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, LinearAlgebragr(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#292".LorenzExample{Main.var"##WeaveSandBox#292".
var"###ParameterizedDiffEqFunction#294", Main.var"##WeaveSandBox#292".var"#
##ParameterizedTGradFunction#295", Main.var"##WeaveSandBox#292".var"###Para
meterizedJacobianFunction#296", 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
2.66prob = 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
0.0032623422047798056
0.00905768704232478
0.016955555290324203
0.02768838157490834
0.04185394254810964
0.06023706929459432
0.08368090418086754
⋮
29.458011533857974
29.540337322491876
29.60852266181987
29.6799878490824
29.75517147616987
29.82511904397534
29.885820680730024
29.942806967936278
30.0
u: 362-element Vector{Vector{Float64}}:
[1.0, 0.0, 0.0]
[0.9996434557625105, 0.0009988049817849054, 1.7814349300524496e-8]
[0.9961045497425811, 0.010965399721242273, 2.1469572398550234e-6]
[0.9693597399881168, 0.08976883341704892, 0.00014379720463291655]
[0.9242070114022924, 0.24227916339361602, 0.0010460977999503748]
[0.8800496190914562, 0.4387143392377057, 0.0034240472081709093]
[0.84833345707203, 0.6915265721541136, 0.008487272847904601]
[0.8494996923231211, 1.0144878257055865, 0.018211861884682275]
[0.9138892814330075, 1.4424793180152062, 0.036694607831848025]
[1.0888203535352743, 2.0521984876534165, 0.07402929498872184]
⋮
[13.37523333791337, 18.060381179266496, 27.881595463212278]
[13.955616018381155, 9.968378021166673, 38.019750799076256]
[9.495839034590919, 1.7259535630125686, 35.80175620177522]
[4.603344839542985, -0.793002814198357, 29.675553569350267]
[1.776592259424738, -0.5623195021537521, 24.139883212993965]
[0.7417289005296333, -0.07479695797803172, 20.01660906268533]
[0.45273432726039287, 0.24270813103202074, 17.034619521176904]
[0.4255982102181717, 0.5161131130830211, 14.647156800033866]
[0.5442412780662025, 0.8686754748547896, 12.597537036980794]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
0.0032623422047798056
0.00905768704232478
0.016955555290324203
0.02768838157490834
0.04185394254810964
0.06023706929459432
0.08368090418086754
⋮
29.458011533857974
29.540337322491876
29.60852266181987
29.6799878490824
29.75517147616987
29.82511904397534
29.885820680730024
29.942806967936278
30.0
u: 362-element Vector{StaticArraysCore.SVector{3, Float64}}:
[1.0, 0.0, 0.0]
[0.9996434557625105, 0.0009988049817849054, 1.7814349300524496e-8]
[0.9961045497425811, 0.010965399721242273, 2.1469572398550234e-6]
[0.9693597399881168, 0.08976883341704892, 0.00014379720463291655]
[0.9242070114022924, 0.24227916339361602, 0.0010460977999503748]
[0.8800496190914562, 0.4387143392377057, 0.0034240472081709093]
[0.84833345707203, 0.6915265721541136, 0.008487272847904601]
[0.8494996923231211, 1.0144878257055865, 0.018211861884682275]
[0.9138892814330075, 1.4424793180152062, 0.036694607831848025]
[1.0888203535352743, 2.0521984876534165, 0.07402929498872184]
⋮
[13.37523333791337, 18.060381179266496, 27.881595463212278]
[13.955616018381155, 9.968378021166673, 38.019750799076256]
[9.495839034590919, 1.7259535630125686, 35.80175620177522]
[4.603344839542985, -0.793002814198357, 29.675553569350267]
[1.776592259424738, -0.5623195021537521, 24.139883212993965]
[0.7417289005296333, -0.07479695797803172, 20.01660906268533]
[0.45273432726039287, 0.24270813103202074, 17.034619521176904]
[0.4255982102181717, 0.5161131130830211, 14.647156800033866]
[0.5442412780662025, 0.8686754748547896, 12.597537036980794]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}:
-8.85217 -8.08603 -8.39116 -9.72382 … 11.5624 2.98546 0.47296
-8.20962 -7.99019 -6.67531 -9.98188 15.5606 1.08227 0.789943
29.0078 25.0478 28.269 26.4865 25.0532 25.349 12.7473Plots 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)))27435.345857 seconds (2.87 M allocations: 179.127 MiB, 0.00% gc time, 0.01%
compilation time)
27458.325887 seconds (9.67 M allocations: 607.667 MiB, 0.00% gc time, 0.02%
compilation time: 2% 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 0.4307 0.0000 0.0000 NaN NaN NaN
⋯
sigma1.2 0.3746 0.0000 0.0000 NaN NaN NaN
⋯
sigma1.3 0.8635 0.0000 0.0000 NaN NaN NaN
⋯
theta_1 7.6330 0.0000 0.0000 NaN NaN NaN
⋯
theta_2 39.6308 0.0000 0.0000 NaN NaN NaN
⋯
theta_3 2.7601 0.0000 0.0000 NaN NaN NaN
⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
sigma1.1 0.4307 0.4307 0.4307 0.4307 0.4307
sigma1.2 0.3746 0.3746 0.3746 0.3746 0.3746
sigma1.3 0.8635 0.8635 0.8635 0.8635 0.8635
theta_1 7.6330 7.6330 7.6330 7.6330 7.6330
theta_2 39.6308 39.6308 39.6308 39.6308 39.6308
theta_3 2.7601 2.7601 2.7601 2.7601 2.7601Direct 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)456.825342 seconds (987.43 M allocations: 185.818 GiB, 4.76% gc time, 8.09%
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 = 437.45 seconds
Compute duration = 437.45 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 0.3226 0.0000 0.0000 23.5517 33.2178 1.3423
⋯
σ 6.1625 0.0000 0.0000 NaN NaN NaN
⋯
ρ 33.0272 0.0000 0.0000 NaN NaN NaN
⋯
β 2.1994 0.0000 0.0000 313.5952 NaN 1.0002
⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 0.3226 0.3226 0.3226 0.3226 0.3226
σ 6.1625 6.1625 6.1625 6.1625 6.1625
ρ 33.0272 33.0272 33.0272 33.0272 33.0272
β 2.1994 2.1994 2.1994 2.1994 2.1994Using 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)])236.756006 seconds (838.47 M allocations: 51.735 GiB, 3.03% gc time, 9.88%
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 = 233.06 seconds
Compute duration = 233.06 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] 10.0210 1.0101 0.6967 2.5495 26.5890 2.1089
⋯
theta[2] 23.9578 0.1118 0.0766 2.6982 32.0121 1.9810
⋯
theta[3] 2.4202 0.0542 0.0371 2.5631 24.3631 2.1043
⋯
σ[1] 5.5870 0.1879 0.1018 2.6653 12.8395 2.0207
⋯
σ[2] 7.8030 1.7086 1.1628 2.5126 25.3117 2.0669
⋯
σ[3] 8.3781 0.7174 0.4832 2.5937 15.7215 2.1080
⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
theta[1] 8.9197 8.9728 10.2091 11.1071 11.3399
theta[2] 23.8287 23.8514 23.8970 24.0748 24.0888
theta[3] 2.3609 2.3677 2.3840 2.4758 2.4843
σ[1] 5.0721 5.5182 5.5968 5.6558 6.0222
σ[2] 5.8025 6.3744 7.1622 9.5196 9.8533
σ[3] 7.3876 7.7314 8.2818 8.8765 9.5153Using DynamicHMC.jl backend
@time bayesian_result_dynamichmc = dynamichmc_inference(prob,Tsit5(),t,data,priors;solve_kwargs = (reltol=1e-8,abstol=1e-8,))630.036196 seconds (161.22 M allocations: 31.273 GiB, 0.77% gc time, 1.60%
compilation time)
(posterior = NamedTuple{(:parameters, :σ), Tuple{Vector{Float64}, Vector{Fl
oat64}}}[(parameters = [5.892885119396755, 20.254768703211592, 1.3960370552
705414], σ = [4.581535285413694, 6.894597727232037, 8.761118444572379]), (p
arameters = [5.89092620006707, 20.25862449653875, 1.3958734427452093], σ =
[4.582088450660769, 6.8935218679985395, 8.755013874553747]), (parameters =
[5.890456309207671, 20.262430728225326, 1.3959562341772502], σ = [4.5863972
21935279, 6.882120988064264, 8.759272315133982]), (parameters = [5.88559846
8497951, 20.276157401537503, 1.396096240789092], σ = [4.595252289976552, 6.
869124767664774, 8.736386203216705]), (parameters = [5.8924087182984, 20.19
2225876848706, 1.3917876001443956], σ = [4.456696536426733, 6.8254680568652
18, 8.784355855584737]), (parameters = [5.960731555071015, 20.1140141921062
1, 1.3959931307806401], σ = [4.487290826773641, 7.108976604980974, 8.763470
726004385]), (parameters = [5.9585606201148735, 20.137490450408855, 1.39705
865917747], σ = [4.476362672504596, 7.081596483160127, 8.754855515434611]),
(parameters = [5.853375160594397, 20.241676896127714, 1.389719314589237],
σ = [4.907927942435779, 6.852682661394892, 8.902402444120282]), (parameters
= [5.745634467388232, 20.44164343871836, 1.3885276352661746], σ = [4.84118
50536503485, 6.8262393003308075, 8.829827658525257]), (parameters = [5.7446
24133533806, 20.43960821639364, 1.3884749426463618], σ = [4.840081568176277
, 6.82605353173021, 8.830916741438054]) … (parameters = [5.26603249435250
6, 21.59488292813381, 1.425097154420673], σ = [6.106160777109951, 6.4273573
15218294, 10.953836996449448]), (parameters = [5.268381320450744, 21.591747
361104996, 1.425323227875803], σ = [6.103026167056847, 6.425463730109816, 1
0.952548380196212]), (parameters = [5.270122259893227, 21.592426623456817,
1.4252774390885616], σ = [6.1007421262145165, 6.425468673030976, 10.9508402
56968267]), (parameters = [5.263606823087966, 21.59768075243562, 1.42514559
6871239], σ = [6.105529956650002, 6.418774058708283, 10.947147053084539]),
(parameters = [5.263538066083068, 21.597353784583383, 1.4251497656158438],
σ = [6.105439117519761, 6.419080389757018, 10.94683264528125]), (parameters
= [5.138992215979233, 21.653615353862225, 1.422861383934352], σ = [6.09734
6985729417, 6.503603917680418, 10.798013591750083]), (parameters = [5.14384
7694013023, 21.659359653923534, 1.42374594934479], σ = [6.133130057560993,
6.716550966642757, 10.672791474460025]), (parameters = [5.144800780295417,
21.604505582683714, 1.418026492790116], σ = [6.099501388516848, 6.624912996
000554, 10.46807913571942]), (parameters = [5.220231167223067, 21.514425543
413143, 1.414074409136081], σ = [6.077013678382942, 6.547159024684756, 10.4
13021423468583]), (parameters = [5.216308997293642, 21.52443183211594, 1.41
47581266001996], σ = [6.068710857615087, 6.482805816869188, 10.525228046165
209])], posterior_matrix = [1.773745711248102 1.7734132348880138 … 1.652541
6858226332 1.6517900631484048; 3.008390257345395 3.0085806039460756 … 3.068
723665651192 3.0691886542449076; … ; 1.9307381668648076 1.93058211088803 …
1.8790312188668443 1.8691534130787106; 2.17032357311942 2.169626550494269 …
2.343057082922823 2.3537750464277987], tree_statistics = DynamicHMC.TreeSt
atisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-309.89921295788713, 7, turning
at positions 11:18, 0.126189390396227, 135, DynamicHMC.Directions(0x5393308
a)), DynamicHMC.TreeStatisticsNUTS(-311.0295762920189, 6, turning at positi
ons -49:-64, 0.1984437999648175, 79, DynamicHMC.Directions(0xfce2d60f)), Dy
namicHMC.TreeStatisticsNUTS(-309.5930522922027, 5, divergence at position -
51, 0.96078431372549, 51, DynamicHMC.Directions(0x06bf3780)), DynamicHMC.Tr
eeStatisticsNUTS(-308.68367050853414, 7, turning at positions -135:-142, 0.
7628586387595354, 183, DynamicHMC.Directions(0xa7b3f029)), DynamicHMC.TreeS
tatisticsNUTS(-307.7359746950645, 10, reached maximum depth without diverge
nce or turning, 0.9998699075038789, 1023, DynamicHMC.Directions(0xce7f4831)
), DynamicHMC.TreeStatisticsNUTS(-309.27712732704964, 10, reached maximum d
epth without divergence or turning, 0.9997698354430756, 1023, DynamicHMC.Di
rections(0xb6e6b869)), DynamicHMC.TreeStatisticsNUTS(-308.1311656214998, 10
, reached maximum depth without divergence or turning, 0.9999977317533036,
1023, DynamicHMC.Directions(0x58f05602)), DynamicHMC.TreeStatisticsNUTS(-31
2.7086072379815, 10, reached maximum depth without divergence or turning, 0
.9999018234722643, 1023, DynamicHMC.Directions(0xde76d195)), DynamicHMC.Tre
eStatisticsNUTS(-309.60125571570774, 10, reached maximum depth without dive
rgence or turning, 0.9993758889398268, 1023, DynamicHMC.Directions(0x7e0b98
a6)), DynamicHMC.TreeStatisticsNUTS(-312.1852357324753, 2, turning at posit
ions -4:-7, 0.5709669570231022, 7, DynamicHMC.Directions(0x7850e068)) … D
ynamicHMC.TreeStatisticsNUTS(-311.488012740343, 4, turning at positions -3:
12, 0.46643496564107173, 15, DynamicHMC.Directions(0x95256afc)), DynamicHMC
.TreeStatisticsNUTS(-311.86984708201345, 4, turning at positions 18:21, 0.8
659252494102501, 23, DynamicHMC.Directions(0x873050bd)), DynamicHMC.TreeSta
tisticsNUTS(-313.5017302661109, 4, divergence at position -24, 0.3792359148
7147684, 29, DynamicHMC.Directions(0x25199ae5)), DynamicHMC.TreeStatisticsN
UTS(-314.72176642638846, 6, divergence at position -116, 0.2494163381042602
, 122, DynamicHMC.Directions(0x93ddb906)), DynamicHMC.TreeStatisticsNUTS(-3
08.84858256537996, 2, turning at positions -2:1, 0.354059725166937, 3, Dyna
micHMC.Directions(0x87e7bd35)), DynamicHMC.TreeStatisticsNUTS(-308.84642978
95002, 10, reached maximum depth without divergence or turning, 0.998914482
7961876, 1023, DynamicHMC.Directions(0x327babbb)), DynamicHMC.TreeStatistic
sNUTS(-313.13735241482846, 9, turning at positions -727:-758, 0.91149672936
85995, 799, DynamicHMC.Directions(0x12673029)), DynamicHMC.TreeStatisticsNU
TS(-310.9463077461589, 10, reached maximum depth without divergence or turn
ing, 0.9837860425660939, 1023, DynamicHMC.Directions(0xbff7d777)), DynamicH
MC.TreeStatisticsNUTS(-310.2654642444168, 10, reached maximum depth without
divergence or turning, 0.999271197523369, 1023, DynamicHMC.Directions(0x4a
b7070d)), DynamicHMC.TreeStatisticsNUTS(-311.52630430601346, 9, turning at
positions -504:-631, 0.914874341903769, 895, DynamicHMC.Directions(0x7301cd
08))], κ = Gaussian kinetic energy (Diagonal), √diag(M⁻¹): [0.0532750199058
3436, 0.02394947376390009, 0.012359315098273886, 0.08220281796805226, 0.071
02907592739113, 0.042736344355664], ϵ = 0.0007261671904163304)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 1.9.4
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
JULIA_IMAGE_THREADS = 1
Package Information:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/BayesianInference/Project.toml`
⌃ [6e4b80f9] BenchmarkTools v1.3.2
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⌃ [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
[91a5bcdd] Plots v1.39.0
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[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:
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[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.