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Lorenz Bayesian Parameter Estimation Benchmarks

Parameter estimation of Lorenz Equation using DiffEqBayes.jl

using DiffEqBayes
using DiffEqCallbacks, StaticArrays
using Distributions, StanSample, DynamicHMC, Turing
using OrdinaryDiffEq, RecursiveArrayTools, ParameterizedFunctions, DiffEqCallbacks
using Plots, LinearAlgebra
gr(fmt=:png)
Plots.GRBackend()

Initializing the problem

g1 = @ode_def LorenzExample begin
  dx = σ*(y-x)
  dy = x*(ρ-z) - y
  dz = x*y - β*z
end σ ρ β
(::Main.var"##WeaveSandBox#225".LorenzExample{Main.var"##WeaveSandBox#225".
var"###ParameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".var"#
##ParameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###Para
meterizedJacobianFunction#229", Nothing, Nothing, ModelingToolkit.ODESystem
}) (generic function with 1 method)
r0 = [1.0; 0.0; 0.0]
tspan = (0.0, 30.0)
p = [10.0,28.0,2.66]
3-element Vector{Float64}:
 10.0
 28.0
  2.66
prob = ODEProblem(g1, r0, tspan, p)
sol = solve(prob,Tsit5())
retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 362-element Vector{Float64}:
  0.0
  3.5678604836301404e-5
  0.0003924646531993154
  0.003262343160292866
  0.00905768915231668
  0.016955558260817218
  0.027688386680734336
  0.04185394923402222
  0.060237081190933954
  0.08368091876192292
  ⋮
 29.454408593999744
 29.535835262016416
 29.605800898814937
 29.680544237315484
 29.76351894459027
 29.830453975921273
 29.895187958428064
 29.951353159401716
 30.0
u: 362-element Vector{Vector{Float64}}:
 [1.0, 0.0, 0.0]
 [0.9996434557625105, 0.0009988049817849054, 1.7814349300524274e-8]
 [0.9961045497425811, 0.010965399721242273, 2.1469572398550344e-6]
 [0.969359731583511, 0.08976885926574524, 0.00014379728741456088]
 [0.9242069970136711, 0.24227921748230874, 0.0010460982665403552]
 [0.8800496059816251, 0.4387144111226294, 0.003424048327994956]
 [0.8483334490657588, 0.6915266898669253, 0.008487275727945722]
 [0.8494997033541278, 1.014487977850381, 0.018211867322766993]
 [0.9138893443162335, 1.4424796048698445, 0.03669462235325829]
 [1.088820494006628, 2.0521989010862804, 0.07402932469846656]
 ⋮
 [12.961137512020898, 18.279918361318895, 26.25897074938116]
 [14.392465492943021, 11.508987143103287, 37.65923554892938]
 [10.152335863282124, 2.269979011509871, 36.547101446703415]
 [4.808585699119702, -0.8952979613542543, 30.11368867713037]
 [1.6024263579730906, -0.6846122220278174, 23.951371949023542]
 [0.6084324252084657, -0.25780977194812565, 20.01475398456359]
 [0.2684879085098022, -0.002836416861666546, 16.8456113650265]
 [0.18738369956893544, 0.14299558298777929, 14.508658451450762]
 [0.19493344255581885, 0.2638385897822422, 12.749361976073427]
sr0 = SA[1.0; 0.0; 0.0]
tspan = (0.0, 30.0)
sp = SA[10.0,28.0,2.66]
sprob = ODEProblem{false,SciMLBase.FullSpecialize}(g1, sr0, tspan, sp)
sol = solve(sprob,Tsit5())
retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 362-element Vector{Float64}:
  0.0
  3.5678604836301404e-5
  0.0003924646531993154
  0.003262343160292866
  0.00905768915231668
  0.016955558260817218
  0.027688386680734336
  0.04185394923402222
  0.060237081190933954
  0.08368091876192292
  ⋮
 29.45440860820997
 29.535835277148024
 29.605800914738243
 29.680544254084552
 29.76351899890051
 29.830454031907585
 29.89518801188915
 29.95135321067184
 30.0
u: 362-element Vector{StaticArraysCore.SVector{3, Float64}}:
 [1.0, 0.0, 0.0]
 [0.9996434557625105, 0.0009988049817849054, 1.7814349300524274e-8]
 [0.9961045497425811, 0.010965399721242273, 2.1469572398550344e-6]
 [0.969359731583511, 0.08976885926574524, 0.00014379728741456088]
 [0.9242069970136711, 0.24227921748230874, 0.0010460982665403552]
 [0.8800496059816251, 0.4387144111226294, 0.003424048327994956]
 [0.8483334490657588, 0.6915266898669252, 0.008487275727945722]
 [0.8494997033541277, 1.014487977850381, 0.018211867322766986]
 [0.9138893443162334, 1.4424796048698445, 0.03669462235325828]
 [1.088820494006628, 2.05219890108628, 0.07402932469846653]
 ⋮
 [12.961138489125066, 18.279918309485804, 26.258974000912552]
 [14.392464847216962, 11.508984146832098, 37.65923672513932]
 [10.152334234091272, 2.2699772912331264, 36.5470998219813]
 [4.808584535722784, -0.8952980440122978, 30.11368685429087]
 [1.6024250955199508, -0.6846116829208884, 23.95136813347255]
 [0.6084320079771826, -0.2578093152137886, 20.014750756773317]
 [0.2684878928983248, -0.002836030553965167, 16.845608772737304]
 [0.18738386874531499, 0.14299602936335054, 14.508656308449002]
 [0.19493371295291068, 0.2638390569115199, 12.749361835984178]

Generating data for bayesian estimation of parameters from the obtained solutions using the Tsit5 algorithm by adding random noise to it.

t = collect(range(1, stop=30, length=30))
sig = 0.49
data = convert(Array, VectorOfArray([(sol(t[i]) + sig*randn(3)) for i in 1:length(t)]))
3×30 Matrix{Float64}:
 -8.60221  -7.8308   -7.83975   -9.73928  …  10.7223   3.54911  -0.317459
 -9.03699  -8.64908  -6.95781  -10.1946      15.307    1.32129   0.361791
 27.7355   25.1761   28.9517    26.6683      25.2028  26.002    13.2363

Plots of the generated data and the actual data.

Plots.scatter(t, data[1,:],markersize=4,color=:purple)
Plots.scatter!(t, data[2,:],markersize=4,color=:yellow)
Plots.scatter!(t, data[3,:],markersize=4,color=:black)
plot!(sol)

Uncertainty Quantification plot is used to decide the tolerance for the differential equation.

cb = AdaptiveProbIntsUncertainty(5)
monte_prob = EnsembleProblem(prob)
sim = solve(monte_prob,Tsit5(),trajectories=100,callback=cb,reltol=1e-5,abstol=1e-5)
plot(sim,vars=(0,1),linealpha=0.4)

cb = AdaptiveProbIntsUncertainty(5)
monte_prob = EnsembleProblem(prob)
sim = solve(monte_prob,Tsit5(),trajectories=100,callback=cb,reltol=1e-6,abstol=1e-6)
plot(sim,vars=(0,1),linealpha=0.4)

cb = AdaptiveProbIntsUncertainty(5)
monte_prob = EnsembleProblem(prob)
sim = solve(monte_prob,Tsit5(),trajectories=100,callback=cb,reltol=1e-8,abstol=1e-8)
plot(sim,vars=(0,1),linealpha=0.4)

priors = [truncated(Normal(10,2),1,15),truncated(Normal(30,5),1,45),truncated(Normal(2.5,0.5),1,4)]
3-element Vector{Distributions.Truncated{Distributions.Normal{Float64}, Dis
tributions.Continuous, Float64, Float64, Float64}}:
 Truncated(Distributions.Normal{Float64}(μ=10.0, σ=2.0); lower=1.0, upper=1
5.0)
 Truncated(Distributions.Normal{Float64}(μ=30.0, σ=5.0); lower=1.0, upper=4
5.0)
 Truncated(Distributions.Normal{Float64}(μ=2.5, σ=0.5); lower=1.0, upper=4.
0)

Using Stan.jl backend

Lorenz equation is a chaotic system hence requires very low tolerance to be estimated in a reasonable way, we use 1e-8 obtained from the uncertainty plots. Use of truncated priors is necessary to prevent Stan from stepping into negative and other improbable areas.

@time bayesian_result_stan = stan_inference(prob,t,data,priors; delta = 0.65, reltol=1e-8,abstol=1e-8, vars=(DiffEqBayes.StanODEData(), InverseGamma(2, 3)))
2180.442163 seconds (2.17 M allocations: 148.250 MiB, 0.00% gc time, 0.08% 
compilation time)
2202.962588 seconds (6.75 M allocations: 460.316 MiB, 0.00% gc time, 0.25% 
compilation time: <1% of which was recompilation)
Chains MCMC chain (1000×6×1 Array{Float64, 3}):

Iterations        = 1:1:1000
Number of chains  = 1
Samples per chain = 1000
parameters        = sigma1.1, sigma1.2, sigma1.3, theta_1, theta_2, theta_3
internals         = 

Summary Statistics
  parameters      mean       std      mcse   ess_bulk   ess_tail      rhat 
  e ⋯
      Symbol   Float64   Float64   Float64    Float64    Float64   Float64 
    ⋯

    sigma1.1    4.9489    0.4572    0.1103    17.2519    37.6045    1.0914 
    ⋯
    sigma1.2    7.0738    0.7434    0.1521    28.0525   113.5502    1.0277 
    ⋯
    sigma1.3    7.1863    0.6196    0.1163    23.7609    69.3491    1.0713 
    ⋯
     theta_1    5.2102    0.1518    0.0481     7.6518    19.9568    1.1693 
    ⋯
     theta_2   22.0352    0.0780    0.0250     9.6522    33.4941    1.0344 
    ⋯
     theta_3    1.4975    0.0287    0.0097     7.3043    39.1249    1.0936 
    ⋯
                                                                1 column om
itted

Quantiles
  parameters      2.5%     25.0%     50.0%     75.0%     97.5%
      Symbol   Float64   Float64   Float64   Float64   Float64

    sigma1.1    4.1745    4.7141    4.8851    5.1467    5.8759
    sigma1.2    5.5874    6.5167    7.3347    7.5843    8.3335
    sigma1.3    5.9099    6.9393    7.1756    7.5189    8.5925
     theta_1    4.8563    5.1265    5.2503    5.3275    5.3857
     theta_2   21.9117   21.9886   22.0164   22.0934   22.1995
     theta_3    1.4712    1.4813    1.4854    1.5016    1.5778

Direct Turing.jl

@model function fitlv(data, prob)
    # Prior distributions.
    α ~ InverseGamma(2, 3)
    σ ~ truncated(Normal(10, 2), 1, 15)
    ρ ~ truncated(Normal(30, 5), 1, 45)
    β ~ truncated(Normal(2.5, 0.5), 1, 4)

    # Simulate Lotka-Volterra model. 
    p = SA[σ, ρ, β]
    _prob = remake(prob, p = p)
    predicted = solve(_prob, Vern9(); saveat=t)

    # Observations.
    for i in 1:length(predicted)
        data[:, i] ~ MvNormal(predicted[i], α^2 * I)
    end

    return nothing
end

model = fitlv(data, sprob)

@time chain = sample(model, Turing.NUTS(0.65), 10000; progress=false)
100.478309 seconds (225.68 M allocations: 40.962 GiB, 5.04% gc time, 21.56%
 compilation time)
Chains MCMC chain (10000×16×1 Array{Float64, 3}):

Iterations        = 1001:1:11000
Number of chains  = 1
Samples per chain = 10000
Wall duration     = 92.74 seconds
Compute duration  = 92.74 seconds
parameters        = α, σ, ρ, β
internals         = lp, n_steps, is_accept, acceptance_rate, log_density, h
amiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, 
tree_depth, numerical_error, step_size, nom_step_size

Summary Statistics
  parameters      mean       std      mcse   ess_bulk   ess_tail      rhat 
  e ⋯
      Symbol   Float64   Float64   Float64    Float64    Float64   Float64 
    ⋯

           α    7.2303    0.3938    0.0792    24.9858    33.1335    1.1241 
    ⋯
           σ    5.9531    0.0359    0.0069    25.9645    33.9060    1.3009 
    ⋯
           ρ   23.2424    0.2843    0.0584    23.7966    46.1054    1.0634 
    ⋯
           β    1.8073    0.0471    0.0097    24.4725    27.5327    1.2820 
    ⋯
                                                                1 column om
itted

Quantiles
  parameters      2.5%     25.0%     50.0%     75.0%     97.5%
      Symbol   Float64   Float64   Float64   Float64   Float64

           α    6.4063    6.9531    7.3869    7.5262    7.7012
           σ    5.8776    5.9347    5.9516    5.9758    6.0178
           ρ   22.6368   23.0581   23.3896   23.4460   23.5274
           β    1.7054    1.8080    1.8195    1.8277    1.8987

Using Turing.jl backend

@time bayesian_result_turing = turing_inference(prob, Vern9(), t, data, priors; reltol=1e-8, abstol=1e-8, likelihood=(u, p, t, σ) -> MvNormal(u, Diagonal((σ) .^ 2 .* ones(length(u)))), likelihood_dist_priors=[InverseGamma(2, 3), InverseGamma(2, 3), InverseGamma(2, 3)])
47.283401 seconds (166.48 M allocations: 10.412 GiB, 2.97% gc time, 37.50%
 compilation time)
Chains MCMC chain (1000×18×1 Array{Float64, 3}):

Iterations        = 501:1:1500
Number of chains  = 1
Samples per chain = 1000
Wall duration     = 44.78 seconds
Compute duration  = 44.78 seconds
parameters        = theta[1], theta[2], theta[3], σ[1], σ[2], σ[3]
internals         = lp, n_steps, is_accept, acceptance_rate, log_density, h
amiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, 
tree_depth, numerical_error, step_size, nom_step_size

Summary Statistics
  parameters      mean       std      mcse   ess_bulk   ess_tail      rhat 
  e ⋯
      Symbol   Float64   Float64   Float64    Float64    Float64   Float64 
    ⋯

    theta[1]    7.5233    0.3008    0.1551     6.0450    17.4679    1.1542 
    ⋯
    theta[2]   22.9403    0.1346    0.0687     4.3001    28.5545    1.2654 
    ⋯
    theta[3]    2.0609    0.0329    0.0212     2.5168    15.5271    2.1160 
    ⋯
        σ[1]    6.8272    0.1600    0.0561    10.9176    58.9159    1.0234 
    ⋯
        σ[2]    7.4490    0.6195    0.3959     2.4365    14.3989    2.0762 
    ⋯
        σ[3]    7.1329    0.3048    0.1457     4.5362    39.5257    1.1739 
    ⋯
                                                                1 column om
itted

Quantiles
  parameters      2.5%     25.0%     50.0%     75.0%     97.5%
      Symbol   Float64   Float64   Float64   Float64   Float64

    theta[1]    7.2917    7.3468    7.3863    7.4717    8.2194
    theta[2]   22.6908   22.8244   22.9657   23.0380   23.1462
    theta[3]    1.9991    2.0359    2.0673    2.0879    2.1084
        σ[1]    6.5998    6.6887    6.7739    6.9777    7.1327
        σ[2]    6.6796    6.9072    7.4271    7.7779    8.8338
        σ[3]    6.6832    6.8734    7.1034    7.4558    7.5540

Using DynamicHMC.jl backend

@time bayesian_result_dynamichmc = dynamichmc_inference(prob,Tsit5(),t,data,priors;solve_kwargs = (reltol=1e-8,abstol=1e-8,))
612.037998 seconds (150.45 M allocations: 29.745 GiB, 0.67% gc time, 1.02% 
compilation time)
(posterior = @NamedTuple{parameters::Vector{Float64}, σ::Vector{Float64}}[(
parameters = [5.554865180085494, 21.63383611295706, 1.6992498549011021], σ 
= [5.611592380362323, 5.8552131154221145, 8.17594167599006]), (parameters =
 [6.091714729248618, 21.717011069868672, 1.5569469286060238], σ = [6.175974
329333978, 9.256778156858223, 6.9253043937851535]), (parameters = [6.199925
107065802, 21.543862275991195, 1.5627624218091738], σ = [6.512408765117676,
 8.635012402194489, 7.4812171157136955]), (parameters = [5.593076356592441,
 22.07216812047315, 1.6120639388527707], σ = [6.280362290093031, 7.23229263
3773132, 7.67960585528417]), (parameters = [5.612633629345708, 21.875047802
83442, 1.664020308036359], σ = [7.049648591866764, 7.084565805411298, 7.222
08422653215]), (parameters = [5.572987443767706, 21.619850367879923, 1.6516
64177005243], σ = [7.822429381982619, 7.190747214847372, 6.739509868016136]
), (parameters = [5.56455399677507, 21.794383719162226, 1.634951708379088],
 σ = [6.8720155317098515, 6.754647038502775, 7.7801175825178115]), (paramet
ers = [5.620114711965501, 22.348317878634578, 1.6523790605129065], σ = [6.7
836796472419545, 7.093541125417478, 7.699827435909344]), (parameters = [5.6
35054704764993, 22.358423242434412, 1.6508423977198399], σ = [6.81393267199
9361, 7.101463233050159, 7.7001662616432185]), (parameters = [5.54940820424
7089, 22.234424203533216, 1.6584449906549938], σ = [7.308924163351382, 7.15
8587157318439, 7.683246178058683])  …  (parameters = [6.310966203678753, 21
.797397383926082, 1.567769970145609], σ = [4.683256252688815, 5.82392795161
6331, 7.011002130700901]), (parameters = [6.3081332227419376, 21.7980428284
35225, 1.567799908715652], σ = [4.684696030012695, 5.821699417933461, 7.007
427232882992]), (parameters = [6.3209779751170405, 21.770252468901706, 1.56
7482371310491], σ = [4.696111068078141, 5.820857123186585, 7.03082635122257
1]), (parameters = [6.3209779751170405, 21.770252468901706, 1.5674823713104
91], σ = [4.696111068078141, 5.820857123186585, 7.030826351222571]), (param
eters = [6.321301704037298, 21.774489716646897, 1.5677875108685815], σ = [4
.687047184805562, 5.81895869012107, 7.016987023926552]), (parameters = [6.3
21301704037298, 21.774489716646897, 1.5677875108685815], σ = [4.68704718480
5562, 5.81895869012107, 7.016987023926552]), (parameters = [6.3210454901890
3, 21.775712304194734, 1.5675013132549345], σ = [4.685221747258433, 5.82135
6686413296, 7.0230479085738775]), (parameters = [6.324564139901641, 21.7646
56616102226, 1.5677348360770245], σ = [4.687643404827513, 5.826005625528433
, 6.988741062871255]), (parameters = [6.324564139901641, 21.764656616102226
, 1.5677348360770245], σ = [4.687643404827513, 5.826005625528433, 6.9887410
62871255]), (parameters = [6.327171067547953, 21.765061105598203, 1.5680284
32475149], σ = [4.682391744330155, 5.825196526086061, 6.989169057117163])],
 posterior_matrix = [1.7146741527854594 1.8069296068269414 … 1.844441121551
6459 1.8448532275024045; 3.0742585757406067 3.078095873711851 … 3.080287397
995 3.0803059825168924; … ; 1.767332395263793 2.225356056891162 … 1.7623316
241269642 1.762192737266511; 2.101195899873045 1.9351820068886247 … 1.94430
04345700277 1.9443616732305096], tree_statistics = DynamicHMC.TreeStatistic
sNUTS[DynamicHMC.TreeStatisticsNUTS(-326.35884892652507, 5, turning at posi
tions -39:-46, 0.5402519867900831, 63, DynamicHMC.Directions(0xe6553751)), 
DynamicHMC.TreeStatisticsNUTS(-316.69126201184736, 10, turning at positions
 -698:325, 0.9988076290883171, 1023, DynamicHMC.Directions(0x35664545)), Dy
namicHMC.TreeStatisticsNUTS(-317.4207234085219, 9, turning at positions 149
:660, 0.9636811598360674, 1023, DynamicHMC.Directions(0xe4e4f694)), Dynamic
HMC.TreeStatisticsNUTS(-316.7290501592837, 9, turning at positions -341:170
, 0.999248997592912, 511, DynamicHMC.Directions(0xb5143aaa)), DynamicHMC.Tr
eeStatisticsNUTS(-315.7372926648861, 9, turning at positions -180:331, 0.99
9628030385721, 511, DynamicHMC.Directions(0x73c68d4b)), DynamicHMC.TreeStat
isticsNUTS(-318.63541447413763, 7, turning at positions 191:206, 0.99400125
24546168, 207, DynamicHMC.Directions(0x78f1e1fe)), DynamicHMC.TreeStatistic
sNUTS(-317.76750028781385, 9, turning at positions -194:317, 0.981114204643
8337, 511, DynamicHMC.Directions(0xb477953d)), DynamicHMC.TreeStatisticsNUT
S(-314.5588993302546, 8, turning at positions -145:110, 0.9217265323116614,
 255, DynamicHMC.Directions(0x0a5a106e)), DynamicHMC.TreeStatisticsNUTS(-31
6.29902632166795, 6, turning at positions 41:48, 0.41988012183706336, 111, 
DynamicHMC.Directions(0xdafd8640)), DynamicHMC.TreeStatisticsNUTS(-316.3736
966571559, 7, turning at positions -138:-153, 0.8509088539170357, 175, Dyna
micHMC.Directions(0xc51f5716))  …  DynamicHMC.TreeStatisticsNUTS(-307.40595
057596664, 2, turning at positions 3:6, 0.5391353030528737, 7, DynamicHMC.D
irections(0x414ee446)), DynamicHMC.TreeStatisticsNUTS(-307.10770503880667, 
2, divergence at position 5, 0.2875129131051729, 7, DynamicHMC.Directions(0
xb5953955)), DynamicHMC.TreeStatisticsNUTS(-309.267596659324, 4, turning at
 positions -2:13, 0.2294413627325506, 15, DynamicHMC.Directions(0x3d6956fd)
), DynamicHMC.TreeStatisticsNUTS(-309.8467113888505, 0, divergence at posit
ion -1, 0.0, 1, DynamicHMC.Directions(0x23b201ca)), DynamicHMC.TreeStatisti
csNUTS(-306.9978339576681, 4, divergence at position -11, 0.926590192402939
8, 19, DynamicHMC.Directions(0xbaf75a88)), DynamicHMC.TreeStatisticsNUTS(-3
09.0485507210135, 3, turning at positions -7:0, 0.0011510237180031933, 7, D
ynamicHMC.Directions(0x22449bb8)), DynamicHMC.TreeStatisticsNUTS(-308.43574
02038905, 3, divergence at position -11, 0.1838173056454343, 12, DynamicHMC
.Directions(0x079bd2c1)), DynamicHMC.TreeStatisticsNUTS(-308.2743979956188,
 4, divergence at position -16, 0.40406791668567166, 21, DynamicHMC.Directi
ons(0x696a5de5)), DynamicHMC.TreeStatisticsNUTS(-308.79968309536093, 3, div
ergence at position -2, 0.11247311624440146, 9, DynamicHMC.Directions(0x0b8
cac07)), DynamicHMC.TreeStatisticsNUTS(-308.6566863071676, 2, divergence at
 position -5, 0.2, 5, DynamicHMC.Directions(0x79cc8b38))], κ = Gaussian kin
etic energy (Diagonal), √diag(M⁻¹): [0.09065436440431678, 0.031237859841619
125, 0.03769054477309786, 0.12923649076550886, 0.09321145506197823, 0.11468
930316731393], ϵ = 0.005160717301756368)

Conclusion

Due to the chaotic nature of Lorenz Equation, it is a very hard problem to estimate as it has the property of exponentially increasing errors. Its uncertainty plot points to its chaotic behaviour and goes awry for different values of tolerance, we use 1e-8 as the tolerance as it makes its uncertainty small enough to be trusted in (0,30) time span.

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/BayesianInference","DiffEqBayesLorenz.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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/BayesianInference/Project.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`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.

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⌃ [dd4b983a] LZO_jll v2.10.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
⌃ [d4300ac3] Libgcrypt_jll v1.8.7+0
⌃ [7e76a0d4] Libglvnd_jll v1.6.0+0
⌃ [7add5ba3] Libgpg_error_jll v1.42.0+0
⌃ [94ce4f54] Libiconv_jll v1.16.1+2
⌃ [4b2f31a3] Libmount_jll v2.35.0+0
⌅ [89763e89] Libtiff_jll v4.5.1+1
⌃ [38a345b3] Libuuid_jll v2.36.0+0
⌅ [856f044c] MKL_jll v2023.2.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
⌅ [458c3c95] OpenSSL_jll v1.1.22+0
⌃ [efe28fd5] OpenSpecFun_jll v0.5.5+0
⌃ [91d4177d] Opus_jll v1.3.2+0
⌃ [30392449] Pixman_jll v0.42.2+0
⌅ [c0090381] Qt6Base_jll v6.4.2+3
⌅ [f50d1b31] Rmath_jll v0.4.0+0
⌃ [a2964d1f] Wayland_jll v1.21.0+0
⌃ [2381bf8a] Wayland_protocols_jll v1.25.0+0
⌅ [02c8fc9c] XML2_jll v2.10.3+0
⌃ [aed1982a] XSLT_jll v1.1.34+0
⌃ [ffd25f8a] XZ_jll v5.4.4+0
⌃ [4f6342f7] Xorg_libX11_jll v1.8.6+0
⌃ [0c0b7dd1] Xorg_libXau_jll v1.0.11+0
⌃ [935fb764] Xorg_libXcursor_jll v1.2.0+4
⌃ [a3789734] Xorg_libXdmcp_jll v1.1.4+0
⌃ [1082639a] Xorg_libXext_jll v1.3.4+4
⌃ [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
⌃ [a51aa0fd] Xorg_libXi_jll v1.7.10+4
⌃ [d1454406] Xorg_libXinerama_jll v1.1.4+4
⌃ [ec84b674] Xorg_libXrandr_jll v1.5.2+4
⌃ [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
⌃ [14d82f49] Xorg_libpthread_stubs_jll v0.1.1+0
⌃ [c7cfdc94] Xorg_libxcb_jll v1.15.0+0
⌃ [cc61e674] Xorg_libxkbfile_jll v1.1.2+0
  [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+0
⌃ [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
⌃ [c5fb5394] Xorg_xtrans_jll v1.5.0+0
⌃ [8f1865be] ZeroMQ_jll v4.3.4+0
⌃ [3161d3a3] Zstd_jll v1.5.5+0
⌅ [214eeab7] fzf_jll v0.29.0+0
⌃ [a4ae2306] libaom_jll v3.4.0+0
⌃ [0ac62f75] libass_jll v0.15.1+0
⌃ [f638f0a6] libfdk_aac_jll v2.0.2+0
⌃ [b53b4c65] libpng_jll v1.6.38+0
⌃ [a9144af2] libsodium_jll v1.0.20+0
⌃ [f27f6e37] libvorbis_jll v1.3.7+1
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
⌃ [d8fb68d0] xkbcommon_jll v1.4.1+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.9.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics v1.9.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.0.2+0
  [deac9b47] LibCURL_jll v7.84.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.2+0
  [14a3606d] MozillaCACerts_jll v2022.10.11
  [4536629a] OpenBLAS_jll v0.3.21+4
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.42.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+6
  [83775a58] Zlib_jll v1.2.13+0
  [8e850b90] libblastrampoline_jll v5.8.0+0
  [8e850ede] nghttp2_jll v1.48.0+0
  [3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.