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: t_nounits as t, D_nounits as D
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
        e = 0.0  # Additional parameter (if needed)
    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)
  x(t) [defaults to 1.0]
  y(t) [defaults to 1.0]
Parameters (5): see parameters(f)
  a [defaults to 1.5]
  d [defaults to 1.0]
  b [defaults to 1.0]
  e [defaults to 0.0]
  ⋮

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
prob = ODEProblem(f, u0, tspan,p)
tspan2 = (0.0, 3.0)                     # sample of 3000 observations over the (0,30) timespan
prob_short = ODEProblem(f, u0, tspan2,p)

ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
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  0.984964  0.969865  0.954715  0.939523  …  3.23047e-10  3.12995e-10
 1.0  1.00997   1.01989   1.02976   1.03956      1.55363      1.55363

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))
@btime res1 = solve(optprob, BBO_adaptive_de_rand_1_bin(); maxiters = 7e3)
# Lower tolerance could lead to smaller fitness (more accuracy)

1.194 s (2614253 allocations: 365.42 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.500214965467566
 1.0000024750180068
 3.0001750483239555
 2.637318438652911

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))
@btime res1 = solve(optprob, BBO_adaptive_de_rand_1_bin(); maxiters = 7e3)
# Change in tolerance makes it worse

1.219 s (2613157 allocations: 365.26 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.499974225730644
 0.9999952271858773
 2.999973052611709
 2.230546883489635

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))
@btime res1 = solve(optprob, BBO_adaptive_de_rand_1_bin(); maxiters = 7e3)
# using the moe accurate Vern9() reduces the fitness marginally and leads to some increase in time taken

1.716 s (2652560 allocations: 370.04 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.5000042547804053
 1.0000157391806568
 3.0000489057559867
 0.1405331999454995

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)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)

370.191 ms (578188 allocations: 79.50 MiB)
retcode: Failure
u: 4-element Vector{Float64}:
 2.9629629629570617
 1.112635269019771
 4.444444444446411
 5.183661027276527

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

1.806 s (2480017 allocations: 341.07 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.5000000007104524
 1.000000000066415
 3.00000000072833
 8.609730365585019

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

2.551 s (3760116 allocations: 517.13 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 3.2101820934868086
 3.102703061056422
 6.23284316667356
 3.0228330952260194

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

2.742 s (3760136 allocations: 517.13 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.41717752397541985
 1.0152957354634846
 2.1590654457868057
 0.3327333284030529

Now local optimization algorithms are used to check the global ones, these use the local constraints, different initial values and time step

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)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)

22.004 ms (32988 allocations: 4.52 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.5000000007152077
 1.0000000000665352
 3.0000000007324514
 1.3140958126026705

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

356.796 ms (516524 allocations: 71.02 MiB)
retcode: Failure
u: 4-element Vector{Float64}:
 1.50000000071043
 1.0000000000664235
 3.000000000728334
 1.3358196081664195

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

33.718 s (21641385 allocations: 4.22 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 2.0
 1.625
 2.625
 1.125

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

2.761 s (3760144 allocations: 517.13 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.6469631689746635
 0.9994654455200698
 3.117789184493826
 1.5182360928459269

opt = Opt(:LN_NEWUOA_BOUND, 4)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)

198.261 ms (154304 allocations: 21.21 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.5487464181140709
 0.999840139235765
 3.0391418493128373
 1.3642113819198016

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

19.411 ms (31768 allocations: 4.38 MiB)
retcode: Failure
u: 4-element Vector{Float64}:
 1.0
 0.5
 3.5
 1.5

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

2.421 s (3760136 allocations: 517.13 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.5000005835389816
 0.9999999986401436
 3.000000468839309
 1.8838550099135751

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

6.769 s (7370358 allocations: 1.20 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.0
 0.5
 3.5
 1.5

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

875.323 μs (1235 allocations: 177.48 KiB)
retcode: Failure
u: 4-element Vector{Float64}:
 1.0
 0.5
 3.5
 1.5

Now the longer problem is solved for a global solution

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

@btime res1 = solve(optprob, BBO_adaptive_de_rand_1_bin(), maxiters = 4e3)

10.602 s (12704588 allocations: 1.66 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.4885687659607436
 1.0011714661862494
 2.99416289933156
 9.296455885807822

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

5.275 s (5886150 allocations: 783.86 MiB)
retcode: Failure
u: 4-element Vector{Float64}:
 2.5514403292122063
 1.604938271606906
 4.938271604944172
 8.353909465014675

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

30.086 s (33879834 allocations: 4.41 GiB)
retcode: Failure
u: 4-element Vector{Float64}:
 1.5000000005815561
 1.0000000000683134
 3.000000000619062
 0.0

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

126.998 s (153970386 allocations: 20.03 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 2.7654422322632435
 5.137708728727137
 4.831283307687359
 0.1431635686737227

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

54.391 s (61600130 allocations: 8.01 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 5.164220594437419
 0.8056708844950282
 5.576026825177544
 6.963544609522616

Local problem

obj = build_loss_objective(prob,Vern9(),L2Loss(t,data),Optimization.AutoForwardDiff(),tstops=t,reltol=1e-9,abstol=1e-9)
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)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)

54.861 ms (86248 allocations: 11.85 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.5000000007152077
 1.0000000000665352
 3.0000000007324514
 1.3140958126026705

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

357.405 ms (516524 allocations: 71.02 MiB)
retcode: Failure
u: 4-element Vector{Float64}:
 1.50000000071043
 1.0000000000664235
 3.000000000728334
 1.3358196081664195

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

975.504 μs (1014 allocations: 147.74 KiB)
retcode: Failure
u: 4-element Vector{Float64}:
 4.021234894406051
 3.899461050223107
 3.412580878071708
 1.5000471446870371

Parameter estimation on the longer sample proves to be extremely challenging for some of the global optimizers. A few give the accurate values, BlacBoxOptim also performs quite well while others seem to struggle with accuracy a lot.

Conclusion

In general we observe that lower tolerance lead to higher accuracy but too low tolerance could affect the convergence time drastically. Also fitting a shorter timespan seems to be easier in comparison (quite intuitively). NLOpt methods seem to give great accuracy in the shorter problem with a lot of the algorithms giving 0 fitness, BBO performs very well on it with marginal change with tol values. In case of global optimization of the longer problem there is some difference in the performance amongst the algorithms with LD_SLSQPGN_ESCHGN_ISRESGN_ORIG_DIRECT_L performing among the worse, BBO also gives a bit high fitness in comparison. QuadDIRECT gives accurate results in the case of the shorter problem but doesn't perform very well in the longer problem case.

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 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 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-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/ParameterEstimation/Project.toml`
  [6e4b80f9] BenchmarkTools v1.6.0
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  [1130ab10] DiffEqParamEstim v2.2.0
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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`
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-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/ParameterEstimation/Manifest.toml`
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⌃ [1019f520] JLFzf v0.1.9
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⌃ [ccbc3e58] JumpProcesses v9.14.1
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⌃ [ba0b0d4f] Krylov v0.9.9
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⌃ [5078a376] LazyArrays v2.4.0
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⌃ [bdcacae8] LoopVectorization v0.12.171
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⌃ [961ee093] ModelingToolkit v9.61.0
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⌃ [d8a4904e] MutableArithmetics v1.6.2
⌃ [d41bc354] NLSolversBase v7.8.3
⌃ [76087f3c] NLopt v1.1.2
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⌃ [be0214bd] NonlinearSolveBase v1.4.0
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⌃ [9a2c21bd] NonlinearSolveQuasiNewton v1.1.0
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⌃ [bca83a33] OptimizationBase v2.4.0
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⌃ [50262376] OrdinaryDiffEqDefault v1.2.0
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⌃ [e0540318] OrdinaryDiffEqExponentialRK v1.2.0
⌃ [becaefa8] OrdinaryDiffEqExtrapolation v1.3.0
⌃ [5960d6e9] OrdinaryDiffEqFIRK v1.6.0
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⌃ [9f002381] OrdinaryDiffEqIMEXMultistep v1.2.0
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⌃ [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.3.0
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⌃ [5dd0a6cf] OrdinaryDiffEqPDIRK v1.2.0
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⌃ [43230ef6] OrdinaryDiffEqRosenbrock v1.4.0
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⌃ [669c94d9] OrdinaryDiffEqSSPRK v1.2.0
⌃ [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.2.0
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⌃ [fa646aed] OrdinaryDiffEqSymplecticRK v1.1.0
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⌃ [47a9eef4] SparseDiffTools v2.23.1
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⌃ [d1185830] SymbolicUtils v3.11.0
⌃ [0c5d862f] Symbolics v6.25.0
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⌃ [6e34b625] Bzip2_jll v1.0.8+4
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⌃ [36c8627f] Pango_jll v1.55.5+0
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⌅ [c0090381] Qt6Base_jll v6.7.1+1
⌅ [629bc702] Qt6Declarative_jll v6.7.1+2
⌅ [ce943373] Qt6ShaderTools_jll v6.7.1+1
⌃ [e99dba38] Qt6Wayland_jll v6.7.1+1
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⌃ [8f1865be] ZeroMQ_jll v4.3.5+3
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⌃ [b53b4c65] libpng_jll v1.6.45+1
⌃ [a9144af2] libsodium_jll v1.0.20+3
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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.