FitzHugh-Nagumo Parameter Estimation Benchmarks

Parameter estimation of FitzHugh-Nagumo model using optimisation methods

using ParameterizedFunctions, OrdinaryDiffEq, DiffEqParamEstim, Optimization
using OptimizationBBO, OptimizationNLopt, ForwardDiff, Plots, 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, 1), (0, 1), (
    0, 1), (0, 1)]
glo_bounds = Tuple{Float64, Float64}[(0, 5), (0, 5), (
    0, 5), (0, 5)]
loc_init = [0.5, 0.5, 0.5, 0.5]
glo_init = [2.5, 2.5, 2.5, 2.5]

4-element Vector{Float64}:
 2.5
 2.5
 2.5
 2.5

@mtkmodel FitzHughNagumo begin
    @parameters begin
        a = 0.7      # Parameter for excitability
        b = 0.8      # Recovery rate parameter
        τinv = 0.08  # Inverse of the time constant
        l = 0.5      # External stimulus
    end
    @variables begin
        v(t) = 1.0  # Membrane potential with initial condition
        w(t) = 1.0  # Recovery variable with initial condition
    end
    @equations begin
        D(v) ~ v - v^3 / 3 - w + l
        D(w) ~ τinv * (v + a - b * w)
    end
end

@mtkbuild fitz = FitzHughNagumo()

Model fitz:
Equations (2):
  2 standard: see equations(fitz)
Unknowns (2): see unknowns(fitz)
  w(t)
  v(t)
Parameters (4): see parameters(fitz)
  a
  b
  τinv
  l

p = [0.7, 0.8, 0.08, 0.5]              # Parameters used to construct the dataset
r0 = [1.0; 1.0]                     # initial value
tspan = (0.0, 30.0)                 # sample of 3000 observations over the (0,30) timespan
# unknowns(fitz) is ordered [w, v] rather than in declaration order, so the
# initial condition and parameters are keyed by name instead of positionally.
varmap = merge(Dict([fitz.v, fitz.w] .=> r0), Dict([fitz.a, fitz.b, fitz.τinv, fitz.l] .=> p))
prob = ODEProblem(fitz, varmap, tspan)
tspan2 = (0.0, 3.0)                 # sample of 300 observations with a timestep of 0.01
prob_short = ODEProblem(fitz, 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, Vern9(), saveat = t_short, reltol = 1e-9, abstol = 1e-9)
data_short = convert(Array, data_sol_short) # This operation produces column major dataset obs as columns, equations as rows
data_sol = solve(prob, Vern9(), saveat = time_points, reltol = 1e-9, abstol = 1e-9)
data = convert(Array, data_sol)

2×3001 Matrix{Float64}:
 1.0  1.00072  1.00144  1.00216  1.00289  …  -0.229157  -0.228976  -0.22879
3
 1.0  1.00166  1.00332  1.00497  1.00661     -0.65759   -0.655923  -0.65424
8

Plot of the solution

Short Solution

plot(data_sol_short)

Longer Solution

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, glo_init, lb = first.(glo_bounds), ub = last.(glo_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

2.939 s (21007281 allocations: 801.07 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.741189416168004
 0.8469918997996646
 0.08078465527568746
 0.5000773147121458

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

2.964 s (21242248 allocations: 808.52 MiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.427662976533689
 0.8982231688470127
 0.13309188594957277
 0.5009075845204596

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, glo_init, lb = first.(glo_bounds), ub = last.(glo_bounds))
res1 = @btime solve(optprob, BBO_adaptive_de_rand_1_bin(), maxiters = 7e3)
short_solutions["BBO (Vern9 1e-9)"] = res1

4.813 s (42376102 allocations: 1.11 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.269167742190918
 0.8675330265078323
 0.17189058217583497
 0.49966553376298745

Using NLopt

Global Optimisation

obj_short = build_loss_objective(prob_short, Vern9(), L2Loss(t_short, data_short),
    tstops = t_short, reltol = 1e-9, abstol = 1e-9)

SciMLBase.OptimizationFunction{true, SciMLBase.NoAD, DiffEqParamEstim.var"#
37#38"{Nothing, typeof(DiffEqParamEstim.STANDARD_PROB_GENERATOR), Base.Pair
s{Symbol, Any, Nothing, @NamedTuple{tstops::Vector{Float64}, reltol::Float6
4, abstol::Float64}}, SciMLBase.ODEProblem{Vector{Float64}, Tuple{Float64, 
Float64}, true, ModelingToolkitBase.MTKParameters{Vector{Float64}, Vector{F
loat64}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}, SciMLBase.ODEFunction{true, S
ciMLBase.AutoDespecialize, ModelingToolkitBase.GeneratedFunctionWrapper{Tup
le{2, 3, true}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:___mtk
unknowns___, :___mtkparameters___, :__argₛᵧₘ12968198168038750593), Modeling
ToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xb4
dec7fc, 0x0eef51a9, 0xe6d646f8, 0xd5b37c85, 0x7ae33bfd), Nothing}, RuntimeG
eneratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ1401282876548370056, :
___mtkunknowns___, :___mtkparameters___, :__argₛᵧₘ12968198168038750593), Mo
delingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag",
 (0xcc4caf47, 0xa22b65af, 0xd2ae4f08, 0x45b5d405, 0xd97af149), Nothing}}, L
inearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Noth
ing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Modelin
gToolkitBase.ObservedFunctionCache{ModelingToolkitBase.System, Nothing}, No
thing, ModelingToolkitBase.System, Union{Nothing, SciMLBase.OverrideInitDat
a}, Union{Nothing, SciMLBase.ODENLStepData}}, Base.Pairs{Symbol, Union{}, N
othing, @NamedTuple{}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqVerner
.Vern9{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCo
re.trivial_limiter!), FastBroadcast.Serial, Val{true}}, DiffEqParamEstim.L2
Loss{Vector{Float64}, Matrix{Float64}, Nothing, Nothing, Nothing, Nothing},
 Nothing, Tuple{}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, N
othing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLB
ase.DEFAULT_OBSERVED_NO_TIME), Nothing, Nothing, Nothing, Nothing, Nothing,
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Nothing, typeof(DiffEqParamEstim.STANDARD_PROB_GENERATOR), Base.Pairs{Symbo
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ol::Float64}}, SciMLBase.ODEProblem{Vector{Float64}, Tuple{Float64, Float64
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, Tuple{}, Tuple{}, Tuple{}, Tuple{}}, SciMLBase.ODEFunction{true, SciMLBas
e.AutoDespecialize, ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3
, true}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:___mtkunknown
s___, :___mtkparameters___, :__argₛᵧₘ12968198168038750593), ModelingToolkit
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dFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ1401282876548370056, :___mtku
nknowns___, :___mtkparameters___, :__argₛᵧₘ12968198168038750593), ModelingT
oolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xcc4
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gebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, No
thing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolki
tBase.ObservedFunctionCache{ModelingToolkitBase.System, Nothing}, Nothing, 
ModelingToolkitBase.System, Union{Nothing, SciMLBase.OverrideInitData}, Uni
on{Nothing, SciMLBase.ODENLStepData}}, Base.Pairs{Symbol, Union{}, Nothing,
 @NamedTuple{}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqVerner.Vern9{
typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.triv
ial_limiter!), FastBroadcast.Serial, Val{true}}, DiffEqParamEstim.L2Loss{Ve
ctor{Float64}, Matrix{Float64}, Nothing, Nothing, Nothing, Nothing}, Nothin
g, Tuple{}}(nothing, DiffEqParamEstim.STANDARD_PROB_GENERATOR, Base.Pairs{S
ymbol, Any, Nothing, @NamedTuple{tstops::Vector{Float64}, reltol::Float64, 
abstol::Float64}}(:tstops => [0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07
, 0.08, 0.09  …  2.91, 2.92, 2.93, 2.94, 2.95, 2.96, 2.97, 2.98, 2.99, 3.0]
, :reltol => 1.0e-9, :abstol => 1.0e-9), SciMLBase.ODEProblem{Vector{Float6
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ODEFunction{true, SciMLBase.AutoDespecialize, ModelingToolkitBase.Generated
FunctionWrapper{Tuple{2, 3, true}, RuntimeGeneratedFunctions.RuntimeGenerat
edFunction{(:___mtkunknowns___, :___mtkparameters___, :__argₛᵧₘ129681981680
38750593), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#
_RGF_ModTag", (0xb4dec7fc, 0x0eef51a9, 0xe6d646f8, 0xd5b37c85, 0x7ae33bfd),
 Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ140
1282876548370056, :___mtkunknowns___, :___mtkparameters___, :__argₛᵧₘ129681
98168038750593), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase
.var"#_RGF_ModTag", (0xcc4caf47, 0xa22b65af, 0xd2ae4f08, 0x45b5d405, 0xd97a
f149), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Not
hing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothin
g, Nothing, ModelingToolkitBase.ObservedFunctionCache{ModelingToolkitBase.S
ystem, Nothing}, Nothing, ModelingToolkitBase.System, Union{Nothing, SciMLB
ase.OverrideInitData}, Union{Nothing, SciMLBase.ODENLStepData}}, Base.Pairs
{Symbol, Union{}, Nothing, @NamedTuple{}}, SciMLBase.StandardODEProblem}(Sc
iMLBase.ODEFunction{true, SciMLBase.AutoDespecialize, ModelingToolkitBase.G
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eGeneratedFunction{(:___mtkunknowns___, :___mtkparameters___, :__argₛᵧₘ1296
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elingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3, true}, RuntimeGenerat
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eGeneratedFunction{(:___mtkunknowns___, :___mtkparameters___, :__argₛᵧₘ1296
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se.var"#_RGF_ModTag", (0xb4dec7fc, 0x0eef51a9, 0xe6d646f8, 0xd5b37c85, 0x7a
e33bfd), Nothing}(nothing), RuntimeGeneratedFunctions.RuntimeGeneratedFunct
ion{(:__argₛᵧₘ1401282876548370056, :___mtkunknowns___, :___mtkparameters___
, :__argₛᵧₘ12968198168038750593), ModelingToolkitBase.var"#_RGF_ModTag", Mo
delingToolkitBase.var"#_RGF_ModTag", (0xcc4caf47, 0xa22b65af, 0xd2ae4f08, 0
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ol}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, n
othing, nothing, nothing, nothing, nothing, ModelingToolkitBase.ObservedFun
ctionCache{ModelingToolkitBase.System, Nothing}(Model fitz:
Equations (2):
  2 standard: see equations(fitz)
Unknowns (2): see unknowns(fitz)
  w(t)
  v(t)
Parameters (4): see parameters(fitz)
  a
  b
  τinv
  l, Dict{Any, Any}(), false, false, ModelingToolkitBase, false, nothing), 
nothing, Model fitz:
Equations (2):
  2 standard: see equations(fitz)
Unknowns (2): see unknowns(fitz)
  w(t)
  v(t)
Parameters (4): see parameters(fitz)
  a
  b
  τinv
  l, SciMLBase.OverrideInitData{SciMLBase.NonlinearProblem{Nothing, true, M
odelingToolkitBase.MTKParameters{Vector{Float64}, StaticArraysCore.SVector{
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rapper{Tuple{2, 2, true}, RuntimeGeneratedFunctions.RuntimeGeneratedFunctio
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olkitBase.System, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Nothing, @
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ase.NonlinearFunction{true, SciMLBase.AutoDespecialize, ModelingToolkitBase
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}(ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 2, true}, RuntimeGe
neratedFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :___mtkparam
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1282876548370056, :___mtkunknowns___, :___mtkparameters___), ModelingToolki
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edFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :___mtkparameters
___), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_
ModTag", (0xe0a7f55e, 0xcb0f8c13, 0x5afbe72e, 0x825c4ce9, 0x3418c4a5), Noth
ing}(nothing), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧ
ₘ1401282876548370056, :___mtkunknowns___, :___mtkparameters___), ModelingTo
olkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xac9d
d6f8, 0xf3d8414c, 0x01f2cade, 0x451b4ee8, 0x29c649f2), Nothing}(nothing)), 
LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothin
g, nothing, nothing, nothing, nothing, nothing, nothing, ModelingToolkitBas
e.ObservedFunctionCache{ModelingToolkitBase.System, Nothing}(Model fitz:
Parameters (9): see parameters(fitz)
  t
  a
  b
  τinv
  ⋮
Observed (4): see observed(fitz), Dict{Any, Any}(), false, false, ModelingT
oolkitBase, false, nothing), nothing, Model fitz:
Parameters (9): see parameters(fitz)
  t
  a
  b
  τinv
  ⋮
Observed (4): see observed(fitz), nothing, nothing), nothing, ModelingToolk
itBase.MTKParameters{Vector{Float64}, StaticArraysCore.SVector{0, Float64},
 Tuple{}, Tuple{}, Tuple{}, Tuple{}}([0.0, 0.7, 0.8, 0.08, 0.5, 1.0, 0.0, 0
.0, 1.0], Float64[], (), (), (), ()), SciMLBase.StandardNonlinearProblem(),
 nothing, nothing, Base.Pairs{Symbol, Union{}, Nothing, @NamedTuple{}}()), 
ModelingToolkitBase.update_initializeprob!, ModelingToolkitBase.Initializat
ionMap{true, typeof(identity), ModelingToolkitBase.PromoteToTunableEltype{M
odelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.Par
ameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}, ModelingToolkitBase
.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Nothing}, F
loat64}}(identity, ModelingToolkitBase.PromoteToTunableEltype{ModelingToolk
itBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.ParameterIndex{
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dex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Nothing}, Float64}(Mode
lingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.Parame
terIndex{SciMLStructures.Tunable, UnitRange{Int64}}, ModelingToolkitBase.Pa
rameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Nothing}((Mode
lingToolkitBase.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}(S
ciMLStructures.Tunable(), 9:9, false), ModelingToolkitBase.ParameterIndex{S
ciMLStructures.Tunable, UnitRange{Int64}}(SciMLStructures.Tunable(), 6:6, f
alse)), (2,), nothing))), ModelingToolkitBase.var"#initprobpmap_split#770"{
ModelingToolkitBase.MTKParametersReconstructor{ComposedFunction{ModelingToo
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ParamsByTemplate{true, Tuple{ModelingToolkitBase.ParameterIndex{SciMLStruct
ures.Tunable, UnitRange{Int64}}}, 1, Nothing}}, ComposedFunction{ModelingTo
olkitBase.PConstructorApplicator{typeof(identity)}, ModelingToolkitBase.Cop
yParamsByTemplate{true, Tuple{ModelingToolkitBase.ParameterIndex{SciMLStruc
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Tuple{}}, Returns{Tuple{}}}}(ModelingToolkitBase.MTKParametersReconstructor
{ComposedFunction{ModelingToolkitBase.PConstructorApplicator{typeof(identit
y)}, ModelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBa
se.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Nothing}}
, ComposedFunction{ModelingToolkitBase.PConstructorApplicator{typeof(identi
ty)}, ModelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitB
ase.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Nothing}
}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}(ModelingToolkitBas
e.PConstructorApplicator{typeof(identity)}(identity) ∘ ModelingToolkitBase.
CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.ParameterIndex{SciMLSt
ructures.Tunable, UnitRange{Int64}}}, 1, Nothing}((ModelingToolkitBase.Para
meterIndex{SciMLStructures.Tunable, UnitRange{Int64}}(SciMLStructures.Tunab
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{true, Tuple{ModelingToolkitBase.ParameterIndex{SciMLStructures.Tunable, Un
itRange{Int64}}}, 1, Nothing}((ModelingToolkitBase.ParameterIndex{SciMLStru
ctures.Tunable, UnitRange{Int64}}(SciMLStructures.Tunable(), 6:9, false),),
 (4,), nothing), Returns{Tuple{}}(()), Returns{Tuple{}}(()), Returns{Tuple{
}}(()), 0)), ModelingToolkitBase.InitializationMetadata{ModelingToolkitBase
.ReconstructInitializeprob{ModelingToolkitBase.MTKParametersReconstructor{C
omposedFunction{ModelingToolkitBase.PConstructorApplicator{typeof(identity)
}, ModelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase
.IndepVarTemplate, ModelingToolkitBase.ParameterIndex{SciMLStructures.Tunab
le, UnitRange{Int64}}, ModelingToolkitBase.ParameterIndex{SciMLStructures.I
nitials, UnitRange{Int64}}}, 1, Nothing}}, Returns{StaticArraysCore.SVector
{0, Float64}}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}, Compo
sedFunction{typeof(identity), ModelingToolkitBase.ObservedWrapper{true, Mod
elingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3, true}, RuntimeGenerat
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nction{(:x1, :x2, :x3, :x4), ModelingToolkitBase.var"#_RGF_ModTag", Modelin
gToolkitBase.var"#_RGF_ModTag", (0xb896e553, 0xa118fcf5, 0xbfeb9719, 0xa096
e58a, 0x357542a4), Nothing}}}}}, ModelingToolkitBase.GetUpdatedU0{Nothing, 
SymbolicIndexingInterface.MultipleParametersGetter{SymbolicIndexingInterfac
e.IndexerNotTimeseries, Vector{SymbolicIndexingInterface.GetParameterIndex{
ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, Int64}}}, Noth
ing}}, ModelingToolkitBase.SetInitialUnknowns{SymbolicIndexingInterface.Mul
tipleSetters{Vector{SymbolicIndexingInterface.ParameterHookWrapper{Symbolic
IndexingInterface.SetParameterIndex{ModelingToolkitBase.ParameterIndex{SciM
LStructures.Initials, Int64}}, SymbolicUtils.BasicSymbolicImpl.var"typeof(B
asicSymbolicImpl)"{SymbolicUtils.SymReal}}}}}}(ModelingToolkitBase.AtomicAr
rayDict{SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{Symb
olicUtils.SymReal}, Dict{SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSy
mbolicImpl)"{SymbolicUtils.SymReal}, SymbolicUtils.BasicSymbolicImpl.var"ty
peof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}}}(v(t) => 1.0, Initial(v(t)
) => false, Initial(wˍt(t)) => false, b => 0.8, Initial(vˍt(t)) => false, a
 => 0.7, τinv => 0.08, l => 0.5, Initial(w(t)) => false, w(t) => 1.0…), Mod
elingToolkitBase.AtomicArrayDict{SymbolicUtils.BasicSymbolicImpl.var"typeof
(BasicSymbolicImpl)"{SymbolicUtils.SymReal}, Dict{SymbolicUtils.BasicSymbol
icImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}, SymbolicUtils
.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}}}(
), Symbolics.Equation[], true, true, ModelingToolkitBase.ReconstructInitial
izeprob{ModelingToolkitBase.MTKParametersReconstructor{ComposedFunction{Mod
elingToolkitBase.PConstructorApplicator{typeof(identity)}, ModelingToolkitB
ase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.IndepVarTemplate, 
ModelingToolkitBase.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64
}}, ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, UnitRange{
Int64}}}, 1, Nothing}}, Returns{StaticArraysCore.SVector{0, Float64}}, Retu
rns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}, ComposedFunction{typeof(
identity), ModelingToolkitBase.ObservedWrapper{true, ModelingToolkitBase.Ge
neratedFunctionWrapper{Tuple{2, 3, true}, RuntimeGeneratedFunctions.Runtime
GeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :__argₛᵧₘ12968198168
038750593), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"
#_RGF_ModTag", (0x05e7e591, 0x9a09c886, 0x0267af3f, 0x585cfe8b, 0x75f79d7b)
, Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:x1, :x2, :
x3, :x4), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_
RGF_ModTag", (0xb896e553, 0xa118fcf5, 0xbfeb9719, 0xa096e58a, 0x357542a4), 
Nothing}}}}}(ModelingToolkitBase.MTKParametersReconstructor{ComposedFunctio
n{ModelingToolkitBase.PConstructorApplicator{typeof(identity)}, ModelingToo
lkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.IndepVarTempl
ate, ModelingToolkitBase.ParameterIndex{SciMLStructures.Tunable, UnitRange{
Int64}}, ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, UnitR
ange{Int64}}}, 1, Nothing}}, Returns{StaticArraysCore.SVector{0, Float64}},
 Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}(ModelingToolkitBase.
PConstructorApplicator{typeof(identity)}(identity) ∘ ModelingToolkitBase.Co
pyParamsByTemplate{true, Tuple{ModelingToolkitBase.IndepVarTemplate, Modeli
ngToolkitBase.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}, Mo
delingToolkitBase.ParameterIndex{SciMLStructures.Initials, UnitRange{Int64}
}}, 1, Nothing}((ModelingToolkitBase.IndepVarTemplate(), ModelingToolkitBas
e.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}(SciMLStructures
.Tunable(), 1:4, false), ModelingToolkitBase.ParameterIndex{SciMLStructures
.Initials, UnitRange{Int64}}(SciMLStructures.Initials(), 1:4, false)), (9,)
, nothing), Returns{StaticArraysCore.SVector{0, Float64}}(Float64[]), Retur
ns{Tuple{}}(()), Returns{Tuple{}}(()), Returns{Tuple{}}(()), 0), identity ∘
 ModelingToolkitBase.ObservedWrapper{true, ModelingToolkitBase.GeneratedFun
ctionWrapper{Tuple{2, 3, true}, RuntimeGeneratedFunctions.RuntimeGeneratedF
unction{(:__mtk_arg_1, :___mtkparameters___, :__argₛᵧₘ12968198168038750593)
, ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModT
ag", (0x05e7e591, 0x9a09c886, 0x0267af3f, 0x585cfe8b, 0x75f79d7b), Nothing}
, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:x1, :x2, :x3, :x4), 
ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag
", (0xb896e553, 0xa118fcf5, 0xbfeb9719, 0xa096e58a, 0x357542a4), Nothing}}}
(ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3, true}, RuntimeGen
eratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters__
_, :__argₛᵧₘ12968198168038750593), ModelingToolkitBase.var"#_RGF_ModTag", M
odelingToolkitBase.var"#_RGF_ModTag", (0x05e7e591, 0x9a09c886, 0x0267af3f, 
0x585cfe8b, 0x75f79d7b), Nothing}, RuntimeGeneratedFunctions.RuntimeGenerat
edFunction{(:x1, :x2, :x3, :x4), ModelingToolkitBase.var"#_RGF_ModTag", Mod
elingToolkitBase.var"#_RGF_ModTag", (0xb896e553, 0xa118fcf5, 0xbfeb9719, 0x
a096e58a, 0x357542a4), Nothing}}(RuntimeGeneratedFunctions.RuntimeGenerated
Function{(:__mtk_arg_1, :___mtkparameters___, :__argₛᵧₘ12968198168038750593
), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_Mod
Tag", (0x05e7e591, 0x9a09c886, 0x0267af3f, 0x585cfe8b, 0x75f79d7b), Nothing
}(nothing), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:x1, :x2, :
x3, :x4), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_
RGF_ModTag", (0xb896e553, 0xa118fcf5, 0xbfeb9719, 0xa096e58a, 0x357542a4), 
Nothing}(nothing)))), ModelingToolkitBase.GetUpdatedU0{Nothing, SymbolicInd
exingInterface.MultipleParametersGetter{SymbolicIndexingInterface.IndexerNo
tTimeseries, Vector{SymbolicIndexingInterface.GetParameterIndex{ModelingToo
lkitBase.ParameterIndex{SciMLStructures.Initials, Int64}}}, Nothing}}(Bool[
0, 0], nothing, SymbolicIndexingInterface.MultipleParametersGetter{Symbolic
IndexingInterface.IndexerNotTimeseries, Vector{SymbolicIndexingInterface.Ge
tParameterIndex{ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials
, Int64}}}, Nothing}(SymbolicIndexingInterface.GetParameterIndex{ModelingTo
olkitBase.ParameterIndex{SciMLStructures.Initials, Int64}}[SymbolicIndexing
Interface.GetParameterIndex{ModelingToolkitBase.ParameterIndex{SciMLStructu
res.Initials, Int64}}(ModelingToolkitBase.ParameterIndex{SciMLStructures.In
itials, Int64}(SciMLStructures.Initials(), 4, false)), SymbolicIndexingInte
rface.GetParameterIndex{ModelingToolkitBase.ParameterIndex{SciMLStructures.
Initials, Int64}}(ModelingToolkitBase.ParameterIndex{SciMLStructures.Initia
ls, Int64}(SciMLStructures.Initials(), 1, false))], nothing)), ModelingTool
kitBase.SetInitialUnknowns{SymbolicIndexingInterface.MultipleSetters{Vector
{SymbolicIndexingInterface.ParameterHookWrapper{SymbolicIndexingInterface.S
etParameterIndex{ModelingToolkitBase.ParameterIndex{SciMLStructures.Initial
s, Int64}}, SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{
SymbolicUtils.SymReal}}}}}(SymbolicIndexingInterface.MultipleSetters{Vector
{SymbolicIndexingInterface.ParameterHookWrapper{SymbolicIndexingInterface.S
etParameterIndex{ModelingToolkitBase.ParameterIndex{SciMLStructures.Initial
s, Int64}}, SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{
SymbolicUtils.SymReal}}}}(SymbolicIndexingInterface.ParameterHookWrapper{Sy
mbolicIndexingInterface.SetParameterIndex{ModelingToolkitBase.ParameterInde
x{SciMLStructures.Initials, Int64}}, SymbolicUtils.BasicSymbolicImpl.var"ty
peof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}}[SymbolicIndexingInterface.
ParameterHookWrapper{SymbolicIndexingInterface.SetParameterIndex{ModelingTo
olkitBase.ParameterIndex{SciMLStructures.Initials, Int64}}, SymbolicUtils.B
asicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}}(Sym
bolicIndexingInterface.SetParameterIndex{ModelingToolkitBase.ParameterIndex
{SciMLStructures.Initials, Int64}}(ModelingToolkitBase.ParameterIndex{SciML
Structures.Initials, Int64}(SciMLStructures.Initials(), 4, false)), Initial
(w(t))), SymbolicIndexingInterface.ParameterHookWrapper{SymbolicIndexingInt
erface.SetParameterIndex{ModelingToolkitBase.ParameterIndex{SciMLStructures
.Initials, Int64}}, SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymboli
cImpl)"{SymbolicUtils.SymReal}}(SymbolicIndexingInterface.SetParameterIndex
{ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, Int64}}(Model
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res.Initials(), 1, false)), Initial(v(t)))]), [4, 1]), ModelingToolkitBase.
MissingGuessValue.var"typeof(MissingGuessValue)"(ModelingToolkitBase.Missin
gGuessValue.var"##Storage#HashedRandom"())), Val{true}()), nothing), [1.0, 
1.0], (0.0, 3.0), ModelingToolkitBase.MTKParameters{Vector{Float64}, Vector
{Float64}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}([0.7, 0.8, 0.08, 0.5], [1.0,
 0.0, 0.0, 1.0], (), (), (), ()), Base.Pairs{Symbol, Union{}, Nothing, @Nam
edTuple{}}(), SciMLBase.StandardODEProblem()), OrdinaryDiffEqVerner.Vern9{t
ypeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivi
al_limiter!), FastBroadcast.Serial, Val{true}}(OrdinaryDiffEqCore.trivial_l
imiter!, OrdinaryDiffEqCore.trivial_limiter!, FastBroadcast.Serial(), Val{t
rue}()), DiffEqParamEstim.L2Loss{Vector{Float64}, Matrix{Float64}, Nothing,
 Nothing, Nothing, Nothing}([0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07,
 0.08, 0.09  …  2.91, 2.92, 2.93, 2.94, 2.95, 2.96, 2.97, 2.98, 2.99, 3.0],
 [1.0 1.000720435211627 … 1.22193331177933 1.222601511669354; 1.0 1.0016630
55982029 … 1.1134077677576417 1.11271910687873], nothing, nothing, nothing,
 nothing, nothing), nothing, ()), SciMLBase.NoAD(), nothing, nothing, nothi
ng, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing,
 nothing, nothing, SciMLBase.DEFAULT_OBSERVED_NO_TIME, nothing, nothing, no
thing, nothing, nothing, nothing, nothing, nothing, nothing, nothing)

opt = Opt(:GN_ORIG_DIRECT_L, 4)
optprob = OptimizationProblem(obj_short, glo_init, lb = first.(glo_bounds), ub = last.(glo_bounds))
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["GN_ORIG_DIRECT_L"] = res1

6.855 s (59865528 allocations: 1.56 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.1920438957477088
 1.1316872427984634
 1.1111111111112206
 0.509577685189625

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

6.997 s (59740224 allocations: 1.56 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.7000901073520914
 0.8000114439120442
 0.0799933943628506
 0.5000000467429484

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

6.859 s (59770053 allocations: 1.56 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 4.388686241701901
 4.205524545334197
 0.08531487804176452
 0.5165286699818172

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

6.931 s (59770053 allocations: 1.56 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 2.9654095992371836
 0.4371327199712715
 0.020866076018565223
 0.49924445431268333

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

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, loc_init, lb = first.(loc_bounds), ub = last.(loc_bounds))

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

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

1.265 s (11028088 allocations: 294.58 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 0.7000000000188623
 0.8000000000060236
 0.07999999999893552
 0.5000000000000322

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

631.491 ms (5748224 allocations: 153.55 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.0
 1.0
 0.07355092571547887
 0.5004047023109518

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

8.768 s (71095530 allocations: 1.90 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.700000000015133
 0.8000000000055394
 0.07999999999920873
 0.5000000000000321

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

6.925 s (59740084 allocations: 1.56 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.1829436856114406
 0.8339606715478033
 0.19453032306019252
 0.5003613601073567

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

623.582 ms (4325260 allocations: 115.54 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 0.24815174877228105
 0.4389927966494055
 0.08539344966317435
 0.49911205692389365

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

621.506 ms (5246030 allocations: 140.13 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 0.7000000000151287
 0.8000000000059342
 0.0799999999992364
 0.5000000000000221

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

6.989 s (59740066 allocations: 1.56 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.7000016809870933
 0.8000002415243684
 0.07999987937160621
 0.5000000003805862

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

16.693 s (119380443 allocations: 3.39 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.2194953414908776
 0.7035126977082891
 0.13312823531746235
 0.499708318694746

Each run's estimate, retcode, loss and largest componentwise distance from the true parameters [0.7, 0.8, 0.08, 0.5] 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=[0.741, 0.847, 0.081, 0.5]  retcode=Ma
xIters  loss=1.1195882041168969e-5  max|est-true|=0.047
BBO (Tsit5 reltol=1e-9)     estimate=[0.428, 0.898, 0.133, 0.501]  retcode=
MaxIters  loss=0.00089388888536599  max|est-true|=0.272
BBO (Vern9 1e-9)            estimate=[0.269, 0.868, 0.172, 0.5]  retcode=Ma
xIters  loss=0.0008014784470582032  max|est-true|=0.431
GN_CRS2_LM                  estimate=[0.7, 0.8, 0.08, 0.5]  retcode=MaxIter
s  loss=1.944197726102618e-12  max|est-true|=9.01e-5
GN_ESCH                     estimate=[2.965, 0.437, 0.021, 0.499]  retcode=
MaxIters  loss=0.0003175856994032462  max|est-true|=2.27
GN_ISRES                    estimate=[4.389, 4.206, 0.085, 0.517]  retcode=
MaxIters  loss=0.06205001570957812  max|est-true|=3.69
GN_ORIG_DIRECT_L            estimate=[0.192, 1.132, 1.111, 0.51]  retcode=M
axIters  loss=0.11016600767987259  max|est-true|=1.03
LD_MMA                      estimate=[0.219, 0.704, 0.133, 0.5]  retcode=Ma
xIters  loss=0.00010819461748085794  max|est-true|=0.481
LD_SLSQP                    estimate=[0.7, 0.8, 0.08, 0.5]  retcode=MaxIter
s  loss=4.724821748139161e-25  max|est-true|=1.51e-11
LN_BOBYQA                   estimate=[0.7, 0.8, 0.08, 0.5]  retcode=Success
  loss=4.760436845857627e-25  max|est-true|=1.89e-11
LN_COBYLA                   estimate=[0.183, 0.834, 0.195, 0.5]  retcode=Ma
xIters  loss=0.0007443388864289124  max|est-true|=0.517
LN_NELDERMEAD               estimate=[1.0, 1.0, 0.074, 0.5]  retcode=Succes
s  loss=8.965505337532214e-5  max|est-true|=0.3
LN_NEWUOA_BOUND             estimate=[0.248, 0.439, 0.085, 0.499]  retcode=
Success  loss=0.00042978296411709017  max|est-true|=0.452
LN_PRAXIS                   estimate=[0.7, 0.8, 0.08, 0.5]  retcode=Success
  loss=5.36754271940152e-25  max|est-true|=1.51e-11
LN_SBPLX                    estimate=[0.7, 0.8, 0.08, 0.5]  retcode=MaxIter
s  loss=5.041392563911391e-16  max|est-true|=1.68e-6
lowest loss: LD_SLSQP (loss = 4.724821748139161e-25)
closest to the true parameters: LN_PRAXIS (max|est-true| = 1.51287871119620
83e-11)

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.

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

23.209 s (224309522 allocations: 4.64 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.9426760379710977
 1.0652914786154668
 0.08911992261002938
 0.5218250949015131

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

56.580 s (546202158 allocations: 11.28 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 1.1111111111111116
 1.1111111111111107
 0.10059442158207468
 0.5761316872427985

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

113.987 s (1091640234 allocations: 22.55 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.6999999999966329
 0.8000000000028302
 0.08000000000032029
 0.4999999999984416

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

284.194 s (2729250063 allocations: 56.38 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 0.7000000041562078
 0.7999999958605696
 0.07999999988501204
 0.5000000014225955

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

114.366 s (1091700063 allocations: 22.55 GiB)
retcode: MaxIters
u: 4-element Vector{Float64}:
 2.488012852232664
 2.5910137157546655
 0.12210544064426454
 0.4774460979954282

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

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

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

3.615 s (34932564 allocations: 738.99 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 0.6999999999966582
 0.8000000000027985
 0.08000000000032159
 0.499999999998461

opt = Opt(:LN_NELDERMEAD, 4)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-9)
long_solutions["LN_NELDERMEAD"] = res1

3.364 s (32477100 allocations: 687.05 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 1.0
 1.0
 1.0
 0.8656987233224229

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

304.730 ms (2728974 allocations: 61.56 MiB)
retcode: Success
u: 4-element Vector{Float64}:
 0.999947033693279
 1.0
 0.9999478176213619
 0.8656431187791497

Each run's estimate, retcode, loss and largest componentwise distance from the true parameters [0.7, 0.8, 0.08, 0.5] 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=[0.943, 1.065, 0.089, 0.522]  retcode=
MaxIters  loss=64.27936324010106  max|est-true|=0.265
GN_CRS2_LM                  estimate=[0.7, 0.8, 0.08, 0.5]  retcode=MaxIter
s  loss=7.134691498914623e-19  max|est-true|=3.37e-12
GN_ESCH                     estimate=[2.488, 2.591, 0.122, 0.477]  retcode=
MaxIters  loss=704.5697798658714  max|est-true|=1.79
GN_ISRES                    estimate=[0.7, 0.8, 0.08, 0.5]  retcode=MaxIter
s  loss=1.850013957747529e-13  max|est-true|=4.16e-9
GN_ORIG_DIRECT_L            estimate=[1.111, 1.111, 0.101, 0.576]  retcode=
MaxIters  loss=81.06091854552861  max|est-true|=0.411
LD_SLSQP                    estimate=[1.0, 1.0, 1.0, 0.866]  retcode=Succes
s  loss=3160.449965970858  max|est-true|=0.92
LN_BOBYQA                   estimate=[0.7, 0.8, 0.08, 0.5]  retcode=Success
  loss=7.134817955780692e-19  max|est-true|=3.34e-12
LN_NELDERMEAD               estimate=[1.0, 1.0, 1.0, 0.866]  retcode=Succes
s  loss=3160.4055222879815  max|est-true|=0.92
lowest loss: GN_CRS2_LM (loss = 7.134691498914623e-19)
closest to the true parameters: LN_BOBYQA (max|est-true| = 3.34177130412172
1e-12)

Conclusion

The parameters used to generate the data are [0.7, 0.8, 0.08, 0.5]. The summary tables above print each run's estimate, retcode, loss and largest componentwise distance from these parameters on the short (300-observation) and longer (3000-observation) problems, including which run has the lowest loss and which lands closest. The global optimizers search glo_bounds starting from glo_init, while the local optimizers search the narrower loc_bounds starting from loc_init; 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","FitzHughNagumoParameterEstimation.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:

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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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⌃ [53ae85a6] SciMLStructures v1.10.5
  [7e506255] ScopedValues v1.6.2
  [6c6a2e73] Scratch v1.3.0
  [91c51154] SentinelArrays v1.4.10
  [efcf1570] Setfield v1.1.2
  [992d4aef] Showoff v1.1.1
  [05bca326] SimpleDiffEq v1.18.0
⌃ [727e6d20] SimpleNonlinearSolve v2.14.3
⌃ [510db2f7] SimpleOptimization v2.0.1
  [699a6c99] SimpleTraits v0.9.6
  [ed01d8cd] Sobol v1.5.0
  [a2af1166] SortingAlgorithms v1.2.3
  [a57abbd0] SparseColumnPivotedQR v2.1.8
⌃ [9f842d2f] SparseConnectivityTracer v1.2.3
⌃ [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
⌃ [90137ffa] StaticArrays v1.9.20
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  [82ae8749] StatsAPI v1.8.0
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  [69024149] StringEncodings v0.3.7
⌅ [892a3eda] StringManipulation v0.5.0
  [53d494c1] StructIO v0.3.1
  [2efcf032] SymbolicIndexingInterface v0.3.55
⌃ [19f23fe9] SymbolicLimits v1.2.1
⌃ [d1185830] SymbolicUtils v4.46.5
⌃ [0c5d862f] Symbolics v7.39.2
  [3783bdb8] TableTraits v1.0.1
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  [ed4db957] TaskLocalValues v0.1.3
  [62fd8b95] TensorCore v0.1.1
⌃ [8ea1fca8] TermInterface v2.0.0
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⌃ [a759f4b9] TimerOutputs v1.2.1
  [e689c965] Tracy v0.1.6
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⌃ [5c2747f8] URIs v1.7.0
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⌅ [013be700] UnsafeAtomics v0.3.2
  [41fe7b60] Unzip v0.2.0
  [d30d5f5c] WeakCacheSets v0.1.0
  [44d3d7a6] Weave v0.10.12
⌃ [ddb6d928] YAML v0.4.16 [loaded: v0.4.17]
  [700de1a5] ZygoteRules v0.2.8
  [6e34b625] Bzip2_jll v1.0.9+0
⌃ [83423d85] Cairo_jll v1.18.7+0
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⌅ [7cc45869] Enzyme_jll v0.0.293+0
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⌃ [2e619515] Expat_jll v2.8.4+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
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⌃ [d7e528f0] FreeType2_jll v2.14.3+1
⌃ [559328eb] FriBidi_jll v1.0.17+0
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⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
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⌃ [7746bdde] Glib_jll v2.88.3+0
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⌃ [2e76f6c2] HarfBuzz_jll v100.14004.0+0
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⌅ [dad2f222] LLVMExtra_jll v0.0.47+0
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⌅ [e9f186c6] Libffi_jll v3.4.7+0
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⌃ [4b2f31a3] Libmount_jll v2.42.0+0
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⌃ [38a345b3] Libuuid_jll v2.42.0+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
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  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.4+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
⌃ [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.8+0
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⌃ [a51aa0fd] Xorg_libXi_jll v1.8.4+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
⌃ [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
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  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
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  [2def613f] Xorg_xcb_util_jll v0.4.1+0
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  [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
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⌅ [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
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  [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
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  [0dad84c5] ArgTools v1.1.2
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  [ade2ca70] Dates v1.11.0
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  [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
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  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.3.1+2
  [deac9b47] LibCURL_jll v8.15.0+0
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  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
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  [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.