Lotka-Volterra Parameter Estimation Benchmarks

Parameter estimation of Lotka Volterra model using optimisation methods

using ParameterizedFunctions, OrdinaryDiffEq, DiffEqParamEstim, Optimization, ForwardDiff
using OptimizationBBO, OptimizationNLopt, Plots, RecursiveArrayTools, BenchmarkTools
using ModelingToolkit
using ModelingToolkitBase
using SciCompDSL
using ModelingToolkit: @mtkbuild, D_nounits as D, t_nounits as t
gr(fmt = :png)

Plots.GRBackend()

loc_bounds = Tuple{Float64, Float64}[(0, 5), (0, 5), (
    0, 5), (0, 5)]
glo_bounds = Tuple{Float64, Float64}[(0, 10), (0, 10), (
    0, 10), (0, 10)]
loc_init = [1, 0.5, 3.5, 1.5]
glo_init = [5.0, 5.0, 5.0, 5.0]

4-element Vector{Float64}:
 5.0
 5.0
 5.0
 5.0

@mtkmodel LotkaVolterraTest begin
    @parameters begin
        a = 1.5  # Growth rate of prey
        b = 1.0  # Predation rate
        c = 3.0  # Death rate of predators
        d = 1.0  # Reproduction rate of predators
    end
    @variables begin
        x(t) = 1.0  # Population of prey with initial condition
        y(t) = 1.0  # Population of predators with initial condition
    end
    @equations begin
        D(x) ~ a * x - b * x * y
        D(y) ~ -c * y + d * x * y
    end
end

@mtkbuild f = LotkaVolterraTest()

Model f:
Equations (2):
  2 standard: see equations(f)
Unknowns (2): see unknowns(f)
  y(t)
  x(t)
Parameters (4): see parameters(f)
  a
  b
  c
  d

u0 = [1.0, 1.0]                          #initial values
tspan = (0.0, 10.0)
p = [1.5, 1.0, 3.0, 1.0]                   #parameters used, these need to be estimated from the data
tspan = (0.0, 30.0)                     # sample of 3000 observations over the (0,30) timespan
# unknowns(f) is ordered [y, x] rather than in declaration order, so the
# initial condition and parameters are keyed by name instead of positionally.
varmap = merge(Dict([f.x, f.y] .=> u0), Dict([f.a, f.b, f.c, f.d] .=> p))
prob = ODEProblem(f, varmap, tspan)
tspan2 = (0.0, 3.0)                     # sample of 300 observations over the (0,3) timespan
prob_short = ODEProblem(f, varmap, tspan2)

ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
Initialization status: FULLY_DETERMINED
Non-trivial mass matrix: false
timespan: (0.0, 3.0)
u0: 2-element Vector{Float64}:
 1.0
 1.0

dt = 30.0/3000
tf = 30.0
tinterval = 0:dt:tf
time_points = collect(tinterval)

3001-element Vector{Float64}:
  0.0
  0.01
  0.02
  0.03
  0.04
  0.05
  0.06
  0.07
  0.08
  0.09
  ⋮
 29.92
 29.93
 29.94
 29.95
 29.96
 29.97
 29.98
 29.99
 30.0

h = 0.01
M = 300
tstart = 0.0
tstop = tstart + M * h
tinterval_short = 0:h:tstop
t_short = collect(tinterval_short)

301-element Vector{Float64}:
 0.0
 0.01
 0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0.09
 ⋮
 2.92
 2.93
 2.94
 2.95
 2.96
 2.97
 2.98
 2.99
 3.0

#Generate Data
data_sol_short = solve(prob_short, Tsit5(), saveat = t_short, reltol = 1e-9, abstol = 1e-9)
data_short = convert(Array, data_sol_short)
data_sol = solve(prob, Tsit5(), saveat = time_points, reltol = 1e-9, abstol = 1e-9)
data = convert(Array, data_sol)

2×3001 Matrix{Float64}:
 1.0  0.980224  0.960888  0.941986  0.923508  …  0.785597  0.770673  0.7560
92
 1.0  1.00511   1.01045   1.01601   1.02179      1.07814   1.08595   1.0939
8

Plot of the solution

Short Solution

p1 = plot(data_sol_short)

Longer Solution

p2 = plot(data_sol)

Local Solution from the short data set

obj_short = build_loss_objective(prob_short, Tsit5(), L2Loss(t_short, data_short), tstops = t_short)
optprob = OptimizationProblem(obj_short, loc_init, lb = first.(loc_bounds), ub = last.(loc_bounds))
short_solutions = Dict{String, Any}()
res1 = @btime solve(optprob, BBO_adaptive_de_rand_1_bin(); maxiters = 7e3)
short_solutions["BBO (Tsit5 default tol)"] = res1

2.944 s (21056068 allocations: 795.31 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.4974634639604811
 0.9979609554388449
 3.009417537054618
 1.0032492247357683

obj_short = build_loss_objective(
    prob_short, Tsit5(), L2Loss(t_short, data_short), tstops = t_short, reltol = 1e-9)
optprob = OptimizationProblem(obj_short, loc_init, lb = first.(loc_bounds), ub = last.(loc_bounds))
res1 = @btime solve(optprob, BBO_adaptive_de_rand_1_bin(); maxiters = 7e3)
short_solutions["BBO (Tsit5 reltol=1e-9)"] = res1

2.940 s (20997144 allocations: 792.90 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.499259300837089
 0.9999727721802977
 3.003418532683453
 1.0008263617525566

obj_short = build_loss_objective(prob_short, Vern9(), L2Loss(t_short, data_short),
    tstops = t_short, reltol = 1e-9, abstol = 1e-9)
optprob = OptimizationProblem(obj_short, loc_init, lb = first.(loc_bounds), ub = last.(loc_bounds))
res1 = @btime solve(optprob, BBO_adaptive_de_rand_1_bin(); maxiters = 7e3)
short_solutions["BBO (Vern9 1e-9)"] = res1

4.675 s (42602846 allocations: 1.11 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.5001274434188954
 1.0000323718431998
 2.99863576526574
 0.9996403032257236

Using NLopt

Global Optimisation first

obj_short = build_loss_objective(prob_short, Vern9(), L2Loss(t_short, data_short),
    Optimization.AutoForwardDiff(), tstops = t_short, reltol = 1e-9, abstol = 1e-9)
optprob = OptimizationProblem(obj_short, glo_init, lb = first.(glo_bounds), ub = last.(glo_bounds))

OptimizationProblem. In-place: true
u0: 4-element Vector{Float64}:
 5.0
 5.0
 5.0
 5.0

opt = Opt(:GN_ORIG_DIRECT_L, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["GN_ORIG_DIRECT_L"] = res1

6.572 s (59570574 allocations: 1.54 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.3219306476960178
 1.1111111111111114
 4.444444444444443
 1.4814814814814825

opt = Opt(:GN_CRS2_LM, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["GN_CRS2_LM"] = res1

6.622 s (59678772 allocations: 1.54 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.5000000000703086
 1.0000000000849358
 2.9999999995081157
 0.9999999999252926

opt = Opt(:GN_ISRES, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["GN_ISRES"] = res1

6.611 s (59613263 allocations: 1.54 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.3766824465728058
 1.0076071741244537
 3.8381507219096687
 1.273221838394185

opt = Opt(:GN_ESCH, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["GN_ESCH"] = res1

6.644 s (59637183 allocations: 1.54 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.1441830931655061
 0.8950492094307001
 5.5928007703892755
 1.9579319412802705

Now local optimization algorithms are used to verify the global ones. These use the local bounds (loc_bounds) and initial values (loc_init).

optprob = OptimizationProblem(obj_short, loc_init, lb = first.(loc_bounds), ub = last.(loc_bounds))

OptimizationProblem. In-place: true
u0: 4-element Vector{Float64}:
 1.0
 0.5
 3.5
 1.5

opt = Opt(:LN_BOBYQA, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_BOBYQA"] = res1

325.087 ms (3179520 allocations: 84.24 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.5000000000703886
 1.0000000000849636
 2.9999999995077036
 0.9999999999251571

opt = Opt(:LN_NELDERMEAD, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_NELDERMEAD"] = res1

305.373 ms (3030955 allocations: 80.30 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.50000000007048
 1.0000000000849136
 2.999999999507337
 0.9999999999250451

opt = Opt(:LD_SLSQP, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LD_SLSQP"] = res1

8.120 s (70530530 allocations: 1.87 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.500000000070235
 1.0000000000849147
 2.9999999995084474
 0.9999999999254001

opt = Opt(:LN_COBYLA, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_COBYLA"] = res1

6.638 s (59540084 allocations: 1.54 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.4999991475913865
 0.999999751348316
 3.00000440952992
 1.0000013905975

# LN_NEWUOA_BOUND is pathologically slow on this objective with NLopt_jll 2.11
# (https://github.com/jump-dev/NLopt.jl/issues/280); it is given a 300 s
# wall-time budget so the page terminates.
opt = Opt(:LN_NEWUOA_BOUND, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12, maxtime = 300)
short_solutions["LN_NEWUOA_BOUND"] = res1

380.690 s (7930813 allocations: 210.13 MiB)
retcode: MaxTime
u: 4-element Vector{Float64}:
 1.4999957298677546
 0.9999954574279393
 3.000007111770159
 1.0000010025281418

opt = Opt(:LN_PRAXIS, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_PRAXIS"] = res1

121.170 ms (1202912 allocations: 31.87 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.5000000000701403
 1.0000000000849765
 2.9999999995087068
 0.9999999999255875

opt = Opt(:LN_SBPLX, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_SBPLX"] = res1

6.675 s (59540066 allocations: 1.54 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.4999999301104459
 0.9999999814386067
 3.000000353466034
 1.0000001103210938

opt = Opt(:LD_MMA, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LD_MMA"] = res1

14.929 s (118980443 allocations: 3.35 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.5000000000653453
 1.0000000000836686
 2.999999999534401
 0.9999999999334743

opt = Opt(:LD_TNEWTON_PRECOND_RESTART, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LD_TNEWTON_PRECOND_RESTART"] = res1

51.705 ms (464474 allocations: 13.46 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.500000000070224
 1.0000000000849025
 2.999999999508518
 0.9999999999254164

Each run's estimate, retcode, loss and largest componentwise distance from the true parameters [1.5, 1.0, 3.0, 1.0] on the short problem, together with the lowest-loss and closest runs:

for (name, sol) in sort!(collect(short_solutions); by = first)
    println(rpad(name, 28),
        "estimate=", round.(sol.u; digits = 3),
        "  retcode=", sol.retcode,
        "  loss=", sol.objective,
        "  max|est-true|=", round(maximum(abs.(sol.u .- p)); sigdigits = 3))
end
best_loss, best_loss_name = minimum((sol.objective, name) for (name, sol) in short_solutions)
best_dist, best_dist_name = minimum((maximum(abs.(sol.u .- p)), name) for (name, sol) in short_solutions)
println("lowest loss: ", best_loss_name, " (loss = ", best_loss, ")")
println("closest to the true parameters: ", best_dist_name, " (max|est-true| = ", best_dist, ")")

BBO (Tsit5 default tol)     estimate=[1.497, 0.998, 3.009, 1.003]  retcode=
MaxIters  loss=0.005807785621693533  max|est-true|=0.00942
BBO (Tsit5 reltol=1e-9)     estimate=[1.499, 1.0, 3.003, 1.001]  retcode=Ma
xIters  loss=0.0019291499610795842  max|est-true|=0.00342
BBO (Vern9 1e-9)            estimate=[1.5, 1.0, 2.999, 1.0]  retcode=MaxIte
rs  loss=0.00019885543019526047  max|est-true|=0.00136
GN_CRS2_LM                  estimate=[1.5, 1.0, 3.0, 1.0]  retcode=MaxIters
  loss=1.6660903123215364e-16  max|est-true|=4.92e-10
GN_ESCH                     estimate=[1.144, 0.895, 5.593, 1.958]  retcode=
MaxIters  loss=159.95973656835872  max|est-true|=2.59
GN_ISRES                    estimate=[1.377, 1.008, 3.838, 1.273]  retcode=
MaxIters  loss=20.005115384039982  max|est-true|=0.838
GN_ORIG_DIRECT_L            estimate=[1.322, 1.111, 4.444, 1.481]  retcode=
MaxIters  loss=73.06637193676285  max|est-true|=1.44
LD_MMA                      estimate=[1.5, 1.0, 3.0, 1.0]  retcode=MaxIters
  loss=1.666288300811678e-16  max|est-true|=4.66e-10
LD_SLSQP                    estimate=[1.5, 1.0, 3.0, 1.0]  retcode=MaxIters
  loss=1.6660903116233407e-16  max|est-true|=4.92e-10
LD_TNEWTON_PRECOND_RESTART  estimate=[1.5, 1.0, 3.0, 1.0]  retcode=Success 
 loss=1.6660913092696734e-16  max|est-true|=4.91e-10
LN_BOBYQA                   estimate=[1.5, 1.0, 3.0, 1.0]  retcode=Success 
 loss=1.6660908153090768e-16  max|est-true|=4.92e-10
LN_COBYLA                   estimate=[1.5, 1.0, 3.0, 1.0]  retcode=MaxIters
  loss=5.810693336044264e-10  max|est-true|=4.41e-6
LN_NELDERMEAD               estimate=[1.5, 1.0, 3.0, 1.0]  retcode=Success 
 loss=1.6660919106479706e-16  max|est-true|=4.93e-10
LN_NEWUOA_BOUND             estimate=[1.5, 1.0, 3.0, 1.0]  retcode=MaxTime 
 loss=1.3302718723246076e-7  max|est-true|=7.11e-6
LN_PRAXIS                   estimate=[1.5, 1.0, 3.0, 1.0]  retcode=Success 
 loss=1.6660923073246894e-16  max|est-true|=4.91e-10
LN_SBPLX                    estimate=[1.5, 1.0, 3.0, 1.0]  retcode=MaxIters
  loss=3.857624352958902e-12  max|est-true|=3.53e-7
lowest loss: LD_SLSQP (loss = 1.6660903116233407e-16)
closest to the true parameters: LD_MMA (max|est-true| = 4.6559911481836025e
-10)

Now the longer problem is solved for a global solution

The Vern9 solver with reltol=1e-9 and abstol=1e-9 is used, and the dataset is increased to 3000 observations per variable with the same integration time step of 0.01.

t_concrete = collect(0.0:dt:tf)
obj = build_loss_objective(prob, Vern9(), L2Loss(t_concrete, data),
    tstops = t_concrete, reltol = 1e-9, abstol = 1e-9)
optprob = OptimizationProblem(obj, glo_init, lb = first.(glo_bounds), ub = last.(glo_bounds))

OptimizationProblem. In-place: true
u0: 4-element Vector{Float64}:
 5.0
 5.0
 5.0
 5.0

long_solutions = Dict{String, Any}()
res1 = @btime solve(optprob, BBO_adaptive_de_rand_1_bin(), maxiters = 4e3)
long_solutions["BBO_adaptive_de_rand_1_bin"] = res1

23.283 s (229206756 allocations: 4.73 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 9.08472428867907
 9.89690216571772
 0.44158725581912694
 0.1389183130261622

opt = Opt(:GN_ORIG_DIRECT_L, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
long_solutions["GN_ORIG_DIRECT_L"] = res1

55.281 s (547530784 allocations: 11.30 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.5587857860572962
 0.3703703703710262
 7.777777777777123
 5.212620027435056

opt = Opt(:GN_CRS2_LM, 4)
res1 = @btime solve(optprob, opt, maxiters = 20000, xtol_rel = 1e-12)
long_solutions["GN_CRS2_LM"] = res1

111.552 s (1091110875 allocations: 22.51 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 6.166783974559788
 4.60170461793676
 2.5977536421699083
 1.23502988041467

opt = Opt(:GN_ISRES, 4)
res1 = @btime solve(optprob, opt, maxiters = 50000, xtol_rel = 1e-12)
long_solutions["GN_ISRES"] = res1

277.941 s (2728758005 allocations: 56.30 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.9330368233935324
 3.9116084555489117
 9.955555673263145
 3.576130109478881

opt = Opt(:GN_ESCH, 4)
res1 = @btime solve(optprob, opt, maxiters = 20000, xtol_rel = 1e-12)
long_solutions["GN_ESCH"] = res1

111.573 s (1091369925 allocations: 22.52 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.098189954547478
 0.7597725306290161
 4.480015432832095
 1.6771818210798026

Each global run's estimate, retcode, loss and largest componentwise distance from the true parameters [1.5, 1.0, 3.0, 1.0] on the longer problem, together with the lowest-loss and closest runs:

for (name, sol) in sort!(collect(long_solutions); by = first)
    println(rpad(name, 28),
        "estimate=", round.(sol.u; digits = 3),
        "  retcode=", sol.retcode,
        "  loss=", sol.objective,
        "  max|est-true|=", round(maximum(abs.(sol.u .- p)); sigdigits = 3))
end
best_loss, best_loss_name = minimum((sol.objective, name) for (name, sol) in long_solutions)
best_dist, best_dist_name = minimum((maximum(abs.(sol.u .- p)), name) for (name, sol) in long_solutions)
println("lowest loss: ", best_loss_name, " (loss = ", best_loss, ")")
println("closest to the true parameters: ", best_dist_name, " (max|est-true| = ", best_dist, ")")

BBO_adaptive_de_rand_1_bin  estimate=[9.085, 9.897, 0.442, 0.139]  retcode=
MaxIters  loss=7326.221865555986  max|est-true|=8.9
GN_CRS2_LM                  estimate=[6.167, 4.602, 2.598, 1.235]  retcode=
MaxIters  loss=20816.300381031822  max|est-true|=4.67
GN_ESCH                     estimate=[1.098, 0.76, 4.48, 1.677]  retcode=Ma
xIters  loss=1873.430670155313  max|est-true|=1.48
GN_ISRES                    estimate=[0.933, 3.912, 9.956, 3.576]  retcode=
MaxIters  loss=19049.129729926986  max|est-true|=6.96
GN_ORIG_DIRECT_L            estimate=[0.559, 0.37, 7.778, 5.213]  retcode=M
axIters  loss=14653.110751552666  max|est-true|=4.78
lowest loss: GN_ESCH (loss = 1873.430670155313)
closest to the true parameters: GN_ESCH (max|est-true| = 1.4800154328320954
)

Local problem

Each local solve on the long problem is given a 300 s wall-time budget so the page terminates even when an optimizer does not converge.

obj = build_loss_objective(prob, Vern9(), L2Loss(time_points, data), Optimization.AutoForwardDiff(),
    tstops = time_points, reltol = 1e-9, abstol = 1e-9)
optprob = OptimizationProblem(obj, loc_init, lb = first.(loc_bounds), ub = last.(loc_bounds))

OptimizationProblem. In-place: true
u0: 4-element Vector{Float64}:
 1.0
 0.5
 3.5
 1.5

opt = Opt(:LN_BOBYQA, 4)
local_solutions = Dict{String, Any}()
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12, maxtime = 300)
local_solutions["LN_BOBYQA"] = res1

4.862 s (48069207 allocations: 1015.65 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.4999999994913764
 1.0000000002369105
 3.0000000017581714
 1.0000000007511933

opt = Opt(:LN_NELDERMEAD, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12, maxtime = 300)
local_solutions["LN_NELDERMEAD"] = res1

3.181 s (31319069 allocations: 661.74 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.4999999994927005
 1.0000000002375609
 3.000000001754503
 1.0000000007499836

opt = Opt(:LD_SLSQP, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12, maxtime = 300)
local_solutions["LD_SLSQP"] = res1

68.117 s (646153924 allocations: 13.68 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 3.2597117225055077
 2.7238409647671933
 0.8581704906144543
 0.40449683944949555

for (name, sol) in sort!(collect(local_solutions); by = first)
    println(rpad(name, 28),
        "estimate=", round.(sol.u; digits = 3),
        "  retcode=", sol.retcode,
        "  loss=", sol.objective,
        "  max|est-true|=", round(maximum(abs.(sol.u .- p)); sigdigits = 3),
        "  strictly inside loc_bounds: ", all(first.(loc_bounds) .< sol.u .< last.(loc_bounds)))
end
best_loss, best_loss_name = minimum((sol.objective, name) for (name, sol) in local_solutions)
best_dist, best_dist_name = minimum((maximum(abs.(sol.u .- p)), name) for (name, sol) in local_solutions)
println("lowest loss: ", best_loss_name, " (loss = ", best_loss, ")")
println("closest to the true parameters: ", best_dist_name, " (max|est-true| = ", best_dist, ")")

LD_SLSQP                    estimate=[3.26, 2.724, 0.858, 0.404]  retcode=M
axIters  loss=21569.713608932525  max|est-true|=2.14  strictly inside loc_b
ounds: true
LN_BOBYQA                   estimate=[1.5, 1.0, 3.0, 1.0]  retcode=Success 
 loss=1.2706803443895793e-14  max|est-true|=1.76e-9  strictly inside loc_bo
unds: true
LN_NELDERMEAD               estimate=[1.5, 1.0, 3.0, 1.0]  retcode=Success 
 loss=1.2706827828703252e-14  max|est-true|=1.75e-9  strictly inside loc_bo
unds: true
lowest loss: LN_BOBYQA (loss = 1.2706803443895793e-14)
closest to the true parameters: LN_NELDERMEAD (max|est-true| = 1.7545032093
835289e-9)

The two tables above summarize the longer-problem runs: each global run's estimate, retcode, loss and distance from the true parameters [1.5, 1.0, 3.0, 1.0], and each local run's estimate, retcode, loss, distance and whether the estimate lies strictly inside loc_bounds. LD_SLSQP uses the objective's gradient, which is why obj is built with Optimization.AutoForwardDiff().

Conclusion

Tighter ODE solver tolerances make each objective evaluation more expensive: the three BBO runs on the short problem use the default Tsit5 tolerance, Tsit5 at reltol = 1e-9 and Vern9 at reltol = abstol = 1e-9, and the printed timings and the distance column in the summary table show what each setting buys. LN_NEWUOA_BOUND on the short problem is given a 300 s wall-time budget because NLopt_jll 2.11's newuoa is pathologically slow on this objective (see the note above). The summary tables print each run's estimate, retcode, loss and distance from the true parameters [1.5, 1.0, 3.0, 1.0] on both the short and the longer problem, including which run has the lowest loss and which lands closest; the global optimizers are stochastic, so the distances and winners differ from run to run.

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/ParameterEstimation","LotkaVolterraParameterEstimation.jmd")

Computer Information:

Julia Version 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 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
  LLVM: libLLVM-18.1.7 (ORCJIT, znver2)
  GC: Built with stock GC
Threads: 128 default, 1 interactive, 128 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/ParameterEstimation/Project.toml`
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Info Packages marked with ⌃ have new versions available and may be upgradable.
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.

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  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
⌃ [35661453] Xorg_xkbcomp_jll v1.4.7+0
⌃ [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
⌃ [a4ae2306] libaom_jll v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.134+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
⌃ [b53b4c65] libpng_jll v1.6.58+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
⌃ [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.7.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.12.1
  [de0858da] Printf v1.11.0
  [9abbd945] Profile v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.12.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.3.1+2
  [deac9b47] LibCURL_jll v8.15.0+0
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.6+0
  [efcefdf7] PCRE2_jll v10.44.0+1
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.7.0+0
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.