Lotka-Volterra Bayesian Parameter Estimation Benchmarks
Parameter Estimation of Lotka-Volterra Equation using DiffEqBayes.jl
using DiffEqBayes, StanSample, DynamicHMC, Turingusing Distributions, BenchmarkTools, StaticArrays
using OrdinaryDiffEq, RecursiveArrayTools, ParameterizedFunctions
using Plots, LinearAlgebragr(fmt=:png)Plots.GRBackend()Initializing the problem
f = @ode_def LotkaVolterraTest begin
dx = a*x - b*x*y
dy = -c*y + d*x*y
end a b c d(::Main.var"##WeaveSandBox#225".LotkaVolterraTest{Main.var"##WeaveSandBox#2
25".var"###ParameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".v
ar"###ParameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###
ParameterizedJacobianFunction#229", Nothing, Nothing, ModelingToolkit.ODESy
stem}) (generic function with 1 method)u0 = [1.0,1.0]
tspan = (0.0,10.0)
p = [1.5,1.0,3.0,1,0]5-element Vector{Float64}:
1.5
1.0
3.0
1.0
0.0prob = ODEProblem(f, u0, tspan, p)
sol = solve(prob,Tsit5())retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 34-element Vector{Float64}:
0.0
0.0776084743154256
0.23264513699277584
0.4291185174543143
0.6790821987497083
0.9444046158046306
1.2674601546021105
1.6192913303893046
1.9869754428624007
2.2640902393538296
⋮
7.584863345264154
7.978068981329682
8.48316543760351
8.719248247740158
8.949206788834692
9.200185054623292
9.438029017301554
9.711808134779586
10.0
u: 34-element Vector{Vector{Float64}}:
[1.0, 1.0]
[1.0454942346944578, 0.8576684823217128]
[1.1758715885138271, 0.6394595703175443]
[1.419680960717083, 0.4569962601282089]
[1.8767193950080012, 0.3247334292791134]
[2.588250064553348, 0.26336255535952197]
[3.860708909220769, 0.2794458098285261]
[5.750812667710401, 0.522007253793458]
[6.8149789991301635, 1.9177826328390826]
[4.392999292571394, 4.1946707928506015]
⋮
[2.6142539677883248, 0.26416945387526314]
[4.24107612719179, 0.3051236762922018]
[6.791123785297775, 1.1345287797146668]
[6.26537067576476, 2.741693507540315]
[3.780765111887945, 4.431165685863443]
[1.816420140681737, 4.064056625315896]
[1.1465021407690763, 2.791170661621642]
[0.9557986135403417, 1.623562295185047]
[1.0337581256020802, 0.9063703842885995]su0 = SA[1.0,1.0]
sp = SA[1.5,1.0,3.0,1,0]
sprob = ODEProblem{false,SciMLBase.FullSpecialize}(f, su0, tspan, sp)
sol = solve(sprob,Tsit5())retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 34-element Vector{Float64}:
0.0
0.0776084743154256
0.23264513699277584
0.4291185174543143
0.6790821987497083
0.9444046158046306
1.2674601546021105
1.6192913303893046
1.9869754428624007
2.2640902393538296
⋮
7.584863345264154
7.978068981329682
8.48316543760351
8.719248247740158
8.949206788834692
9.200185054623292
9.438029017301554
9.711808134779586
10.0
u: 34-element Vector{StaticArraysCore.SVector{2, Float64}}:
[1.0, 1.0]
[1.0454942346944578, 0.8576684823217128]
[1.1758715885138271, 0.6394595703175443]
[1.419680960717083, 0.4569962601282089]
[1.8767193950080012, 0.3247334292791134]
[2.588250064553348, 0.26336255535952197]
[3.860708909220769, 0.2794458098285261]
[5.750812667710401, 0.522007253793458]
[6.8149789991301635, 1.9177826328390826]
[4.392999292571394, 4.1946707928506015]
⋮
[2.6142539677883248, 0.26416945387526314]
[4.241076127191789, 0.30512367629220183]
[6.791123785297779, 1.1345287797146653]
[6.265370675764766, 2.7416935075403135]
[3.7807651118879293, 4.431165685863457]
[1.8164201406817235, 4.064056625315901]
[1.146502140769069, 2.791170661621637]
[0.9557986135403385, 1.6235622951850437]
[1.033758125602079, 0.9063703842885992]We take the solution data obtained and add noise to it to obtain data for using in the Bayesian Inference of the parameters
t = collect(range(1,stop=10,length=10))
sig = 0.49
data = convert(Array, VectorOfArray([(sol(t[i]) + sig*randn(2)) for i in 1:length(t)]))2×10 Matrix{Float64}:
1.99225 7.08929 0.9339 1.74927 … 3.84984 3.83629 -0.243501
0.938425 1.68661 2.36932 -0.379958 -0.587858 4.56227 1.08317Plots of the actual data and generated data
scatter(t, data[1,:], lab="#prey (data)")
scatter!(t, data[2,:], lab="#predator (data)")
plot!(sol)
priors = [truncated(Normal(1.5,0.5),0.5,2.5),truncated(Normal(1.2,0.5),0,2),truncated(Normal(3.0,0.5),1,4),truncated(Normal(1.0,0.5),0,2)]4-element Vector{Distributions.Truncated{Distributions.Normal{Float64}, Dis
tributions.Continuous, Float64, Float64, Float64}}:
Truncated(Distributions.Normal{Float64}(μ=1.5, σ=0.5); lower=0.5, upper=2.
5)
Truncated(Distributions.Normal{Float64}(μ=1.2, σ=0.5); lower=0.0, upper=2.
0)
Truncated(Distributions.Normal{Float64}(μ=3.0, σ=0.5); lower=1.0, upper=4.
0)
Truncated(Distributions.Normal{Float64}(μ=1.0, σ=0.5); lower=0.0, upper=2.
0)Stan.jl backend
The solution converges for tolerance values lower than 1e-3, lower tolerance leads to better accuracy in result but is accompanied by longer warmup and sampling time, truncated normal priors are used for preventing Stan from stepping into negative values.
@btime bayesian_result_stan = stan_inference(prob,t,data,priors,num_samples=10_000,print_summary=false,delta = 0.65, vars = (DiffEqBayes.StanODEData(), InverseGamma(2, 3)))33.050041 seconds (2.13 M allocations: 144.962 MiB, 1.42% gc time, 6.02% c
ompilation time)
32.300621 seconds (674 allocations: 56.625 KiB)
25.812665 seconds (674 allocations: 56.625 KiB)
25.728564 seconds (675 allocations: 56.922 KiB)
44.489 s (260872 allocations: 31.95 MiB)
Chains MCMC chain (10000×6×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
parameters = sigma1.1, sigma1.2, theta_1, theta_2, theta_3, theta_4
internals =
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
sigma1.1 0.6944 0.1941 0.0032 3399.1221 2461.8497 1.000
0 ⋯
sigma1.2 0.6907 0.1918 0.0030 4399.8382 4427.5426 0.999
9 ⋯
theta_1 1.4586 0.0976 0.0020 2236.6745 2731.2775 1.000
1 ⋯
theta_2 1.0844 0.1470 0.0030 3100.7289 2530.3454 0.999
9 ⋯
theta_3 3.1423 0.3029 0.0064 2276.4138 2183.8754 1.000
1 ⋯
theta_4 1.0425 0.1117 0.0023 2389.9320 3215.6497 1.000
0 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
sigma1.1 0.4245 0.5589 0.6582 0.7888 1.1747
sigma1.2 0.4237 0.5577 0.6561 0.7857 1.1592
theta_1 1.2889 1.3933 1.4502 1.5142 1.6740
theta_2 0.8559 0.9847 1.0645 1.1627 1.4328
theta_3 2.5622 2.9424 3.1344 3.3378 3.7734
theta_4 0.8365 0.9681 1.0369 1.1141 1.2765Direct Turing.jl
@model function fitlv(data, prob)
# Prior distributions.
σ ~ InverseGamma(2, 3)
α ~ truncated(Normal(1.5, 0.5), 0.5, 2.5)
β ~ truncated(Normal(1.2, 0.5), 0, 2)
γ ~ truncated(Normal(3.0, 0.5), 1, 4)
δ ~ truncated(Normal(1.0, 0.5), 0, 2)
# Simulate Lotka-Volterra model.
p = SA[α, β, γ, δ]
_prob = remake(prob, p = p)
predicted = solve(_prob, Tsit5(); saveat=t)
# Observations.
for i in 1:length(predicted)
data[:, i] ~ MvNormal(predicted[i], σ^2 * I)
end
return nothing
end
model = fitlv(data, sprob)
@time chain = sample(model, Turing.NUTS(0.65), 10000; progress=false)49.943596 seconds (137.55 M allocations: 22.088 GiB, 5.86% gc time, 43.16%
compilation time)
Chains MCMC chain (10000×17×1 Array{Float64, 3}):
Iterations = 1001:1:11000
Number of chains = 1
Samples per chain = 10000
Wall duration = 41.98 seconds
Compute duration = 41.98 seconds
parameters = σ, α, β, γ, δ
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 rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
σ 0.6073 0.1162 0.0019 4040.9844 4621.6486 1.000
1 ⋯
α 1.4569 0.0921 0.0020 2026.0851 2226.7652 1.001
0 ⋯
β 1.0741 0.1299 0.0024 3367.5230 3055.1646 1.000
3 ⋯
γ 3.1412 0.2897 0.0064 2077.0964 2320.5783 1.001
6 ⋯
δ 1.0410 0.1039 0.0023 2101.5446 2475.1650 1.001
5 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ 0.4295 0.5259 0.5922 0.6697 0.8752
α 1.2967 1.3929 1.4503 1.5128 1.6581
β 0.8679 0.9865 1.0576 1.1440 1.3873
γ 2.5934 2.9421 3.1278 3.3319 3.7324
δ 0.8438 0.9705 1.0383 1.1095 1.2560Turing.jl backend
@btime bayesian_result_turing = turing_inference(prob, Tsit5(), t, data, priors, num_samples=10_000)20.243 s (102038729 allocations: 16.44 GiB)
Chains MCMC chain (10000×17×1 Array{Float64, 3}):
Iterations = 1001:1:11000
Number of chains = 1
Samples per chain = 10000
Wall duration = 20.16 seconds
Compute duration = 20.16 seconds
parameters = theta[1], theta[2], theta[3], theta[4], σ[1]
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 rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
theta[1] 1.4578 0.0930 0.0021 1930.2613 2900.8278 1.000
0 ⋯
theta[2] 1.0732 0.1263 0.0028 2443.8108 2570.1407 1.000
0 ⋯
theta[3] 3.1380 0.2904 0.0062 2214.6105 2916.5340 1.000
3 ⋯
theta[4] 1.0400 0.1044 0.0023 2108.5399 3146.7209 1.000
0 ⋯
σ[1] 0.6036 0.1132 0.0021 2726.6675 2993.5162 1.002
4 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
theta[1] 1.2987 1.3935 1.4509 1.5124 1.6687
theta[2] 0.8701 0.9873 1.0577 1.1426 1.3622
theta[3] 2.5795 2.9445 3.1306 3.3297 3.7277
theta[4] 0.8382 0.9717 1.0364 1.1072 1.2524
σ[1] 0.4308 0.5221 0.5879 0.6664 0.8737DynamicHMC.jl backend
@btime bayesian_result_dynamichmc = dynamichmc_inference(prob,Tsit5(),t,data,priors,num_samples=10_000)27.361 s (233104229 allocations: 17.01 GiB)
(posterior = @NamedTuple{parameters::Vector{Float64}, σ::Vector{Float64}}[(
parameters = [1.4952787760166542, 0.9795402970255677, 2.9476869823156955, 0
.966927522105304], σ = [0.6367975083763966, 0.5615486428956382]), (paramete
rs = [1.3566398999249143, 1.0536723482907244, 3.5629638838881332, 1.1705831
239609186], σ = [0.6861837909213727, 0.47630448627452154]), (parameters = [
1.511893343588603, 0.9770951810936597, 2.9455923014167644, 0.99478592998037
66], σ = [0.6532440814569386, 0.414580196407776]), (parameters = [1.4401379
739014994, 1.1248037701589886, 3.1695560723488105, 1.0298906938645045], σ =
[0.41689794887511006, 0.5130958923493837]), (parameters = [1.4571804227889
873, 1.1964366834362319, 3.2718781506532513, 1.0062183839968675], σ = [0.57
98441557634343, 0.4498198712896204]), (parameters = [1.3997012245700473, 1.
0759423127064194, 3.355121019701527, 1.0850146236311433], σ = [0.4934294851
394983, 0.46005616949407035]), (parameters = [1.4048750214729608, 0.9913981
056188329, 3.322681495068753, 1.0820882304534323], σ = [0.6095718739702543,
0.49050750486062267]), (parameters = [1.3979592820536173, 0.99810908398219
36, 3.3411701027838414, 1.0931726390336778], σ = [0.6102166288651508, 0.495
88816287734266]), (parameters = [1.4802360319385002, 1.0383783739498296, 3.
0199708751129672, 1.0035915139654212], σ = [0.38770597390933154, 0.44335732
1986018]), (parameters = [1.4856525899054631, 1.0326569747654, 2.9742899735
760413, 1.0055419999237747], σ = [0.36074749931646516, 0.42690559535482375]
) … (parameters = [1.333915233324529, 1.0654903579955377, 3.7241477554555
207, 1.167152066855398], σ = [0.7269589771775857, 0.5650651470716619]), (pa
rameters = [1.4305532736453381, 0.9968962854349178, 3.1038455141597523, 1.0
6890198788187], σ = [0.43643033884725585, 0.4422576703960259]), (parameters
= [1.4277568144051505, 1.0596186331566653, 3.224615985573098, 1.0433044913
549459], σ = [0.44333921616184224, 0.45042300171539607]), (parameters = [1.
566707833571088, 1.068351807654099, 2.7463514699751927, 0.9187607940417547]
, σ = [0.5020529305881937, 0.537792529976105]), (parameters = [1.5585826348
044354, 1.2273186683285995, 2.82950664018062, 0.9374276830881155], σ = [0.4
842653248446857, 0.5803440740699037]), (parameters = [1.4943686999158974, 1
.0064611357490014, 2.924185770595777, 0.9788123605953497], σ = [0.712934755
398517, 0.4865827360542992]), (parameters = [1.30627111467881, 1.0087647332
269427, 3.6218712490809173, 1.1929947285057547], σ = [0.4420236118506816, 0
.6448950570840983]), (parameters = [1.4284179039152083, 1.024477456008023,
3.183550993982444, 1.0631724302656136], σ = [0.8053138167546255, 0.41157171
94466109]), (parameters = [1.435221117079158, 1.0436777819340681, 3.1806754
595214723, 1.0672144093703546], σ = [0.7913043826236178, 0.4142254610581592
7]), (parameters = [1.5088950276559596, 1.0406641232919223, 2.9747348867805
39, 0.9636783551927511], σ = [0.4198076856685422, 0.49928136909878856])], p
osterior_matrix = [0.40231266171080304 0.30501098080668365 … 0.361318925898
0667 0.4113776131883844; -0.020671902039938315 0.052281536791318056 … 0.042
750803827751246 0.03985908943616285; … ; -0.4513035571851263 -0.37660977034
454335 … -0.23407257787149943 -0.8679585638343164; -0.577056878287546 -0.74
1697952209558 … -0.8813448614652692 -0.6945854762138415], tree_statistics =
DynamicHMC.TreeStatisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-22.3716327588
17107, 5, turning at positions 42:57, 0.9901458448801421, 63, DynamicHMC.Di
rections(0x5c545af9)), DynamicHMC.TreeStatisticsNUTS(-25.93300321496061, 6,
turning at positions -19:44, 0.9544061834679402, 63, DynamicHMC.Directions
(0x2dc7ce2c)), DynamicHMC.TreeStatisticsNUTS(-24.555202453387864, 5, turnin
g at positions 4:35, 0.9866727684750237, 63, DynamicHMC.Directions(0x817ab5
23)), DynamicHMC.TreeStatisticsNUTS(-23.884970669958662, 6, turning at posi
tions -29:34, 0.9587120515542572, 63, DynamicHMC.Directions(0xbd207362)), D
ynamicHMC.TreeStatisticsNUTS(-24.7132207802753, 5, turning at positions -9:
22, 0.9606865969827195, 31, DynamicHMC.Directions(0x1a072f96)), DynamicHMC.
TreeStatisticsNUTS(-26.242648205808102, 5, turning at positions -14:-45, 0.
9994033791092954, 63, DynamicHMC.Directions(0x77b4a952)), DynamicHMC.TreeSt
atisticsNUTS(-21.19732285437598, 5, turning at positions -8:23, 0.982758497
3542746, 31, DynamicHMC.Directions(0x7a9f3ff7)), DynamicHMC.TreeStatisticsN
UTS(-21.61933622455548, 3, turning at positions -6:1, 0.8791506842743703, 7
, DynamicHMC.Directions(0xc2cd19e1)), DynamicHMC.TreeStatisticsNUTS(-20.394
181261868738, 6, turning at positions -6:57, 0.9998293766539543, 63, Dynami
cHMC.Directions(0xeb2b1cb9)), DynamicHMC.TreeStatisticsNUTS(-21.26774159906
5396, 4, turning at positions -6:9, 0.9284776714404738, 15, DynamicHMC.Dire
ctions(0xd7d39639)) … DynamicHMC.TreeStatisticsNUTS(-24.12367784936122, 5
, turning at positions -22:9, 0.9922834660624564, 31, DynamicHMC.Directions
(0x23bf0a29)), DynamicHMC.TreeStatisticsNUTS(-23.953061378355297, 6, turnin
g at positions -5:58, 0.9894591579783205, 63, DynamicHMC.Directions(0x9ec98
47a)), DynamicHMC.TreeStatisticsNUTS(-20.78418527751442, 4, turning at posi
tions -4:11, 0.9980792211556438, 15, DynamicHMC.Directions(0xdf23551b)), Dy
namicHMC.TreeStatisticsNUTS(-22.33144271357113, 6, turning at positions -18
:45, 0.9369750713863976, 63, DynamicHMC.Directions(0xc0a0a42d)), DynamicHMC
.TreeStatisticsNUTS(-21.251476359898383, 5, turning at positions -23:8, 0.9
904613967158185, 31, DynamicHMC.Directions(0xd8656d88)), DynamicHMC.TreeSta
tisticsNUTS(-22.524269027805516, 5, turning at positions -4:-35, 0.91137435
77183251, 63, DynamicHMC.Directions(0x22a82b5c)), DynamicHMC.TreeStatistics
NUTS(-25.049026099101894, 6, turning at positions -40:-103, 0.9017450111231
021, 127, DynamicHMC.Directions(0xc9080e98)), DynamicHMC.TreeStatisticsNUTS
(-25.501536758349733, 6, turning at positions -57:6, 0.9955105216497803, 63
, DynamicHMC.Directions(0x7da57386)), DynamicHMC.TreeStatisticsNUTS(-21.356
932726721176, 5, turning at positions -14:17, 0.9916521252960447, 31, Dynam
icHMC.Directions(0x7c91ca51)), DynamicHMC.TreeStatisticsNUTS(-21.6731677139
08843, 6, turning at positions -39:24, 0.9999389610896954, 63, DynamicHMC.D
irections(0xf3e036d8))], κ = Gaussian kinetic energy (Diagonal), √diag(M⁻¹)
: [0.07131351740604645, 0.1334176945471142, 0.10321891745411786, 0.11240097
887935113, 0.2903183045121105, 0.267766249298579], ϵ = 0.06305738285550684)Conclusion
Lotka-Volterra Equation is a "predator-prey" model, it models population of two species in which one is the predator (wolf) and the other is the prey (rabbit). It depicts a cyclic behaviour, which is also seen in its Uncertainty Quantification Plots. This behaviour makes it easy to estimate even at very high tolerance values (1e-3).
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","DiffEqBayesLotkaVolterra.jmd")Computer Information:
Julia Version 1.10.5
Commit 6f3fdf7b362 (2024-08-27 14:19 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/BayesianInference/Project.toml`
⌃ [6e4b80f9] BenchmarkTools v1.3.2
⌃ [ebbdde9d] DiffEqBayes v3.6.0
⌅ [459566f4] DiffEqCallbacks v2.29.1
⌃ [31c24e10] Distributions v0.25.100
⌃ [bbc10e6e] DynamicHMC v3.4.6
⌃ [1dea7af3] OrdinaryDiffEq v6.55.0
⌃ [65888b18] ParameterizedFunctions v5.15.0
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⌃ [c1514b29] StanSample v7.4.2
⌃ [90137ffa] StaticArrays v1.6.2
⌅ [fce5fe82] Turing v0.28.3
[37e2e46d] LinearAlgebra
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/BayesianInference/Manifest.toml`
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⌃ [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.