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 biased 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: t_nounits as t, D_nounits as D
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]
tspan = (0.0, 30.0)                 # PODES sample of 3000 observations over the (0,30) timespan
prob = ODEProblem(g1, r0, tspan,p)
tspan2 = (0.0, 3.0)                 # Xiang test sample of 300 observations with a timestep of 0.01
prob_short = ODEProblem(g1, r0, tspan2,p)

ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
timespan: (0.0, 3.0)
u0: 3-element Vector{Float64}:
 1.0
 0.0
 0.0

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}:
 1.0  0.975053     0.953295    …  15.8745  15.8745  15.8745  15.8745
 0.0  0.0982298    0.193153       15.8745  15.8745  15.8745  15.8745
 0.0  0.000443091  0.00157839      9.0      9.0      9.0      9.0

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=t,reltol=1e-12,abstol=1e-12)
plot(interpolation_sol,vars=(1,2,3))

Error: TypeError: non-boolean (Symbolics.Num) used in boolean context
A symbolic expression appeared in a Boolean context. This error arises in s
ituations where Julia expects a Bool, like 
if boolean_condition		 use ifelse(boolean_condition, then branch, else bran
ch)
x && y				 use x & y
boolean_condition ? a : b	 use ifelse(boolean_condition, a, b)
but a symbolic expression appeared instead of a Bool. For help regarding co
ntrol flow with symbolic variables, see https://docs.sciml.ai/ModelingToolk
it/dev/basics/FAQ/#How-do-I-handle-if-statements-in-my-symbolic-forms?

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)
@btime res1 = solve(optprob, BBO_adaptive_de_rand_1_bin(), maxiters = 7e3)
# Tolernace is still too high to get close enough

1.238 s (2615087 allocations: 365.53 MiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.000000011896784
 27.999999679098426
  2.6600000866654425

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)
@btime res1 = solve(optprob, BBO_adaptive_de_rand_1_bin(), maxiters = 7e3)
# With the tolerance lower, it achieves the correct solution in 3.5 seconds.

1.247 s (2616258 allocations: 365.68 MiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.000000314276468
 28.000000010244495
  2.6599997639634707

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)
@btime res1 = solve(optprob, BBO_adaptive_de_rand_1_bin(), maxiters = 7e3)
# With the more accurate solver Vern9 in the solution of the ODE, the convergence is less efficient!

# Fastest BlackBoxOptim: 3.5 seconds

1.764 s (2654650 allocations: 370.31 MiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  9.999999820179724
 28.00000025496668
  2.660000008997058

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, No
thing}, DiffEqParamEstim.var"#29#30"{Nothing, typeof(DiffEqParamEstim.STAND
ARD_PROB_GENERATOR), Base.Pairs{Symbol, Any, Tuple{Symbol, Symbol, Symbol},
 @NamedTuple{tstops::Vector{Float64}, reltol::Float64, abstol::Float64}}, S
ciMLBase.ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, true, Modelin
gToolkit.MTKParameters{Vector{Float64}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}
, SciMLBase.ODEFunction{true, SciMLBase.AutoSpecialize, ModelingToolkit.var
"#f#1091"{RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋arg1, :ˍ₋a
rg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModT
ag", (0x9164ddac, 0x43cdc5b3, 0x75d31def, 0x444336ac, 0xda70202a), Nothing}
, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :ˍ₋arg1, :ˍ₋a
rg2, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModT
ag", (0x745b5079, 0x1445a295, 0xac2e3b8d, 0x50c01577, 0xcec704b8), Nothing}
}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, 
Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingTool
kit.ObservedFunctionCache{ModelingToolkit.ODESystem}, Nothing, ModelingTool
kit.ODESystem, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @Nam
edTuple{}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqVerner.Vern9{typeo
f(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_l
imiter!), Static.False}, DiffEqParamEstim.L2Loss{Vector{Float64}, Matrix{Fl
oat64}, Nothing, Nothing, Nothing}, Nothing, Tuple{}}, Nothing, Nothing, No
thing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothi
ng, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED_NO_TIME), Nothing, 
Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Not
hing}) (generic function with 1 method)

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

2.163 s (3242524 allocations: 445.93 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
 10.000000000079076
 27.999999999605798
  2.659999999934638

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

823.223 ms (1245212 allocations: 171.24 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
 10.000000000010058
 27.99999999994318
  2.659999999993697

opt = Opt(:GN_ISRES, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12) # Accurate to single precision 8.2 seconds

2.501 s (3760136 allocations: 517.13 MiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.000399733229292
 27.99861294666916
  2.65960429735005

opt = Opt(:GN_ESCH, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12) # Approximately accurate, good starting values for local optimization

2.494 s (3760136 allocations: 517.13 MiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  9.453461888135255
 30.033644594261425
  3.115883435403752

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

21.466 ms (32988 allocations: 4.52 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
 10.000000000000089
 27.99999999999836
  2.6600000000010318

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

69.677 ms (107436 allocations: 14.76 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
  9.999999999999293
 28.000000000006864
  2.660000000001213

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

10.925 ms (15083 allocations: 2.53 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
 10.000000000000107
 27.999999999998312
  2.6600000000010358

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

130.683 ms (201060 allocations: 27.63 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
  9.999999999998614
 28.00000000000312
  2.660000000002369

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

57.033 ms (56168 allocations: 7.71 MiB)
retcode: Success
u: 3-element Vector{Float64}:
 10.000004314773646
 27.999981806974528
  2.6599957890239403

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

17.348 ms (32722 allocations: 4.69 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
 10.000000000000073
 27.99999999999839
  2.660000000001071

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

247.086 ms (379652 allocations: 52.20 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
 10.000000000017945
 27.99999999991421
  2.65999999998792

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

93.129 ms (126118 allocations: 21.77 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
  9.999999999980686
 28.000000000097163
  2.6600000000171913

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

13.090 ms (17881 allocations: 3.08 MiB)
retcode: Success
u: 3-element Vector{Float64}:
 10.000000000000492
 27.99999999999682
  2.6600000000007737

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

14.772 ms (20317 allocations: 3.48 MiB)
retcode: Success
u: 3-element Vector{Float64}:
 10.000000000000087
 27.99999999999839
  2.6600000000010335

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]

# BB with Vern9 converges very slowly. The final values are within the NarrowBounds.
obj = build_loss_objective(prob,Vern9(),L2Loss(time_points,data),tstops=time_points,reltol=1e-9,abstol=1e-9)
optprob = OptimizationProblem(obj, GloIniPar, lb = first.(LooserBounds), ub = last.(LooserBounds))
@btime res1 = solve(optprob, BBO_adaptive_de_rand_1_bin(); maxiters = 4e3) # Default adaptive_de_rand_1_bin_radiuslimited 33 sec [10.2183, 24.6711, 2.28969]
#@btime res1 = bboptimize(obj;SearchRange = LooserBounds, Method = :adaptive_de_rand_1_bin, MaxSteps = 4e3) # Method 32 sec [13.2222, 25.8589, 2.56176]
#@btime res1 = bboptimize(obj;SearchRange = LooserBounds, Method = :dxnes, MaxSteps = 2e3) # Method dxnes 119 sec  [16.8648, 24.393, 2.29119]
#@btime res1 = bboptimize(obj;SearchRange = LooserBounds, Method = :xnes, MaxSteps = 2e3) # Method xnes 304 sec  [19.1647, 24.9479, 2.39467]
#@btime res1 = bboptimize(obj;SearchRange = LooserBounds, Method = :de_rand_1_bin_radiuslimited, MaxSteps = 2e3) # Method 44 sec  [13.805, 24.6054, 2.37274]
#@btime res1 = bboptimize(obj;SearchRange = LooserBounds, Method = :generating_set_search, MaxSteps = 2e3) # Method 195 sec [19.1847, 24.9492, 2.39412]

10.302 s (12676751 allocations: 1.65 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
  9.999866005438932
 28.000240000068334
  2.659811261200796

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

16.299 s (19943270 allocations: 2.59 GiB)
retcode: Failure
u: 3-element Vector{Float64}:
 10.000000041311779
 27.999999814143607
  2.6599999567875927

opt = Opt(:GN_CRS2_LM, 3)
@btime res1 = solve(optprob, opt, maxiters = 20000, xtol_rel = 1e-12) # Hit and miss. converge approximately accurate values for local opt.91 seconds

8.762 s (10758710 allocations: 1.40 GiB)
retcode: Failure
u: 3-element Vector{Float64}:
  9.99999999998017
 28.000000000056435
  2.6600000000158017

opt = Opt(:GN_ISRES, 3)
@btime res1 = solve(optprob, opt, maxiters = 50000, xtol_rel = 1e-12) # Approximately accurate within local bounds

81.254 s (100054062 allocations: 13.01 GiB)
retcode: Failure
u: 3-element Vector{Float64}:
  9.99999999997841
 28.00000000008834
  2.6600000000042194

opt = Opt(:GN_ESCH, 3)
@btime res1 = solve(optprob, opt, maxiters = 20000, xtol_rel = 1e-12) # Approximately accurate

49.981 s (61600130 allocations: 8.01 GiB)
retcode: MaxIters
u: 3-element Vector{Float64}:
 10.087612118071565
 28.015446277215144
  2.419258521905909

This parameter estimation on the longer sample proves to be extremely challenging for the global optimizers. BlackBoxOptim is best in optimizing the objective function. All of the global algorithms produces final parameter estimates that could be used as starting values for further refinement with the local optimization algorithms.

opt = Opt(:LN_BOBYQA, 3)
optprob = OptimizationProblem(obj_short, LocIniPar, lb = xlow_bounds, ub = xhigh_bounds)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12) # Claims SUCCESS but does not iterate to the true values.

21.470 ms (32988 allocations: 4.52 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
 10.000000000000089
 27.99999999999836
  2.6600000000010318

opt = Opt(:LN_NELDERMEAD, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12) # Inaccurate final values

69.796 ms (107436 allocations: 14.76 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
  9.999999999999293
 28.000000000006864
  2.660000000001213

opt = Opt(:LD_SLSQP, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12) # Inaccurate final values

10.868 ms (15083 allocations: 2.53 MiB)
retcode: Failure
u: 3-element Vector{Float64}:
 10.000000000000107
 27.999999999998312
  2.6600000000010358

No local optimizer can improve the global solution to the true values.

minimum(root)

Error: UndefVarError: `root` not defined

Conclusion:

1. As expected the Lorenz system is extremely sensitive to initial space values. Starting the integration from r0 = [0.1,0.0,0.0] produces convergence with the short sample of 300 observations. This can be achieved by all the global optimizers as well as most of the local optimizers. Instead starting from r0= [-11.8,-5.1,37.5], as in PODES, with the shorter sample shrinks the number of successful algorithms to 3: BBO, :GN_CRS2_LMand :LD_SLSQP. For the longer sample, all the algorithms fail.

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. Excessively low tolerance in the numerical solution is inefficient and delays the convergence of the estimation.

4. The estimation method and the global versus local optimization make a huge difference in the timings. Here, BBO always find the correct solution for a global optimization setup. For local optimization, most methods in NLopt, like :LN_BOBYQA, solve the problem in <0.05 seconds. This is an algorithm that can scale a local optimization but we are aiming to scale a global optimization.

5. QuadDIRECT performs very well on the shorter problem but doesn't give very great results for the longer in the Lorenz case, more can be read about the algorithm here.

6. Fitting shorter timespans is easier... 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 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
  JULIA_CPU_THREADS = 128
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/ParameterEstimation/Project.toml`
  [6e4b80f9] BenchmarkTools v1.6.0
  [a134a8b2] BlackBoxOptim v0.6.3
  [1130ab10] DiffEqParamEstim v2.2.0
⌃ [31c24e10] Distributions v0.25.117
⌅ [f6369f11] ForwardDiff v0.10.38
⌃ [961ee093] ModelingToolkit v9.61.0
⌃ [76087f3c] NLopt v1.1.2
⌃ [7f7a1694] Optimization v4.1.0
  [3e6eede4] OptimizationBBO v0.4.0
  [4e6fcdb7] OptimizationNLopt v0.3.2
⌃ [1dea7af3] OrdinaryDiffEq v6.90.1
  [65888b18] ParameterizedFunctions v5.17.2
⌃ [91a5bcdd] Plots v1.40.9
⌃ [731186ca] RecursiveArrayTools v3.27.4
  [31c91b34] SciMLBenchmarks v0.1.3
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.

And the full manifest:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/ParameterEstimation/Manifest.toml`
⌃ [47edcb42] ADTypes v1.12.1
  [1520ce14] AbstractTrees v0.4.5
⌃ [7d9f7c33] Accessors v0.1.41
⌃ [79e6a3ab] Adapt v4.1.1
  [66dad0bd] AliasTables v1.1.3
  [ec485272] ArnoldiMethod v0.4.0
  [4fba245c] ArrayInterface v7.18.0
⌃ [4c555306] ArrayLayouts v1.11.0
  [6e4b80f9] BenchmarkTools v1.6.0
  [e2ed5e7c] Bijections v0.1.9
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⌃ [8e7c35d0] BlockArrays v1.3.0
⌃ [70df07ce] BracketingNonlinearSolve v1.1.0
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⌃ [944b1d66] CodecZlib v0.7.6
⌃ [35d6a980] ColorSchemes v3.28.0
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⌅ [a80b9123] CommonMark v0.8.15
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⌃ [f0e56b4a] ConcurrentUtilities v2.4.3
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⌃ [864edb3b] DataStructures v0.18.20
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⌃ [2b5f629d] DiffEqBase v6.161.0
⌃ [459566f4] DiffEqCallbacks v4.2.2
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⌃ [a0c0ee7d] DifferentiationInterface v0.6.32
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⌃ [31c24e10] Distributions v0.25.117
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⌃ [06fc5a27] DynamicQuantities v1.4.0
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⌅ [6b7a57c9] Expronicon v0.8.5
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  [64ca27bc] FindFirstFunctions v1.4.1
⌃ [6a86dc24] FiniteDiff v2.26.2
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⌅ [f6369f11] ForwardDiff v0.10.38
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⌃ [28b8d3ca] GR v0.73.12
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  [d7ba0133] Git v1.3.1
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  [42e2da0e] Grisu v1.0.2
  [cd3eb016] HTTP v1.10.15
  [eafb193a] Highlights v0.5.3
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⌃ [34004b35] HypergeometricFunctions v0.3.27
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⌃ [92d709cd] IrrationalConstants v0.2.2
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⌃ [1019f520] JLFzf v0.1.9
  [692b3bcd] JLLWrappers v1.7.0
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⌃ [ccbc3e58] JumpProcesses v9.14.1
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⌃ [ba0b0d4f] Krylov v0.9.9
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⌃ [23fbe1c1] Latexify v0.16.5
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⌃ [5078a376] LazyArrays v2.4.0
  [1d6d02ad] LeftChildRightSiblingTrees v0.2.0
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  [d3d80556] LineSearches v7.3.0
⌅ [7ed4a6bd] LinearSolve v2.38.0
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⌃ [bdcacae8] LoopVectorization v0.12.171
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  [442fdcdd] Measures v0.3.2
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⌃ [961ee093] ModelingToolkit v9.61.0
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  [102ac46a] MultivariatePolynomials v0.5.7
  [ffc61752] Mustache v1.0.20
⌃ [d8a4904e] MutableArithmetics v1.6.2
⌃ [d41bc354] NLSolversBase v7.8.3
⌃ [76087f3c] NLopt v1.1.2
⌃ [77ba4419] NaNMath v1.1.1
⌃ [8913a72c] NonlinearSolve v4.3.0
⌃ [be0214bd] NonlinearSolveBase v1.4.0
⌃ [5959db7a] NonlinearSolveFirstOrder v1.2.0
⌃ [9a2c21bd] NonlinearSolveQuasiNewton v1.1.0
  [26075421] NonlinearSolveSpectralMethods v1.1.0
⌃ [6fe1bfb0] OffsetArrays v1.15.0
  [4d8831e6] OpenSSL v1.4.3
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⌃ [7f7a1694] Optimization v4.1.0
  [3e6eede4] OptimizationBBO v0.4.0
⌃ [bca83a33] OptimizationBase v2.4.0
  [4e6fcdb7] OptimizationNLopt v0.3.2
⌃ [bac558e1] OrderedCollections v1.7.0
⌃ [1dea7af3] OrdinaryDiffEq v6.90.1
⌃ [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.1.0
⌃ [6ad6398a] OrdinaryDiffEqBDF v1.2.0
⌃ [bbf590c4] OrdinaryDiffEqCore v1.15.1
⌃ [50262376] OrdinaryDiffEqDefault v1.2.0
⌃ [4302a76b] OrdinaryDiffEqDifferentiation v1.3.0
  [9286f039] OrdinaryDiffEqExplicitRK v1.1.0
⌃ [e0540318] OrdinaryDiffEqExponentialRK v1.2.0
⌃ [becaefa8] OrdinaryDiffEqExtrapolation v1.3.0
⌃ [5960d6e9] OrdinaryDiffEqFIRK v1.6.0
  [101fe9f7] OrdinaryDiffEqFeagin v1.1.0
  [d3585ca7] OrdinaryDiffEqFunctionMap v1.1.1
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v1.1.0
⌃ [9f002381] OrdinaryDiffEqIMEXMultistep v1.2.0
  [521117fe] OrdinaryDiffEqLinear v1.1.0
  [1344f307] OrdinaryDiffEqLowOrderRK v1.2.0
⌃ [b0944070] OrdinaryDiffEqLowStorageRK v1.2.1
⌃ [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.3.0
  [c9986a66] OrdinaryDiffEqNordsieck v1.1.0
⌃ [5dd0a6cf] OrdinaryDiffEqPDIRK v1.2.0
  [5b33eab2] OrdinaryDiffEqPRK v1.1.0
  [04162be5] OrdinaryDiffEqQPRK v1.1.0
  [af6ede74] OrdinaryDiffEqRKN v1.1.0
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v1.4.0
⌃ [2d112036] OrdinaryDiffEqSDIRK v1.2.0
⌃ [669c94d9] OrdinaryDiffEqSSPRK v1.2.0
⌃ [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.2.0
  [358294b1] OrdinaryDiffEqStabilizedRK v1.1.0
⌃ [fa646aed] OrdinaryDiffEqSymplecticRK v1.1.0
  [b1df2697] OrdinaryDiffEqTsit5 v1.1.0
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⌃ [91a5bcdd] Plots v1.40.9
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⌃ [d236fae5] PreallocationTools v0.4.24
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.4.3
⌃ [27ebfcd6] Primes v0.5.6
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⌃ [1fd47b50] QuadGK v2.11.1
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⌃ [731186ca] RecursiveArrayTools v3.27.4
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⌃ [ae029012] Requires v1.3.0
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⌃ [0bca4576] SciMLBase v2.72.1
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⌃ [c0aeaf25] SciMLOperators v0.3.12
⌃ [53ae85a6] SciMLStructures v1.6.1
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⌃ [efcf1570] Setfield v1.1.1
  [992d4aef] Showoff v1.0.3
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⌃ [727e6d20] SimpleNonlinearSolve v2.1.0
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  [a2af1166] SortingAlgorithms v1.2.1
⌃ [9f842d2f] SparseConnectivityTracer v0.6.10
⌃ [47a9eef4] SparseDiffTools v2.23.1
⌃ [0a514795] SparseMatrixColorings v0.4.12
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⌃ [aedffcd0] Static v1.1.1
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⌃ [90137ffa] StaticArrays v1.9.10
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⌃ [2efcf032] SymbolicIndexingInterface v0.3.37
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⌃ [d1185830] SymbolicUtils v3.11.0
⌃ [0c5d862f] Symbolics v6.25.0
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⌃ [a759f4b9] TimerOutputs v0.5.26
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⌃ [ddb6d928] YAML v0.4.12
  [c2297ded] ZMQ v1.4.0
⌃ [6e34b625] Bzip2_jll v1.0.8+4
⌃ [83423d85] Cairo_jll v1.18.2+1
  [ee1fde0b] Dbus_jll v1.14.10+0
  [cd4c43a9] Dierckx_jll v0.2.0+0
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⌃ [2e619515] Expat_jll v2.6.4+3
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
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⌅ [d2c73de3] GR_jll v0.73.12+0
  [78b55507] Gettext_jll v0.21.0+0
⌃ [f8c6e375] Git_jll v2.47.1+0
⌃ [7746bdde] Glib_jll v2.82.4+0
  [3b182d85] Graphite2_jll v1.3.14+1
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⌅ [e9f186c6] Libffi_jll v3.2.2+2
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  [7e76a0d4] Libglvnd_jll v1.7.0+0
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  [94ce4f54] Libiconv_jll v1.18.0+0
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  [38a345b3] Libuuid_jll v2.40.3+0
  [856f044c] MKL_jll v2025.0.1+1
⌃ [079eb43e] NLopt_jll v2.9.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
⌃ [458c3c95] OpenSSL_jll v3.0.15+3
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.3.3+0
⌃ [36c8627f] Pango_jll v1.55.5+0
⌅ [30392449] Pixman_jll v0.43.4+0
⌅ [c0090381] Qt6Base_jll v6.7.1+1
⌅ [629bc702] Qt6Declarative_jll v6.7.1+2
⌅ [ce943373] Qt6ShaderTools_jll v6.7.1+1
⌃ [e99dba38] Qt6Wayland_jll v6.7.1+1
  [f50d1b31] Rmath_jll v0.5.1+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
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⌃ [02c8fc9c] XML2_jll v2.13.5+0
  [aed1982a] XSLT_jll v1.1.42+0
  [ffd25f8a] XZ_jll v5.6.4+1
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  [c834827a] Xorg_libSM_jll v1.2.4+0
  [4f6342f7] Xorg_libX11_jll v1.8.6+3
  [0c0b7dd1] Xorg_libXau_jll v1.0.12+0
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  [a3789734] Xorg_libXdmcp_jll v1.1.5+0
  [1082639a] Xorg_libXext_jll v1.3.6+3
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  [a51aa0fd] Xorg_libXi_jll v1.8.2+0
  [d1454406] Xorg_libXinerama_jll v1.1.5+0
  [ec84b674] Xorg_libXrandr_jll v1.5.4+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.11+1
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.2+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.0+3
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  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.6+1
  [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
  [c5fb5394] Xorg_xtrans_jll v1.5.1+0
⌃ [8f1865be] ZeroMQ_jll v4.3.5+3
⌃ [3161d3a3] Zstd_jll v1.5.7+0
  [35ca27e7] eudev_jll v3.2.9+0
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  [f638f0a6] libfdk_aac_jll v2.0.3+0
  [36db933b] libinput_jll v1.18.0+0
⌃ [b53b4c65] libpng_jll v1.6.45+1
⌃ [a9144af2] libsodium_jll v1.0.20+3
  [f27f6e37] libvorbis_jll v1.3.7+2
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⌃ [1317d2d5] oneTBB_jll v2021.12.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
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  [2a0f44e3] Base64
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  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
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  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
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.