Lorenz Parameter Estimation Benchmarks

Estimate the parameters of the Lorenz system from the dataset

Note: If data is generated with a fixed time step method and then is tested against with the same time step, there is a bias introduced since it's no longer about hitting the true solution, rather it's just about retrieving the same values that the ODE was first generated by! Thus this version uses adaptive timestepping for all portions so that way tests are against the true solution.

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

Plots.GRBackend()

Xiang2015Bounds = Tuple{Float64, Float64}[(9, 11), (20, 30), (
    2, 3)] # for local optimizations
xlow_bounds = [9.0, 20.0, 2.0]
xhigh_bounds = [11.0, 30.0, 3.0]
LooserBounds = Tuple{Float64, Float64}[(0, 22), (0, 60), (
    0, 6)] # for global optimization
GloIniPar = [0.0, 0.5, 0.1] # for global optimizations
LocIniPar = [9.0, 20.0, 2.0] # for local optimization

3-element Vector{Float64}:
  9.0
 20.0
  2.0

@mtkmodel LorenzExample begin
    @parameters begin
        σ = 10.0  # Parameter: Prandtl number
        ρ = 28.0  # Parameter: Rayleigh number
        β = 8/3   # Parameter: Geometric factor
    end
    @variables begin
        x(t) = 1.0  # Initial condition for x
        y(t) = 1.0  # Initial condition for y
        z(t) = 1.0  # Initial condition for z
    end
    @equations begin
        D(x) ~ σ * (y - x)
        D(y) ~ x * (ρ - z) - y
        D(z) ~ x * y - β * z
    end
end

@mtkbuild g1 = LorenzExample()
p = [10.0, 28.0, 2.66] # Parameters used to construct the dataset
r0 = [1.0; 0.0; 0.0]                #[-11.8,-5.1,37.5] PODES Initial values of the system in space # [0.1, 0.0, 0.0]
# unknowns(g1) is ordered [z, y, x] rather than in declaration order, so the
# initial condition and parameters are keyed by name instead of positionally.
varmap = merge(Dict([g1.x, g1.y, g1.z] .=> r0), Dict([g1.σ, g1.ρ, g1.β] .=> p))
tspan = (0.0, 30.0)                 # PODES sample of 3000 observations over the (0,30) timespan
prob = ODEProblem(g1, varmap, tspan)
tspan2 = (0.0, 3.0)                 # Xiang test sample of 300 observations with a timestep of 0.01
prob_short = ODEProblem(g1, 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: 3-element Vector{Float64}:
 0.0
 0.0
 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)

3×3001 Matrix{Float64}:
 0.0  0.00126393  0.00465567  0.00983655  …  39.19     39.1699  38.904
 0.0  0.26634     0.51174     0.744654        8.31875   6.7199   5.22868
 1.0  0.917924    0.867919    0.84536        13.8987   13.2896  12.5913

Plot the data

plot(data_sol_short, vars = (1, 2, 3)) # the short solution
plot(data_sol, vars = (1, 2, 3)) # the longer solution
interpolation_sol = solve(prob, Vern7(), saveat = time_points, reltol = 1e-12, abstol = 1e-12)
plot(interpolation_sol, vars = (1, 2, 3))

xyzt = plot(data_sol_short, plotdensity = 10000, lw = 1.5)
xy = plot(data_sol_short, plotdensity = 10000, vars = (1, 2))
xz = plot(data_sol_short, plotdensity = 10000, vars = (1, 3))
yz = plot(data_sol_short, plotdensity = 10000, vars = (2, 3))
xyz = plot(data_sol_short, plotdensity = 10000, vars = (1, 2, 3))
plot(plot(xyzt, xyz), plot(xy, xz, yz, layout = (1, 3), w = 1), layout = (2, 1), size = (
    800, 600))

xyzt = plot(data_sol, plotdensity = 10000, lw = 1.5)
xy = plot(data_sol, plotdensity = 10000, vars = (1, 2))
xz = plot(data_sol, plotdensity = 10000, vars = (1, 3))
yz = plot(data_sol, plotdensity = 10000, vars = (2, 3))
xyz = plot(data_sol, plotdensity = 10000, vars = (1, 2, 3))
plot(plot(xyzt, xyz), plot(xy, xz, yz, layout = (1, 3), w = 1), layout = (2, 1), size = (
    800, 600))

Find a local solution for the three parameters from a short data set

obj_short = build_loss_objective(prob_short, Tsit5(), L2Loss(t_short, data_short), tstops = t_short)
optprob = OptimizationProblem(obj_short, LocIniPar, lb = xlow_bounds, ub = xhigh_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

3.104 s (21422265 allocations: 810.09 MiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  9.99999958167166
 28.000000184610723
  2.6599999509502785

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

4.555 s (33941941 allocations: 1002.75 MiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  9.999999950185428
 28.00000001810664
  2.6599999942497954

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, LocIniPar, lb = xlow_bounds, ub = xhigh_bounds)
res1 = @btime solve(optprob, BBO_adaptive_de_rand_1_bin(), maxiters = 7e3)
short_solutions["BBO (Vern9 1e-9)"] = res1

4.896 s (42398262 allocations: 1.11 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.000000015152246
 27.999999997843137
  2.659999966370605

Using NLopt

First, the global optimization algorithms

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

SciMLBase.OptimizationFunction{true, ADTypes.AutoForwardDiff{nothing, Nothi
ng}, DiffEqParamEstim.var"#37#38"{Nothing, typeof(DiffEqParamEstim.STANDARD
_PROB_GENERATOR), Base.Pairs{Symbol, Any, Nothing, @NamedTuple{tstops::Vect
or{Float64}, reltol::Float64, abstol::Float64}}, SciMLBase.ODEProblem{Vecto
r{Float64}, Tuple{Float64, Float64}, true, ModelingToolkitBase.MTKParameter
s{Vector{Float64}, Vector{Float64}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}, Sc
iMLBase.ODEFunction{true, SciMLBase.AutoDespecialize, ModelingToolkitBase.G
eneratedFunctionWrapper{Tuple{2, 3, true}, RuntimeGeneratedFunctions.Runtim
eGeneratedFunction{(:___mtkunknowns___, :___mtkparameters___, :__argₛᵧₘ1296
8198168038750593), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBa
se.var"#_RGF_ModTag", (0xef05dfea, 0xbf0f5a9f, 0x987e0f45, 0x8d1ed9b8, 0x33
4d458b), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__a
rgₛᵧₘ1401282876548370056, :___mtkunknowns___, :___mtkparameters___, :__argₛ
ᵧₘ12968198168038750593), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToo
lkitBase.var"#_RGF_ModTag", (0x62ee0678, 0xa187627b, 0x2814c1fa, 0xbb69d703
, 0x3a47543e), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Noth
ing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing
, Nothing, Nothing, ModelingToolkitBase.ObservedFunctionCache{ModelingToolk
itBase.System, Nothing}, Nothing, ModelingToolkitBase.System, Union{Nothing
, SciMLBase.OverrideInitData}, Union{Nothing, SciMLBase.ODENLStepData}}, Ba
se.Pairs{Symbol, Union{}, Nothing, @NamedTuple{}}, SciMLBase.StandardODEPro
blem}, OrdinaryDiffEqVerner.Vern9{typeof(OrdinaryDiffEqCore.trivial_limiter
!), typeof(OrdinaryDiffEqCore.trivial_limiter!), FastBroadcast.Serial, Val{
true}}, DiffEqParamEstim.L2Loss{Vector{Float64}, Matrix{Float64}, Nothing, 
Nothing, Nothing, Nothing}, Nothing, Tuple{}}, Nothing, Nothing, Nothing, N
othing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Noth
ing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED_NO_TIME), Nothing, Nothing,
 Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing}(Di
ffEqParamEstim.var"#37#38"{Nothing, typeof(DiffEqParamEstim.STANDARD_PROB_G
ENERATOR), Base.Pairs{Symbol, Any, Nothing, @NamedTuple{tstops::Vector{Floa
t64}, reltol::Float64, abstol::Float64}}, SciMLBase.ODEProblem{Vector{Float
64}, Tuple{Float64, Float64}, true, ModelingToolkitBase.MTKParameters{Vecto
r{Float64}, Vector{Float64}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}, SciMLBase
.ODEFunction{true, SciMLBase.AutoDespecialize, ModelingToolkitBase.Generate
dFunctionWrapper{Tuple{2, 3, true}, RuntimeGeneratedFunctions.RuntimeGenera
tedFunction{(:___mtkunknowns___, :___mtkparameters___, :__argₛᵧₘ12968198168
038750593), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"
#_RGF_ModTag", (0xef05dfea, 0xbf0f5a9f, 0x987e0f45, 0x8d1ed9b8, 0x334d458b)
, Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ14
01282876548370056, :___mtkunknowns___, :___mtkparameters___, :__argₛᵧₘ12968
198168038750593), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBas
e.var"#_RGF_ModTag", (0x62ee0678, 0xa187627b, 0x2814c1fa, 0xbb69d703, 0x3a4
7543e), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, No
thing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothi
ng, Nothing, ModelingToolkitBase.ObservedFunctionCache{ModelingToolkitBase.
System, Nothing}, Nothing, ModelingToolkitBase.System, Union{Nothing, SciML
Base.OverrideInitData}, Union{Nothing, SciMLBase.ODENLStepData}}, Base.Pair
s{Symbol, Union{}, Nothing, @NamedTuple{}}, SciMLBase.StandardODEProblem}, 
OrdinaryDiffEqVerner.Vern9{typeof(OrdinaryDiffEqCore.trivial_limiter!), typ
eof(OrdinaryDiffEqCore.trivial_limiter!), FastBroadcast.Serial, Val{true}},
 DiffEqParamEstim.L2Loss{Vector{Float64}, Matrix{Float64}, Nothing, Nothing
, Nothing, Nothing}, Nothing, Tuple{}}(nothing, DiffEqParamEstim.STANDARD_P
ROB_GENERATOR, Base.Pairs{Symbol, 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), SciMLBa
se.ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, ModelingToolk
itBase.MTKParameters{Vector{Float64}, Vector{Float64}, Tuple{}, Tuple{}, Tu
ple{}, Tuple{}}, SciMLBase.ODEFunction{true, SciMLBase.AutoDespecialize, Mo
delingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3, true}, RuntimeGenera
tedFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :___mtkparameter
s___, :__argₛᵧₘ12968198168038750593), ModelingToolkitBase.var"#_RGF_ModTag"
, ModelingToolkitBase.var"#_RGF_ModTag", (0xef05dfea, 0xbf0f5a9f, 0x987e0f4
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ratedFunction{(:__argₛᵧₘ1401282876548370056, :___mtkunknowns___, :___mtkpar
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odTag", ModelingToolkitBase.var"#_RGF_ModTag", (0x62ee0678, 0xa187627b, 0x2
814c1fa, 0xbb69d703, 0x3a47543e), Nothing}}, LinearAlgebra.UniformScaling{B
ool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothin
g, Nothing, Nothing, Nothing, Nothing, ModelingToolkitBase.ObservedFunction
Cache{ModelingToolkitBase.System, Nothing}, Nothing, ModelingToolkitBase.Sy
stem, Union{Nothing, SciMLBase.OverrideInitData}, Union{Nothing, SciMLBase.
ODENLStepData}}, Base.Pairs{Symbol, Union{}, Nothing, @NamedTuple{}}, SciML
Base.StandardODEProblem}(SciMLBase.ODEFunction{true, SciMLBase.AutoDespecia
lize, ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3, true}, Runti
meGeneratedFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :___mtkp
arameters___, :__argₛᵧₘ12968198168038750593), ModelingToolkitBase.var"#_RGF
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x987e0f45, 0x8d1ed9b8, 0x334d458b), Nothing}, RuntimeGeneratedFunctions.Run
timeGeneratedFunction{(:__argₛᵧₘ1401282876548370056, :___mtkunknowns___, :_
__mtkparameters___, :__argₛᵧₘ12968198168038750593), ModelingToolkitBase.var
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27b, 0x2814c1fa, 0xbb69d703, 0x3a47543e), Nothing}}, LinearAlgebra.UniformS
caling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing
, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkitBase.Observed
FunctionCache{ModelingToolkitBase.System, Nothing}, Nothing, ModelingToolki
tBase.System, Union{Nothing, SciMLBase.OverrideInitData}, Union{Nothing, Sc
iMLBase.ODENLStepData}}(ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{
2, 3, true}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:___mtkunk
nowns___, :___mtkparameters___, :__argₛᵧₘ12968198168038750593), ModelingToo
lkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xef05d
fea, 0xbf0f5a9f, 0x987e0f45, 0x8d1ed9b8, 0x334d458b), Nothing}, RuntimeGene
ratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ1401282876548370056, :___
mtkunknowns___, :___mtkparameters___, :__argₛᵧₘ12968198168038750593), Model
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x62ee0678, 0xa187627b, 0x2814c1fa, 0xbb69d703, 0x3a47543e), Nothing}}(Runti
meGeneratedFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :___mtkp
arameters___, :__argₛᵧₘ12968198168038750593), ModelingToolkitBase.var"#_RGF
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tions.RuntimeGeneratedFunction{(:__argₛᵧₘ1401282876548370056, :___mtkunknow
ns___, :___mtkparameters___, :__argₛᵧₘ12968198168038750593), ModelingToolki
tBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0x62ee0678
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arAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, n
othing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, Mode
lingToolkitBase.ObservedFunctionCache{ModelingToolkitBase.System, Nothing}(
Model g1:
Equations (3):
  3 standard: see equations(g1)
Unknowns (3): see unknowns(g1)
  z(t)
  y(t)
  x(t)
Parameters (3): see parameters(g1)
  σ
  ρ
  β, Dict{Any, Any}(), false, false, ModelingToolkitBase, false, nothing), 
nothing, Model g1:
Equations (3):
  3 standard: see equations(g1)
Unknowns (3): see unknowns(g1)
  z(t)
  y(t)
  x(t)
Parameters (3): see parameters(g1)
  σ
  ρ
  β, SciMLBase.OverrideInitData{SciMLBase.NonlinearProblem{Nothing, true, M
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nctions.RuntimeGeneratedFunction{(:x1, :x2, :x3, :x4), ModelingToolkitBase.
var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xb896e553, 0xa1
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ase.GetUpdatedU0{Nothing, SymbolicIndexingInterface.MultipleParametersGette
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Tuple{}, Tuple{}, Tuple{}}, SciMLBase.NonlinearFunction{true, SciMLBase.Aut
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LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Not
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rvedFunctionCache{ModelingToolkitBase.System, Nothing}, Nothing, ModelingTo
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ase.NonlinearFunction{true, SciMLBase.AutoDespecialize, ModelingToolkitBase
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olkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xe0a7
f55e, 0xcb0f8c13, 0x5afbe72e, 0x825c4ce9, 0x3418c4a5), Nothing}, RuntimeGen
eratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ1401282876548370056, :__
_mtkunknowns___, :___mtkparameters___), ModelingToolkitBase.var"#_RGF_ModTa
g", ModelingToolkitBase.var"#_RGF_ModTag", (0xac9dd6f8, 0xf3d8414c, 0x01f2c
ade, 0x451b4ee8, 0x29c649f2), Nothing}}, LinearAlgebra.UniformScaling{Bool}
, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, N
othing, Nothing, ModelingToolkitBase.ObservedFunctionCache{ModelingToolkitB
ase.System, Nothing}, Nothing, ModelingToolkitBase.System, Nothing, Nothing
}(ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 2, true}, RuntimeGe
neratedFunctions.RuntimeGeneratedFunction{(:___mtkunknowns___, :___mtkparam
eters___), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#
_RGF_ModTag", (0xe0a7f55e, 0xcb0f8c13, 0x5afbe72e, 0x825c4ce9, 0x3418c4a5),
 Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__argₛᵧₘ140
1282876548370056, :___mtkunknowns___, :___mtkparameters___), ModelingToolki
tBase.var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xac9dd6f8
, 0xf3d8414c, 0x01f2cade, 0x451b4ee8, 0x29c649f2), Nothing}}(RuntimeGenerat
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 g1:
Parameters (10): see parameters(g1)
  t
  σ
  ρ
  β
  ⋮
Observed (6): see observed(g1), Dict{Any, Any}(), false, false, ModelingToo
lkitBase, false, nothing), nothing, Model g1:
Parameters (10): see parameters(g1)
  t
  σ
  ρ
  β
  ⋮
Observed (6): see observed(g1), nothing, nothing), nothing, ModelingToolkit
Base.MTKParameters{Vector{Float64}, StaticArraysCore.SVector{0, Float64}, T
uple{}, Tuple{}, Tuple{}, Tuple{}}([0.0, 10.0, 28.0, 2.66, 0.0, 0.0, 0.0, 0
.0, 1.0, 0.0], Float64[], (), (), (), ()), SciMLBase.StandardNonlinearProbl
em(), nothing, nothing, Base.Pairs{Symbol, Union{}, Nothing, @NamedTuple{}}
()), ModelingToolkitBase.update_initializeprob!, ModelingToolkitBase.Initia
lizationMap{true, typeof(identity), ModelingToolkitBase.PromoteToTunableElt
ype{ModelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBas
e.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}, ModelingToolki
tBase.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Nothin
g}, Float64}}(identity, ModelingToolkitBase.PromoteToTunableEltype{Modeling
ToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.ParameterI
ndex{SciMLStructures.Tunable, UnitRange{Int64}}, ModelingToolkitBase.Parame
terIndex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Nothing}, Float64}
(ModelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.P
arameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}, ModelingToolkitBa
se.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Nothing}(
(ModelingToolkitBase.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int6
4}}(SciMLStructures.Tunable(), 6:6, false), ModelingToolkitBase.ParameterIn
dex{SciMLStructures.Tunable, UnitRange{Int64}}(SciMLStructures.Tunable(), 8
:9, false)), (3,), nothing))), ModelingToolkitBase.var"#initprobpmap_split#
770"{ModelingToolkitBase.MTKParametersReconstructor{ComposedFunction{Modeli
ngToolkitBase.PConstructorApplicator{typeof(identity)}, ModelingToolkitBase
.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.ParameterIndex{SciMLS
tructures.Tunable, UnitRange{Int64}}}, 1, Nothing}}, ComposedFunction{Model
ingToolkitBase.PConstructorApplicator{typeof(identity)}, ModelingToolkitBas
e.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.ParameterIndex{SciML
Structures.Tunable, UnitRange{Int64}}}, 1, Nothing}}, Returns{Tuple{}}, Ret
urns{Tuple{}}, Returns{Tuple{}}}}(ModelingToolkitBase.MTKParametersReconstr
uctor{ComposedFunction{ModelingToolkitBase.PConstructorApplicator{typeof(id
entity)}, ModelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingTool
kitBase.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Noth
ing}}, ComposedFunction{ModelingToolkitBase.PConstructorApplicator{typeof(i
dentity)}, ModelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToo
lkitBase.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}}, 1, Not
hing}}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}(ModelingToolk
itBase.PConstructorApplicator{typeof(identity)}(identity) ∘ ModelingToolkit
Base.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.ParameterIndex{Sc
iMLStructures.Tunable, UnitRange{Int64}}}, 1, Nothing}((ModelingToolkitBase
.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int64}}(SciMLStructures.
Tunable(), 2:4, false),), (3,), nothing), ModelingToolkitBase.PConstructorA
pplicator{typeof(identity)}(identity) ∘ ModelingToolkitBase.CopyParamsByTem
plate{true, Tuple{ModelingToolkitBase.ParameterIndex{SciMLStructures.Tunabl
e, UnitRange{Int64}}}, 1, Nothing}((ModelingToolkitBase.ParameterIndex{SciM
LStructures.Tunable, UnitRange{Int64}}(SciMLStructures.Tunable(), 5:10, fal
se),), (6,), nothing), Returns{Tuple{}}(()), Returns{Tuple{}}(()), Returns{
Tuple{}}(()), 0)), ModelingToolkitBase.InitializationMetadata{ModelingToolk
itBase.ReconstructInitializeprob{ModelingToolkitBase.MTKParametersReconstru
ctor{ComposedFunction{ModelingToolkitBase.PConstructorApplicator{typeof(ide
ntity)}, ModelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolk
itBase.IndepVarTemplate, ModelingToolkitBase.ParameterIndex{SciMLStructures
.Tunable, UnitRange{Int64}}, ModelingToolkitBase.ParameterIndex{SciMLStruct
ures.Initials, UnitRange{Int64}}}, 1, Nothing}}, Returns{StaticArraysCore.S
Vector{0, Float64}}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}},
 ComposedFunction{typeof(identity), ModelingToolkitBase.ObservedWrapper{tru
e, ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 3, true}, RuntimeG
eneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters
___, :__argₛᵧₘ12968198168038750593), ModelingToolkitBase.var"#_RGF_ModTag",
 ModelingToolkitBase.var"#_RGF_ModTag", (0x05e7e591, 0x9a09c886, 0x0267af3f
, 0x585cfe8b, 0x75f79d7b), Nothing}, RuntimeGeneratedFunctions.RuntimeGener
atedFunction{(:x1, :x2, :x3, :x4), ModelingToolkitBase.var"#_RGF_ModTag", M
odelingToolkitBase.var"#_RGF_ModTag", (0xb896e553, 0xa118fcf5, 0xbfeb9719, 
0xa096e58a, 0x357542a4), Nothing}}}}}, ModelingToolkitBase.GetUpdatedU0{Not
hing, SymbolicIndexingInterface.MultipleParametersGetter{SymbolicIndexingIn
terface.IndexerNotTimeseries, Vector{SymbolicIndexingInterface.GetParameter
Index{ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, Int64}}}
, Nothing}}, ModelingToolkitBase.SetInitialUnknowns{SymbolicIndexingInterfa
ce.MultipleSetters{Vector{SymbolicIndexingInterface.ParameterHookWrapper{Sy
mbolicIndexingInterface.SetParameterIndex{ModelingToolkitBase.ParameterInde
x{SciMLStructures.Initials, Int64}}, SymbolicUtils.BasicSymbolicImpl.var"ty
peof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}}}}}}(ModelingToolkitBase.At
omicArrayDict{SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)
"{SymbolicUtils.SymReal}, Dict{SymbolicUtils.BasicSymbolicImpl.var"typeof(B
asicSymbolicImpl)"{SymbolicUtils.SymReal}, SymbolicUtils.BasicSymbolicImpl.
var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}}}(Initial(xˍt(t)) => 
false, Initial(x(t)) => false, z(t) => 0.0, Initial(z(t)) => false, σ => 10
.0, y(t) => 0.0, Initial(yˍt(t)) => false, Initial(y(t)) => false, ρ => 28.
0, β => 2.66…), ModelingToolkitBase.AtomicArrayDict{SymbolicUtils.BasicSymb
olicImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}, Dict{Symbol
icUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymR
eal}, SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{Symbol
icUtils.SymReal}}}(), Symbolics.Equation[], true, true, 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
edFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :
__argₛᵧₘ12968198168038750593), ModelingToolkitBase.var"#_RGF_ModTag", Model
ingToolkitBase.var"#_RGF_ModTag", (0x05e7e591, 0x9a09c886, 0x0267af3f, 0x58
5cfe8b, 0x75f79d7b), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFu
nction{(:x1, :x2, :x3, :x4), ModelingToolkitBase.var"#_RGF_ModTag", Modelin
gToolkitBase.var"#_RGF_ModTag", (0xb896e553, 0xa118fcf5, 0xbfeb9719, 0xa096
e58a, 0x357542a4), Nothing}}}}}(ModelingToolkitBase.MTKParametersReconstruc
tor{ComposedFunction{ModelingToolkitBase.PConstructorApplicator{typeof(iden
tity)}, ModelingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolki
tBase.IndepVarTemplate, ModelingToolkitBase.ParameterIndex{SciMLStructures.
Tunable, UnitRange{Int64}}, ModelingToolkitBase.ParameterIndex{SciMLStructu
res.Initials, UnitRange{Int64}}}, 1, Nothing}}, Returns{StaticArraysCore.SV
ector{0, Float64}}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}(M
odelingToolkitBase.PConstructorApplicator{typeof(identity)}(identity) ∘ Mod
elingToolkitBase.CopyParamsByTemplate{true, Tuple{ModelingToolkitBase.Indep
VarTemplate, ModelingToolkitBase.ParameterIndex{SciMLStructures.Tunable, Un
itRange{Int64}}, ModelingToolkitBase.ParameterIndex{SciMLStructures.Initial
s, UnitRange{Int64}}}, 1, Nothing}((ModelingToolkitBase.IndepVarTemplate(),
 ModelingToolkitBase.ParameterIndex{SciMLStructures.Tunable, UnitRange{Int6
4}}(SciMLStructures.Tunable(), 1:3, false), ModelingToolkitBase.ParameterIn
dex{SciMLStructures.Initials, UnitRange{Int64}}(SciMLStructures.Initials(),
 1:6, false)), (10,), nothing), Returns{StaticArraysCore.SVector{0, Float64
}}(Float64[]), Returns{Tuple{}}(()), Returns{Tuple{}}(()), Returns{Tuple{}}
(()), 0), identity ∘ ModelingToolkitBase.ObservedWrapper{true, ModelingTool
kitBase.GeneratedFunctionWrapper{Tuple{2, 3, true}, RuntimeGeneratedFunctio
ns.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :__argₛᵧₘ1
2968198168038750593), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolki
tBase.var"#_RGF_ModTag", (0x05e7e591, 0x9a09c886, 0x0267af3f, 0x585cfe8b, 0
x75f79d7b), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:
x1, :x2, :x3, :x4), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolkitB
ase.var"#_RGF_ModTag", (0xb896e553, 0xa118fcf5, 0xbfeb9719, 0xa096e58a, 0x3
57542a4), Nothing}}}(ModelingToolkitBase.GeneratedFunctionWrapper{Tuple{2, 
3, true}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1,
 :___mtkparameters___, :__argₛᵧₘ12968198168038750593), ModelingToolkitBase.
var"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0x05e7e591, 0x9a
09c886, 0x0267af3f, 0x585cfe8b, 0x75f79d7b), Nothing}, RuntimeGeneratedFunc
tions.RuntimeGeneratedFunction{(:x1, :x2, :x3, :x4), ModelingToolkitBase.va
r"#_RGF_ModTag", ModelingToolkitBase.var"#_RGF_ModTag", (0xb896e553, 0xa118
fcf5, 0xbfeb9719, 0xa096e58a, 0x357542a4), Nothing}}(RuntimeGeneratedFuncti
ons.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :__argₛᵧₘ
12968198168038750593), ModelingToolkitBase.var"#_RGF_ModTag", ModelingToolk
itBase.var"#_RGF_ModTag", (0x05e7e591, 0x9a09c886, 0x0267af3f, 0x585cfe8b, 
0x75f79d7b), Nothing}(nothing), RuntimeGeneratedFunctions.RuntimeGeneratedF
unction{(:x1, :x2, :x3, :x4), ModelingToolkitBase.var"#_RGF_ModTag", Modeli
ngToolkitBase.var"#_RGF_ModTag", (0xb896e553, 0xa118fcf5, 0xbfeb9719, 0xa09
6e58a, 0x357542a4), Nothing}(nothing)))), ModelingToolkitBase.GetUpdatedU0{
Nothing, SymbolicIndexingInterface.MultipleParametersGetter{SymbolicIndexin
gInterface.IndexerNotTimeseries, Vector{SymbolicIndexingInterface.GetParame
terIndex{ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, Int64
}}}, Nothing}}(Bool[0, 0, 0], nothing, SymbolicIndexingInterface.MultiplePa
rametersGetter{SymbolicIndexingInterface.IndexerNotTimeseries, Vector{Symbo
licIndexingInterface.GetParameterIndex{ModelingToolkitBase.ParameterIndex{S
ciMLStructures.Initials, Int64}}}, Nothing}(SymbolicIndexingInterface.GetPa
rameterIndex{ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, I
nt64}}[SymbolicIndexingInterface.GetParameterIndex{ModelingToolkitBase.Para
meterIndex{SciMLStructures.Initials, Int64}}(ModelingToolkitBase.ParameterI
ndex{SciMLStructures.Initials, Int64}(SciMLStructures.Initials(), 2, false)
), SymbolicIndexingInterface.GetParameterIndex{ModelingToolkitBase.Paramete
rIndex{SciMLStructures.Initials, Int64}}(ModelingToolkitBase.ParameterIndex
{SciMLStructures.Initials, Int64}(SciMLStructures.Initials(), 4, false)), S
ymbolicIndexingInterface.GetParameterIndex{ModelingToolkitBase.ParameterInd
ex{SciMLStructures.Initials, Int64}}(ModelingToolkitBase.ParameterIndex{Sci
MLStructures.Initials, Int64}(SciMLStructures.Initials(), 5, false))], 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}}}}}(SymbolicIndexingInterface.Mul
tipleSetters{Vector{SymbolicIndexingInterface.ParameterHookWrapper{Symbolic
IndexingInterface.SetParameterIndex{ModelingToolkitBase.ParameterIndex{SciM
LStructures.Initials, Int64}}, SymbolicUtils.BasicSymbolicImpl.var"typeof(B
asicSymbolicImpl)"{SymbolicUtils.SymReal}}}}(SymbolicIndexingInterface.Para
meterHookWrapper{SymbolicIndexingInterface.SetParameterIndex{ModelingToolki
tBase.ParameterIndex{SciMLStructures.Initials, Int64}}, SymbolicUtils.Basic
SymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}}[Symboli
cIndexingInterface.ParameterHookWrapper{SymbolicIndexingInterface.SetParame
terIndex{ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, Int64
}}, SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{Symbolic
Utils.SymReal}}(SymbolicIndexingInterface.SetParameterIndex{ModelingToolkit
Base.ParameterIndex{SciMLStructures.Initials, Int64}}(ModelingToolkitBase.P
arameterIndex{SciMLStructures.Initials, Int64}(SciMLStructures.Initials(), 
2, false)), Initial(z(t))), SymbolicIndexingInterface.ParameterHookWrapper{
SymbolicIndexingInterface.SetParameterIndex{ModelingToolkitBase.ParameterIn
dex{SciMLStructures.Initials, Int64}}, SymbolicUtils.BasicSymbolicImpl.var"
typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}}(SymbolicIndexingInterfac
e.SetParameterIndex{ModelingToolkitBase.ParameterIndex{SciMLStructures.Init
ials, Int64}}(ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, 
Int64}(SciMLStructures.Initials(), 4, false)), Initial(y(t))), SymbolicInde
xingInterface.ParameterHookWrapper{SymbolicIndexingInterface.SetParameterIn
dex{ModelingToolkitBase.ParameterIndex{SciMLStructures.Initials, Int64}}, S
ymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils
.SymReal}}(SymbolicIndexingInterface.SetParameterIndex{ModelingToolkitBase.
ParameterIndex{SciMLStructures.Initials, Int64}}(ModelingToolkitBase.Parame
terIndex{SciMLStructures.Initials, Int64}(SciMLStructures.Initials(), 5, fa
lse)), Initial(x(t)))]), [2, 4, 5]), ModelingToolkitBase.MissingGuessValue.
var"typeof(MissingGuessValue)"(ModelingToolkitBase.MissingGuessValue.var"##
Storage#HashedRandom"())), Val{true}()), nothing), [0.0, 0.0, 1.0], (0.0, 3
.0), ModelingToolkitBase.MTKParameters{Vector{Float64}, Vector{Float64}, Tu
ple{}, Tuple{}, Tuple{}, Tuple{}}([10.0, 28.0, 2.66], [0.0, 0.0, 0.0, 0.0, 
1.0, 0.0], (), (), (), ()), Base.Pairs{Symbol, Union{}, Nothing, @NamedTupl
e{}}(), SciMLBase.StandardODEProblem()), OrdinaryDiffEqVerner.Vern9{typeof(
OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_lim
iter!), FastBroadcast.Serial, Val{true}}(OrdinaryDiffEqCore.trivial_limiter
!, OrdinaryDiffEqCore.trivial_limiter!, FastBroadcast.Serial(), Val{true}()
), DiffEqParamEstim.L2Loss{Vector{Float64}, Matrix{Float64}, Nothing, Nothi
ng, 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], [0.0 
0.0012639260967322993 … 28.337036460460194 28.165930892029362; 0.0 0.266339
98088990296 … -7.0863707672554055 -6.9951453942460375; 1.0 0.91792445412835
03 … -8.30544480108837 -8.18490008750312], nothing, nothing, nothing, nothi
ng, nothing), nothing, ()), ADTypes.AutoForwardDiff(), nothing, nothing, no
thing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothi
ng, nothing, nothing, SciMLBase.DEFAULT_OBSERVED_NO_TIME, nothing, nothing,
 nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing)

opt = Opt(:GN_ORIG_DIRECT_L, 3)
optprob = OptimizationProblem(obj_short, GloIniPar, lb = first.(LooserBounds), ub = last.(LooserBounds))
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["GN_ORIG_DIRECT_L"] = res1

7.040 s (59821214 allocations: 1.56 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.00000000005247
 28.000000000022816
  2.6600000000080746

opt = Opt(:GN_CRS2_LM, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["GN_CRS2_LM"] = res1

7.362 s (59752428 allocations: 1.56 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.000000000051847
 28.00000000002288
  2.66000000000799

opt = Opt(:GN_ISRES, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["GN_ISRES"] = res1

6.972 s (59957967 allocations: 1.56 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.003186894839686
 27.998505062411514
  2.661060418177894

opt = Opt(:GN_ESCH, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["GN_ESCH"] = res1

7.290 s (59809471 allocations: 1.56 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  9.814194109317777
 28.040475206432863
  2.6694805140464988

Next, the local optimization algorithms that could be used after the global algorithms as a check on the solution and its precision. All the local optimizers are started from LocIniPar and with the narrow bounds of the Xiang2015Paper.

opt = Opt(:LN_BOBYQA, 3)
optprob = OptimizationProblem(obj_short, LocIniPar, lb = xlow_bounds, ub = xhigh_bounds)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_BOBYQA"] = res1

202.557 ms (1911761 allocations: 51.12 MiB)
retcode: Success
u: 3-element Vector{Float64}:
 10.000000000051081
 28.000000000023142
  2.660000000007931

opt = Opt(:LN_NELDERMEAD, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_NELDERMEAD"] = res1

208.670 ms (2013496 allocations: 53.84 MiB)
retcode: Success
u: 3-element Vector{Float64}:
 10.000000000053411
 28.0000000000238
  2.6600000000081927

opt = Opt(:LD_SLSQP, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LD_SLSQP"] = res1

8.676 s (70719798 allocations: 1.90 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.000000000051806
 28.000000000022776
  2.660000000007988

opt = Opt(:LN_COBYLA, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_COBYLA"] = res1

2.699 s (22438865 allocations: 602.30 MiB)
retcode: Success
u: 3-element Vector{Float64}:
 10.000000000040231
 28.000000000025903
  2.660000000007147

opt = Opt(:LN_NEWUOA_BOUND, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_NEWUOA_BOUND"] = res1

451.927 ms (3208122 allocations: 85.79 MiB)
retcode: Success
u: 3-element Vector{Float64}:
  9.999992747331307
 27.999996979392797
  2.659999334666021

opt = Opt(:LN_PRAXIS, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_PRAXIS"] = res1

196.640 ms (1529443 allocations: 40.90 MiB)
retcode: Success
u: 3-element Vector{Float64}:
  9.0
 28.31067230326584
  2.579310946262273

opt = Opt(:LN_SBPLX, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LN_SBPLX"] = res1

536.830 ms (4468618 allocations: 119.49 MiB)
retcode: Success
u: 3-element Vector{Float64}:
 10.000000000058924
 28.000000000020663
  2.660000000008665

opt = Opt(:LD_MMA, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LD_MMA"] = res1

15.679 s (119380437 allocations: 3.42 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.000000000051813
 28.000000000022446
  2.6600000000079653

opt = Opt(:LD_LBFGS, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LD_LBFGS"] = res1

23.947 ms (214961 allocations: 6.29 MiB)
retcode: Success
u: 3-element Vector{Float64}:
 10.000000000052792
 28.000000000022634
  2.6600000000078925

opt = Opt(:LD_TNEWTON_PRECOND_RESTART, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
short_solutions["LD_TNEWTON_PRECOND_RESTART"] = res1

30.311 ms (275020 allocations: 8.11 MiB)
retcode: Success
u: 3-element Vector{Float64}:
 10.000000000051315
 28.00000000002281
  2.6600000000079493

Each run's estimate, retcode, loss and largest componentwise distance from the true parameters [10.0, 28.0, 2.66], 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=[10.0, 28.0, 2.66]  retcode=MaxIters  
loss=4.48840138052368e-11  max|est-true|=4.18e-7
BBO (Tsit5 reltol=1e-9)     estimate=[10.0, 28.0, 2.66]  retcode=MaxIters  
loss=4.363880425955739e-13  max|est-true|=4.98e-8
BBO (Vern9 1e-9)            estimate=[10.0, 28.0, 2.66]  retcode=MaxIters  
loss=3.1862469014764254e-11  max|est-true|=3.36e-8
GN_CRS2_LM                  estimate=[10.0, 28.0, 2.66]  retcode=MaxIters  
loss=2.765663420706971e-18  max|est-true|=5.18e-11
GN_ESCH                     estimate=[9.814, 28.04, 2.669]  retcode=MaxIter
s  loss=21.635830041364915  max|est-true|=0.186
GN_ISRES                    estimate=[10.003, 27.999, 2.661]  retcode=MaxIt
ers  loss=0.022093839491880075  max|est-true|=0.00319
GN_ORIG_DIRECT_L            estimate=[10.0, 28.0, 2.66]  retcode=MaxIters  
loss=2.7661304189470634e-18  max|est-true|=5.25e-11
LD_LBFGS                    estimate=[10.0, 28.0, 2.66]  retcode=Success  l
oss=2.767143177838859e-18  max|est-true|=5.28e-11
LD_MMA                      estimate=[10.0, 28.0, 2.66]  retcode=MaxIters  
loss=2.7666664170231267e-18  max|est-true|=5.18e-11
LD_SLSQP                    estimate=[10.0, 28.0, 2.66]  retcode=MaxIters  
loss=2.765864576836226e-18  max|est-true|=5.18e-11
LD_TNEWTON_PRECOND_RESTART  estimate=[10.0, 28.0, 2.66]  retcode=Success  l
oss=2.766826219444564e-18  max|est-true|=5.13e-11
LN_BOBYQA                   estimate=[10.0, 28.0, 2.66]  retcode=Success  l
oss=2.7662720882359067e-18  max|est-true|=5.11e-11
LN_COBYLA                   estimate=[10.0, 28.0, 2.66]  retcode=Success  l
oss=2.803778105481148e-18  max|est-true|=4.02e-11
LN_NELDERMEAD               estimate=[10.0, 28.0, 2.66]  retcode=Success  l
oss=2.769748214192164e-18  max|est-true|=5.34e-11
LN_NEWUOA_BOUND             estimate=[10.0, 28.0, 2.66]  retcode=Success  l
oss=3.866807398220735e-8  max|est-true|=7.25e-6
LN_PRAXIS                   estimate=[9.0, 28.311, 2.579]  retcode=Success 
 loss=292.75880000812424  max|est-true|=1.0
LN_SBPLX                    estimate=[10.0, 28.0, 2.66]  retcode=Success  l
oss=2.782644973352366e-18  max|est-true|=5.89e-11
lowest loss: GN_CRS2_LM (loss = 2.765663420706971e-18)
closest to the true parameters: LN_COBYLA (max|est-true| = 4.02309296987368
7e-11)

Now let's solve the longer version for a global solution

Notice from the plotting above that this ODE problem is chaotic and tends to diverge over time. In the longer version of parameter estimation, the dataset is increased to 3000 observations per variable with the same integration time step of 0.01. Vern9 solver with reltol=1e-9 and abstol=1e-9 has been established to be accurate on the time interval [0,50]. The long objective is built with Optimization.AutoForwardDiff() because LD_SLSQP below requires a gradient.

# BBO is given a smaller iteration budget on the long problem.
obj = build_loss_objective(prob, Vern9(), L2Loss(time_points, data),
    Optimization.AutoForwardDiff(), tstops = time_points, reltol = 1e-9, abstol = 1e-9)
optprob = OptimizationProblem(obj, GloIniPar, lb = first.(LooserBounds), ub = last.(LooserBounds))
global_solutions = Dict{String, Any}()
res1 = @btime solve(optprob, BBO_adaptive_de_rand_1_bin(); maxiters = 4e3)
global_solutions["BBO_adaptive_de_rand_1_bin"] = res1
#res1 = @btime bboptimize(obj;SearchRange = LooserBounds, Method = :adaptive_de_rand_1_bin, MaxSteps = 4e3)
#res1 = @btime bboptimize(obj;SearchRange = LooserBounds, Method = :dxnes, MaxSteps = 2e3)
#res1 = @btime bboptimize(obj;SearchRange = LooserBounds, Method = :xnes, MaxSteps = 2e3)
#res1 = @btime bboptimize(obj;SearchRange = LooserBounds, Method = :de_rand_1_bin_radiuslimited, MaxSteps = 2e3)
#res1 = @btime bboptimize(obj;SearchRange = LooserBounds, Method = :generating_set_search, MaxSteps = 2e3)

23.461 s (227060553 allocations: 4.69 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  7.566414773430441
 24.483736497674705
  2.024286312941135

# using Evolutionary
# N = 3
# @time result, fitness, cnt = cmaes(obj, N; μ = 3, λ = 12, iterations = 1000) # cmaes( rastrigin, N; μ = 15, λ = P, tol = 1e-8)

opt = Opt(:GN_ORIG_DIRECT_L, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12, maxtime = 300)
global_solutions["GN_ORIG_DIRECT_L"] = res1

56.669 s (547045583 allocations: 11.30 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  8.148148148143822
 25.53527675913541
  2.1353949816278246

opt = Opt(:GN_CRS2_LM, 3)
res1 = @btime solve(optprob, opt, maxiters = 20000, xtol_rel = 1e-12, maxtime = 300)
global_solutions["GN_CRS2_LM"] = res1

113.583 s (1091660877 allocations: 22.56 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  9.857901067373891
 28.085506534550532
  2.637245105224132

opt = Opt(:GN_ISRES, 3)
res1 = @btime solve(optprob, opt, maxiters = 50000, xtol_rel = 1e-12, maxtime = 300)
global_solutions["GN_ISRES"] = res1

283.810 s (2729814672 allocations: 56.40 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  8.818071714916293
 24.299299781865475
  2.209287214248194

opt = Opt(:GN_ESCH, 3)
res1 = @btime solve(optprob, opt, maxiters = 20000, xtol_rel = 1e-12, maxtime = 300)
global_solutions["GN_ESCH"] = res1

113.325 s (1091801984 allocations: 22.56 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 12.517831352922963
 23.160423943678694
  2.1634855434705367

Each global run's estimate, retcode, loss, largest componentwise distance from the true parameters [10.0, 28.0, 2.66], and whether the estimate lies inside the bounds its optimizer searched (LooserBounds) and inside the narrower Xiang2015Bounds used by the local optimizers below:

for (name, sol) in sort!(collect(global_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),
        "  within LooserBounds: ", all(first.(LooserBounds) .<= sol.u .<= last.(LooserBounds)),
        "  within Xiang2015Bounds: ", all(xlow_bounds .<= sol.u .<= xhigh_bounds))
end
best_loss, best_loss_name = minimum((sol.objective, name) for (name, sol) in global_solutions)
best_dist, best_dist_name = minimum((maximum(abs.(sol.u .- p)), name) for (name, sol) in global_solutions)
println("lowest loss: ", best_loss_name, " (loss = ", best_loss, ")")
println("closest to the true parameters: ", best_dist_name, " (max|est-true| = ", best_dist, ")")
println("loss at the true parameters ", p, ": ", obj(p))
println("loss with σ shifted by +1e-6: ", obj([10.0 + 1e-6, 28.0, 2.66]))
println("loss with σ shifted by +1e-4: ", obj([10.0 + 1e-4, 28.0, 2.66]))

BBO_adaptive_de_rand_1_bin  estimate=[7.566, 24.484, 2.024]  retcode=MaxIte
rs  loss=481918.31171202345  max|est-true|=3.52  within LooserBounds: true 
 within Xiang2015Bounds: false
GN_CRS2_LM                  estimate=[9.858, 28.086, 2.637]  retcode=MaxIte
rs  loss=163548.13951903005  max|est-true|=0.142  within LooserBounds: true
  within Xiang2015Bounds: true
GN_ESCH                     estimate=[12.518, 23.16, 2.163]  retcode=MaxIte
rs  loss=471010.9735330795  max|est-true|=4.84  within LooserBounds: true  
within Xiang2015Bounds: false
GN_ISRES                    estimate=[8.818, 24.299, 2.209]  retcode=MaxIte
rs  loss=467881.6191620816  max|est-true|=3.7  within LooserBounds: true  w
ithin Xiang2015Bounds: false
GN_ORIG_DIRECT_L            estimate=[8.148, 25.535, 2.135]  retcode=MaxIte
rs  loss=461239.58985747735  max|est-true|=2.46  within LooserBounds: true 
 within Xiang2015Bounds: false
lowest loss: GN_CRS2_LM (loss = 163548.13951903005)
closest to the true parameters: GN_CRS2_LM (max|est-true| = 0.1420989326261
0862)
loss at the true parameters [10.0, 28.0, 2.66]: 0.0030835739315063656
loss with σ shifted by +1e-6: 1997.473906162449
loss with σ shifted by +1e-4: 346569.7397136992

The printed table shows where each global run stopped on the longer sample. Each NLopt global run keeps its iteration budget and is additionally capped at 300 seconds of wall time via maxtime, so a run that does not converge first reports MaxTime or MaxIters. The loss at the true parameters, printed above, is small but not zero, because tstops at every data point forces the fitting solve onto a slightly different step sequence than the data-generation solve, and the chaotic trajectory amplifies that difference. The perturbation losses printed above are orders of magnitude larger, so the true parameters sit in a deep, narrow minimum on this problem.

opt = Opt(:LN_BOBYQA, 3)
optprob = OptimizationProblem(obj, LocIniPar, lb = xlow_bounds, ub = xhigh_bounds)
local_solutions = Dict{String, Any}()
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
local_solutions["LN_BOBYQA"] = res1

1.834 s (17630070 allocations: 373.02 MiB)
retcode: Success
u: 3-element Vector{Float64}:
  9.862590769081075
 20.58113390812207
  2.0

opt = Opt(:LN_NELDERMEAD, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
local_solutions["LN_NELDERMEAD"] = res1

56.390 s (545836014 allocations: 11.28 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  9.678915633380864
 23.516765371996325
  2.161073807096292

opt = Opt(:LD_SLSQP, 3)
res1 = @btime solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
local_solutions["LD_SLSQP"] = res1

43.827 ms (436700 allocations: 9.95 MiB)
retcode: Success
u: 3-element Vector{Float64}:
  9.248030972411785
 21.230339600399184
  2.105344917871357

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 Xiang2015Bounds: ", all(xlow_bounds .< sol.u .< xhigh_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=[9.248, 21.23, 2.105]  retcode=Success
  loss=549175.2002526054  max|est-true|=6.77  strictly inside Xiang2015Boun
ds: true
LN_BOBYQA                   estimate=[9.863, 20.581, 2.0]  retcode=Success 
 loss=588113.2784254166  max|est-true|=7.42  strictly inside Xiang2015Bound
s: false
LN_NELDERMEAD               estimate=[9.679, 23.517, 2.161]  retcode=MaxIte
rs  loss=404754.5095237799  max|est-true|=4.48  strictly inside Xiang2015Bo
unds: true
lowest loss: LN_NELDERMEAD (loss = 404754.5095237799)
closest to the true parameters: LN_NELDERMEAD (max|est-true| = 4.4832346280
03675)

These local optimizers run on the long objective obj from LocIniPar inside the narrow bounds. The printed table shows each run's estimate, retcode, loss, distance from the true parameters, and whether the estimate lies strictly inside Xiang2015Bounds; on this landscape a Success retcode does not by itself mean the run found the true parameters, and a run stopped on its iteration limit may still sit at a wrong point.

Conclusion

1. The Lorenz system is extremely sensitive to the initial condition; this page generates its data from r0 = [1.0, 0.0, 0.0]. The summary tables above print each run's estimate, retcode, loss and largest componentwise distance from the true parameters [10.0, 28.0, 2.66] on both the short (300-observation) and the longer (3000-observation) sample, including which run has the lowest loss and which lands closest; the long-problem global runs are additionally capped at 300 s of wall time via maxtime. The global optimizers are stochastic, so the distances and winners differ from run to run. Other initial conditions, such as r0 = [-11.8, -5.1, 37.5] from PODES, are not run on this page.
2. When trying to hit the real data, having a low enough tolerance on the numerical solution is key. If the numerical solution is too rough, then we can never actually hone in on the true parameters since even with the true parameters we will erroneously induce numerical error. Maybe this could be adaptive?
3. Tighter tolerances make each objective evaluation more expensive: the three BBO runs on the short problem use the default Tsit5 tolerances, 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.
4. The estimation method and the global versus local optimization make a huge difference in the timings. The chaotic trajectory makes the loss landscape much rougher on the long problem: the printed perturbation losses show even a 1e-6 shift in σ raising the loss by orders of magnitude. The LD_* local methods evaluate a gradient of the objective at every step; the printed timings show the per-run cost on each problem. These methods can scale a local optimization, but we are aiming to scale a global optimization.
5. Fitting the shorter timespan is much cheaper... maybe this can lead to determining a minimal sample size for the optimizers and the estimator to succeed.

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","LorenzParameterEstimation.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 `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/ParameterEstimation/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
  [a134a8b2] BlackBoxOptim v0.6.12
  [a93c6f00] DataFrames v1.8.2
⌃ [bcd4f6db] DelayDiffEq v6.4.1
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⌃ [961ee093] ModelingToolkit v11.43.0
⌃ [7771a370] ModelingToolkitBase v1.70.0
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⌃ [7f7a1694] Optimization v5.9.0
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⌃ [4e6fcdb7] OptimizationNLopt v0.3.18
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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.

And the full manifest:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/ParameterEstimation/Manifest.toml`
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
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