[d30d5f5c] WeakCacheSets v0.1.0
[44d3d7a6] Weave v0.10.12
⌃ [ddb6d928] YAML v0.4.16
[6e34b625] Bzip2_jll v1.0.9+0
[83423d85] Cairo_jll v1.18.7+0
[ee1fde0b] Dbus_jll v1.16.2+0
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⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
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[d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
[61579ee1] Ghostscript_jll v9.55.1+0
[7746bdde] Glib_jll v2.88.3+0
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[c1c5ebd0] LAME_jll v3.100.3+0
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[aae0fff6] LSODA_jll v0.1.2+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
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[e7412a2a] Ogg_jll v1.3.6+0
⌅ [656ef2d0] OpenBLAS32_jll v0.3.24+0
[458c3c95] OpenSSL_jll v3.5.8+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
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[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[f50d1b31] Rmath_jll v0.5.2+0
⌅ [ca45d3f4] SuiteSparse32_jll v5.10.1+0
[fb77eaff] Sundials_jll v7.5.0+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
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[c834827a] Xorg_libSM_jll v1.2.6+0
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[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
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[56f22d72] Artifacts
[2a0f44e3] Base64
[ade2ca70] Dates
[8ba89e20] Distributed
[f43a241f] Downloads v1.6.0
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
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[37e2e46d] LinearAlgebra
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[29816b5a] LibSSH2_jll v1.11.0+1
[c8ffd9c3] MbedTLS_jll v2.28.2+1
[14a3606d] MozillaCACerts_jll v2023.1.10
[4536629a] OpenBLAS_jll v0.3.23+4
[05823500] OpenLibm_jll v0.8.5+0
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.2.1+1
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.11.0+0
[8e850ede] nghttp2_jll v1.52.0+1
[3f19e933] p7zip_jll v17.4.0+2
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`