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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
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
g1 = @ode_def LorenzExample begin
  dx = σ*(y-x)
  dy = x*(ρ-z) - y
  dz = x*y - β*z
end σ ρ β
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
t  = 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=t,reltol=1e-9,abstol=1e-9)
data = convert(Array, data_sol)
3×3001 Matrix{Float64}:
 1.0  0.917924    0.867919    0.84536     …  13.8987   13.2896  12.5913
 0.0  0.26634     0.51174     0.744654        8.31875   6.7199   5.22868
 0.0  0.00126393  0.00465567  0.00983655     39.19     39.1699  38.904

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))

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
1.005 s (2757095 allocations: 389.66 MiB)
u: 3-element Vector{Float64}:
  9.999999587014711
 28.00000020248227
  2.6599999386626263
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.
1.471 s (2754491 allocations: 389.30 MiB)
u: 3-element Vector{Float64}:
 10.000000133984779
 27.999999971652443
  2.660000000766214
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
1.414 s (2793612 allocations: 397.70 MiB)
u: 3-element Vector{Float64}:
 10.00000000095171
 28.00000010070656
  2.659999988891568

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, :reltol, :abstol), Tuple{Vector{Float64}, Float64, Fl
oat64}}}, SciMLBase.ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, tr
ue, Vector{Float64}, Main.var"##WeaveSandBox#292".LorenzExample{Main.var"##
WeaveSandBox#292".var"###ParameterizedDiffEqFunction#294", Main.var"##Weave
SandBox#292".var"###ParameterizedTGradFunction#295", Main.var"##WeaveSandBo
x#292".var"###ParameterizedJacobianFunction#296", Nothing, Nothing, Modelin
gToolkit.ODESystem}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tu
ple{}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEq.Vern9{typeof(Ordinar
yDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.
False}, DiffEqParamEstim.L2Loss{Vector{Float64}, Matrix{Float64}, Nothing, 
Nothing, Nothing}, Nothing, Tuple{}}, Nothing, Nothing, Nothing, Nothing, N
othing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Noth
ing, typeof(SciMLBase.DEFAULT_OBSERVED_NO_TIME), Nothing, Nothing, Nothing,
 Nothing, Nothing, Nothing, Nothing}) (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)
749.374 ms (1428847 allocations: 234.06 MiB)
u: 3-element Vector{Float64}:
 10.000000000174282
 28.000000000007077
  2.6600000000125332
opt = Opt(:GN_CRS2_LM, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
575.226 ms (1094510 allocations: 179.30 MiB)
u: 3-element Vector{Float64}:
 10.00000000003555
 28.000000000023544
  2.6600000000097537
opt = Opt(:GN_ISRES, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12) # Accurate to single precision 8.2 seconds
1.949 s (3670116 allocations: 601.12 MiB)
u: 3-element Vector{Float64}:
  9.999046376250089
 27.999452939283536
  2.6598431076807043
opt = Opt(:GN_ESCH, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12) # Approximately accurate, good starting values for local optimization
1.923 s (3670116 allocations: 601.12 MiB)
u: 3-element Vector{Float64}:
  9.842490324932497
 28.328802389399023
  2.6557217984711556

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)
20.403 ms (40486 allocations: 6.68 MiB)
u: 3-element Vector{Float64}:
 10.000000000050505
 28.000000000023256
  2.6600000000078743
opt = Opt(:LN_NELDERMEAD, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
58.110 ms (115721 allocations: 19.00 MiB)
u: 3-element Vector{Float64}:
 10.00000000005633
 28.00000000003226
  2.660000000009111
opt = Opt(:LD_SLSQP, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
12.752 ms (46215 allocations: 3.97 MiB)
u: 3-element Vector{Float64}:
 10.000000000051957
 28.000000000022848
  2.660000000007989
opt = Opt(:LN_COBYLA, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
423.451 ms (803479 allocations: 131.64 MiB)
u: 3-element Vector{Float64}:
 10.000000000035833
 28.00000000002392
  2.6600000000068387
opt = Opt(:LN_NEWUOA_BOUND, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
63.913 ms (68378 allocations: 11.25 MiB)
u: 3-element Vector{Float64}:
 10.000001737489647
 28.000001355370276
  2.6599999734839916
opt = Opt(:LN_PRAXIS, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
12.845 ms (30442 allocations: 5.22 MiB)
u: 3-element Vector{Float64}:
 10.917133996220539
 22.256728511857908
  2.1272387140667357
opt = Opt(:LN_SBPLX, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
144.501 ms (274632 allocations: 45.03 MiB)
u: 3-element Vector{Float64}:
 10.000000000058924
 28.000000000020663
  2.660000000008665
opt = Opt(:LD_MMA, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
63.043 ms (237532 allocations: 19.36 MiB)
u: 3-element Vector{Float64}:
  9.999999999995449
 28.00000000005174
  2.659999999994011
opt = Opt(:LD_LBFGS, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
9.704 ms (36332 allocations: 3.01 MiB)
u: 3-element Vector{Float64}:
 10.000000000052934
 28.000000000022638
  2.660000000007903
opt = Opt(:LD_TNEWTON_PRECOND_RESTART, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12)
12.341 ms (46392 allocations: 3.83 MiB)
u: 3-element Vector{Float64}:
 10.000000000051458
 28.00000000002282
  2.6600000000079564

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(t,data),tstops=t,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]
7.839 s (12745668 allocations: 1.67 GiB)
u: 3-element Vector{Float64}:
 11.933343384144719
 24.821356584568015
  2.571333775671008
# 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)
4.057 s (6562816 allocations: 876.52 MiB)
u: 3-element Vector{Float64}:
  7.04665993025209
 23.666102233396032
  1.8066012972265462
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
38.114 s (61420089 allocations: 8.01 GiB)
u: 3-element Vector{Float64}:
 19.161150928497484
 24.947198148638012
  2.394785705705444
opt = Opt(:GN_ISRES, 3)
@btime res1 = solve(optprob, opt, maxiters = 50000, xtol_rel = 1e-12) # Approximately accurate within local bounds
94.889 s (153550089 allocations: 20.03 GiB)
u: 3-element Vector{Float64}:
 10.919313360569316
 23.79820094295319
  2.4839357532054365
opt = Opt(:GN_ESCH, 3)
@btime res1 = solve(optprob, opt, maxiters = 20000, xtol_rel = 1e-12) # Approximately accurate
37.999 s (61420089 allocations: 8.01 GiB)
u: 3-element Vector{Float64}:
 21.93708643639748
 16.507773217638782
  0.015131532071195806

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.
20.386 ms (40486 allocations: 6.68 MiB)
u: 3-element Vector{Float64}:
 10.000000000050505
 28.000000000023256
  2.6600000000078743
opt = Opt(:LN_NELDERMEAD, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12) # Inaccurate final values
58.133 ms (115721 allocations: 19.00 MiB)
u: 3-element Vector{Float64}:
 10.00000000005633
 28.00000000003226
  2.660000000009111
opt = Opt(:LD_SLSQP, 3)
@btime res1 = solve(optprob, opt, maxiters = 10000, xtol_rel = 1e-12) # Inaccurate final values
12.759 ms (46215 allocations: 3.97 MiB)
u: 3-element Vector{Float64}:
 10.000000000051957
 28.000000000022848
  2.660000000007989

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 1.9.4
Commit 8e5136fa297 (2023-11-14 08:46 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-14.0.6 (ORCJIT, znver2)
  Threads: 1 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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/ParameterEstimation/Project.toml`
⌃ [6e4b80f9] BenchmarkTools v1.3.2
⌃ [a134a8b2] BlackBoxOptim v0.6.2
⌃ [1130ab10] DiffEqParamEstim v2.0.1
⌃ [31c24e10] Distributions v0.25.100
  [f6369f11] ForwardDiff v0.10.36
⌅ [76087f3c] NLopt v0.6.5
⌃ [7f7a1694] Optimization v3.16.0
⌃ [3e6eede4] OptimizationBBO v0.1.5
⌃ [4e6fcdb7] OptimizationNLopt v0.1.8
⌅ [1dea7af3] OrdinaryDiffEq v6.55.0
⌃ [65888b18] ParameterizedFunctions v5.15.0
  [91a5bcdd] Plots v1.39.0
⌅ [731186ca] RecursiveArrayTools v2.38.7
  [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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/ParameterEstimation/Manifest.toml`
⌃ [47edcb42] ADTypes v0.2.1
⌅ [c3fe647b] AbstractAlgebra v0.31.1
  [1520ce14] AbstractTrees v0.4.4
⌅ [79e6a3ab] Adapt v3.6.2
  [ec485272] ArnoldiMethod v0.2.0
⌃ [4fba245c] ArrayInterface v7.4.11
  [30b0a656] ArrayInterfaceCore v0.1.29
⌃ [6e4b80f9] BenchmarkTools v1.3.2
⌃ [e2ed5e7c] Bijections v0.1.4
⌃ [d1d4a3ce] BitFlags v0.1.7
  [62783981] BitTwiddlingConvenienceFunctions v0.1.5
⌃ [a134a8b2] BlackBoxOptim v0.6.2
⌃ [2a0fbf3d] CPUSummary v0.2.3
  [a9c8d775] CPUTime v1.0.0
  [00ebfdb7] CSTParser v3.3.6
  [49dc2e85] Calculus v0.5.1
⌃ [d360d2e6] ChainRulesCore v1.16.0
  [fb6a15b2] CloseOpenIntervals v0.1.12
⌅ [523fee87] CodecBzip2 v0.7.2
⌃ [944b1d66] CodecZlib v0.7.2
⌃ [35d6a980] ColorSchemes v3.23.0
  [3da002f7] ColorTypes v0.11.4
  [c3611d14] ColorVectorSpace v0.10.0
  [5ae59095] Colors v0.12.10
  [861a8166] Combinatorics v1.0.2
  [a80b9123] CommonMark v0.8.12
  [38540f10] CommonSolve v0.2.4
  [bbf7d656] CommonSubexpressions v0.3.0
⌃ [34da2185] Compat v4.9.0
  [b152e2b5] CompositeTypes v0.1.3
⌃ [f0e56b4a] ConcurrentUtilities v2.2.1
⌃ [8f4d0f93] Conda v1.9.1
  [88cd18e8] ConsoleProgressMonitor v0.1.2
⌃ [187b0558] ConstructionBase v1.5.3
  [d38c429a] Contour v0.6.2
  [adafc99b] CpuId v0.3.1
  [a8cc5b0e] Crayons v4.1.1
  [9a962f9c] DataAPI v1.15.0
⌃ [864edb3b] DataStructures v0.18.15
  [e2d170a0] DataValueInterfaces v1.0.0
  [8bb1440f] DelimitedFiles v1.9.1
  [39dd38d3] Dierckx v0.5.3
⌃ [2b5f629d] DiffEqBase v6.128.2
⌃ [459566f4] DiffEqCallbacks v2.29.1
⌃ [1130ab10] DiffEqParamEstim v2.0.1
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⌃ [d4ead438] SpatialIndexing v0.1.5
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⌃ [1d63c593] LLVMOpenMP_jll v15.0.4+0
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⌅ [e9f186c6] Libffi_jll v3.2.2+1
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⌃ [02c8fc9c] XML2_jll v2.10.4+0
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -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.