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, LinearAlgebra"""Print elapsed sampling time and the Turing chain."""
function display_ess_per_sec(chain, elapsed)
println("Elapsed time: $(round(elapsed; digits=2)) seconds")
display(chain)
end
# Distributions requires a vector mean; sol[:, i] can be a scalar after ODE 7.
_mvn(u, σ) = MvNormal(
let n = length(σ)
μ = u isa AbstractVector ? collect(float.(u)) : [float(u)]
length(μ) == n ? μ : fill(μ[1], n)
end,
Diagonal(σ .^ 2),
)
"""Extract and display Stan's internal timing from its CSV output files."""
function display_stan_timing(stan_result)
sample_files = stan_result.model.sample_file
for (chain_idx, f) in enumerate(sample_files)
isfile(f) || continue
lines = readlines(f)
println("Chain $chain_idx timing (from Stan CSV):")
for line in lines
if startswith(line, "#") && occursin("Elapsed Time", line)
println(" ", strip(line[2:end]))
elseif startswith(line, "#") && occursin("seconds", line)
println(" ", strip(line[2:end]))
end
end
end
endMain.var"##WeaveSandBox#277".display_stan_timinggr(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 dMain.var"##WeaveSandBox#277".LotkaVolterraTest{Main.var"##WeaveSandBox#277"
.var"###ParameterizedDiffEqFunction#279", Main.var"##WeaveSandBox#277".var"
###ParameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"###Par
ameterizedJacobianFunction#281", Nothing, Nothing, ModelingToolkitBase.Syst
em}(Main.var"##WeaveSandBox#277".var"##ParameterizedDiffEqFunction#279", Li
nearAlgebra.UniformScaling{Bool}(true), nothing, Main.var"##WeaveSandBox#27
7".var"##ParameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"
##ParameterizedJacobianFunction#281", nothing, nothing, nothing, nothing, n
othing, nothing, nothing, [:x, :y], :t, nothing, Model ##Parameterized#278:
Equations (2):
2 standard: see equations(##Parameterized#278)
Unknowns (2): see unknowns(##Parameterized#278)
x(t)
y(t)
Parameters (4): see parameters(##Parameterized#278)
a
b
c
d, nothing, nothing)u0 = [1.0, 1.0]
tspan = (0.0, 10.0)
p = [1.5, 1.0, 3.0, 1.0]4-element Vector{Float64}:
1.5
1.0
3.0
1.0prob = 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
0.2326451370670694
0.42911851563726466
0.679082199936808
0.9444046279774128
1.2674601918628516
1.61929140093895
1.9869755481702074
2.2640903679981617
⋮
7.5848624442719235
7.978067891667038
8.483164641366145
8.719247691882519
8.949206449510513
9.200184762926114
9.438028551201125
9.711807820573478
10.0
u: 34-element Vector{Vector{Float64}}:
[1.0, 1.0]
[1.0454942346944578, 0.8576684823217127]
[1.1758715885890039, 0.6394595702308833]
[1.4196809580265157, 0.456996261440507]
[1.876719397626222, 0.32473342884607376]
[2.5882501035146124, 0.26336255403957287]
[3.86070908479701, 0.27944581878759067]
[5.750813064347344, 0.5220073551361064]
[6.814978696356639, 1.9177834056716259]
[4.392997771045281, 4.194671543390715]
⋮
[2.614251082502646, 0.26416954350041094]
[4.241070648057705, 0.3051232653305123]
[6.791121825691655, 1.1345253835487776]
[6.2653749402951835, 2.741688595595249]
[3.7807688120522114, 4.431164521488329]
[1.8164214705303103, 4.064057991958692]
[1.1465028256357626, 2.7911730348239576]
[0.9557986528530791, 1.6235633163407794]
[1.0337581330572256, 0.9063703732076024]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
0.2326451370670694
0.42911851563726466
0.679082199936808
0.9444046279774128
1.2674601918628516
1.61929140093895
1.9869755481702074
2.2640903679981617
⋮
7.5848624442719235
7.978067891667038
8.483164641366145
8.719247691882519
8.949206449510513
9.200184762926114
9.438028551201125
9.711807820573478
10.0
u: 34-element Vector{StaticArraysCore.SVector{2, Float64}}:
[1.0, 1.0]
[1.0454942346944578, 0.8576684823217127]
[1.1758715885890039, 0.6394595702308833]
[1.4196809580265157, 0.456996261440507]
[1.876719397626222, 0.32473342884607376]
[2.5882501035146124, 0.26336255403957287]
[3.86070908479701, 0.27944581878759067]
[5.750813064347344, 0.5220073551361064]
[6.814978696356639, 1.9177834056716259]
[4.392997771045281, 4.194671543390715]
⋮
[2.614251082502646, 0.26416954350041094]
[4.241070648057705, 0.3051232653305123]
[6.791121825691655, 1.1345253835487776]
[6.2653749402951835, 2.741688595595249]
[3.7807688120522114, 4.431164521488329]
[1.8164214705303103, 4.064057991958692]
[1.1465028256357626, 2.7911730348239576]
[0.9557986528530791, 1.6235633163407794]
[1.0337581330572256, 0.9063703732076024]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}:
1.57052 7.42532 1.54653 2.09851 … 3.85492 2.93638 0.496001
0.30677 2.07701 1.20756 -0.48807 0.296766 4.38004 0.988962Plots 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.
We use adapt_delta = 0.85 (Stan's default) consistently across all backends for a fair comparison. Stan infers a separate noise parameter (sigma) per data dimension via the vars specification.
bayesian_result_stan = @time stan_inference(
prob, :rk45, t, data, priors; print_summary = false,
sample_kwargs = Dict(:delta => 0.85, :num_samples => 10_000),
vars = (DiffEqBayes.StanODEData(), InverseGamma(2, 3)))68.402137 seconds (4.93 M allocations: 241.034 MiB, 0.06% gc time, 5.37% c
ompilation time)
98.557345 seconds (34.58 M allocations: 1.707 GiB, 0.41% gc time, 16.60% c
ompilation time: 2% of which was recompilation)
10000×6 DataFrame
Row │ sigma1.1 sigma1.2 theta_1 theta_2 theta_3 theta_4
│ Float64 Float64 Float64 Float64 Float64 Float64
───────┼──────────────────────────────────────────────────────────
1 │ 1.03287 0.614896 1.48615 1.03078 3.01911 1.03344
2 │ 1.20034 0.603566 1.49774 0.990152 2.95901 1.02386
3 │ 1.15486 0.575475 1.55995 1.27544 2.99462 0.919558
4 │ 0.701767 0.619013 1.40277 0.985558 3.22042 1.15966
5 │ 0.889381 0.608258 1.42138 1.05466 3.21898 1.0854
6 │ 0.582359 0.686164 1.58027 1.16413 2.80877 0.942421
7 │ 1.13635 0.727793 1.64861 1.18397 2.57596 0.858025
8 │ 1.37071 0.71311 1.70744 1.3028 2.50044 0.856989
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
9994 │ 0.602241 0.79277 1.49744 1.05226 3.02618 0.999925
9995 │ 0.741272 0.590093 1.51232 1.05388 3.00305 1.00286
9996 │ 0.523739 0.465996 1.51596 1.14397 3.00436 0.983834
9997 │ 0.775749 0.442838 1.54501 1.03539 2.85821 0.987821
9998 │ 0.735861 0.527549 1.49875 1.17745 3.07271 1.00273
9999 │ 0.852995 0.510864 1.53072 1.08398 2.88211 0.995557
10000 │ 0.849238 0.67912 1.55677 1.02848 2.89831 0.93791
9985 rows omittedStan's internal timing (excluding data serialization and CSV parsing):
display_stan_timing(bayesian_result_stan)Chain 1 timing (from Stan CSV):
Elapsed Time: 5.566 seconds (Warm-up)
59.153 seconds (Sampling)
64.719 seconds (Total)Direct Turing.jl
We use a per-dimension noise model (matching Stan) with InverseGamma(2, 3) priors on each σ.
@model function fitlv(data, prob)
# Prior distributions.
σ ~ filldist(InverseGamma(2, 3), 2)
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 axes(data, 2)
data[:, i] ~ _mvn(predicted(t[i]), σ)
end
return nothing
end
model = fitlv(data, sprob)
# Warmup run to compile all code paths before timing
sample(model, Turing.NUTS(0.85), 10; progress = false)
elapsed_turing_direct = @elapsed chain = sample(model, Turing.NUTS(0.85), 10_000; progress = false)
chain╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter = 1001:11000
│
│ → chain = 1:1
│
│
│
│ Parameters (5) ── AbstractPPL.VarName
│
│ Vector{Float64} σ (2,)
│
│ Float64 α, β, γ, δ
│
│
│
│ Extras (14)
│
│ Int64 n_steps, tree_depth
│
│ Bool is_accept, numerical_error
│
│ Float64 acceptance_rate, log_density, hamiltonian_energy,
│
│ hamiltonian_energy_error, max_hamiltonian_energy_error, step_si
ze, │
│ nom_step_size, logprior, loglikelihood, logjoint
│
╰──────────────────────────────────────────────────────────────────────────
────╯display_ess_per_sec(chain, elapsed_turing_direct)Elapsed time: 74.7 seconds
╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter = 1001:11000
│
│ → chain = 1:1
│
│
│
│ Parameters (5) ── AbstractPPL.VarName
│
│ Vector{Float64} σ (2,)
│
│ Float64 α, β, γ, δ
│
│
│
│ Extras (14)
│
│ Int64 n_steps, tree_depth
│
│ Bool is_accept, numerical_error
│
│ Float64 acceptance_rate, log_density, hamiltonian_energy,
│
│ hamiltonian_energy_error, max_hamiltonian_energy_error, step_si
ze, │
│ nom_step_size, logprior, loglikelihood, logjoint
│
╰──────────────────────────────────────────────────────────────────────────
────╯Turing.jl backend
@btime bayesian_result_turing = turing_inference(
prob, Tsit5(), t, data, priors;
sample_args = (sampler = Turing.NUTS(0.85), num_samples = 10_000),
likelihood = (u, p, t, σ) -> _mvn(u, σ),
likelihood_dist_priors = [InverseGamma(2, 3), InverseGamma(2, 3)])82.049 s (379327534 allocations: 22.84 GiB)
╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter = 1001:11000
│
│ → chain = 1:1
│
│
│
│ Parameters (5) ── AbstractPPL.VarName
│
│ Float64 theta[1], theta[2], theta[3], theta[4]
│
│ Vector{Float64} σ (2,)
│
│
│
│ Extras (14)
│
│ Int64 n_steps, tree_depth
│
│ Bool is_accept, numerical_error
│
│ Float64 acceptance_rate, log_density, hamiltonian_energy,
│
│ hamiltonian_energy_error, max_hamiltonian_energy_error, step_si
ze, │
│ nom_step_size, logprior, loglikelihood, logjoint
│
╰──────────────────────────────────────────────────────────────────────────
────╯DynamicHMC.jl backend
@btime bayesian_result_dynamichmc = dynamichmc_inference(
prob, Tsit5(), t, data, priors; num_samples = 10_000)22.874 s (80963249 allocations: 6.84 GiB)
(posterior = [(parameters = [1.4997330454528732, 1.0766664603144227, 3.1079
58861417626, 0.9986481305659076], σ = [0.6772431509510991, 0.38429150420010
03]), (parameters = [1.48549621875879, 1.1273843291760801, 3.09902780236434
2, 0.9987713715813167], σ = [0.6985578194534445, 0.3534701428296056]), (par
ameters = [1.4855292542435536, 1.0907400295993441, 3.0952964558042644, 1.01
42106948835303], σ = [0.6851587339014207, 0.34493354141053445]), (parameter
s = [1.4511064960446687, 1.1009790310093763, 3.285141513522325, 1.057981633
166406], σ = [0.6535647385159334, 0.41094462143373534]), (parameters = [1.4
102856560578643, 1.034253578475981, 3.233181838259186, 1.1088629277389137],
σ = [0.510964363141452, 0.6837762599456201]), (parameters = [1.52313707136
98298, 1.0967035335197335, 2.9752305914807025, 0.9857610112907454], σ = [0.
6300821337635645, 0.2955560749448077]), (parameters = [1.4525868354586051,
0.9729318963646694, 3.1112178823561694, 1.0298672076132134], σ = [0.7028435
129048065, 0.3793852996423001]), (parameters = [1.379504180260236, 1.023201
7341336181, 3.4449149022718637, 1.1936887181304263], σ = [0.760489306710945
5, 0.4153899330652635]), (parameters = [1.5012140215454783, 0.9959563866169
98, 2.971263870062896, 0.9893138501115373], σ = [0.4569258519111059, 0.3906
1780848977806]), (parameters = [1.5138368595666722, 1.1232483747350421, 3.0
26497420047663, 1.0010552685525809], σ = [0.5305894520085112, 0.47806790763
49668]) … (parameters = [1.5366544347742324, 1.157668356245576, 2.9032121
305870877, 0.9664560476891016], σ = [0.3942920134880816, 0.4042499951766433
4]), (parameters = [1.4823443395749414, 1.0286154882463514, 3.1073680956462
084, 1.0468199835390049], σ = [0.6650548059671282, 0.3983984195434067]), (p
arameters = [1.4781182726746618, 1.0325519028909722, 3.071300004209172, 1.0
490997488095226], σ = [0.6659553552605298, 0.3944170854344297]), (parameter
s = [1.5419551022969389, 1.1496144667658472, 2.9228428055398754, 0.95932429
55211871], σ = [0.5546446098571367, 0.37477733461874374]), (parameters = [1
.5489704820105965, 1.0349246959254135, 2.873366306508517, 0.952628244726330
3], σ = [0.513536472285199, 0.30444920135591147]), (parameters = [1.5367234
320454815, 1.118582516151434, 2.893950215034362, 0.9856628832025374], σ = [
0.4527843727413465, 0.27492234536074756]), (parameters = [1.551966933464468
, 1.0585268654234967, 2.80547633429411, 0.9788294836662547], σ = [0.5927828
263700019, 0.39631876455974463]), (parameters = [1.6001678330931588, 1.0476
197522467143, 2.8377418093251996, 0.9160781595349549], σ = [0.8315911451771
557, 0.4456874315251175]), (parameters = [1.5110686454207152, 1.09547822900
73586, 2.9108950585427857, 0.9875478554563955], σ = [0.960106002662192, 0.3
2681716077125667]), (parameters = [1.6894909394516229, 1.099650025104306, 2
.642630031205689, 0.8295772043858739], σ = [0.8214104703072722, 0.641428156
7118212])], posterior_matrix = [0.4052871225715939 0.3957488704839568 … 0.4
12817112715795 0.524427264299954; 0.07386965690800612 0.11990019660073514 …
0.09119100674533044 0.09499197018492814; … ; -0.38972491109708524 -0.35873
732711328415 … -0.04071158117624936 -0.1967323306365223; -0.956353888863356
3 -1.0399562587978644 … -1.1183544058350188 -0.44405809385308587], tree_sta
tistics = DynamicHMC.TreeStatisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-21.3
06256988399912, 4, turning at positions -6:-13, 0.9356154761095345, 23, Dyn
amicHMC.Directions(0x34ff0eaa)), DynamicHMC.TreeStatisticsNUTS(-21.08476792
900201, 3, turning at positions -4:-11, 0.9825603392042023, 15, DynamicHMC.
Directions(0x616d8bc4)), DynamicHMC.TreeStatisticsNUTS(-21.131627817950896,
4, turning at positions -8:-23, 0.9933728441783806, 31, DynamicHMC.Directi
ons(0x44cfec08)), DynamicHMC.TreeStatisticsNUTS(-22.69680237348415, 4, turn
ing at positions 18:25, 0.9408811733129007, 31, DynamicHMC.Directions(0x85b
0d939)), DynamicHMC.TreeStatisticsNUTS(-23.84576680967108, 4, turning at po
sitions 0:15, 0.9904992325648224, 15, DynamicHMC.Directions(0x5bfb714f)), D
ynamicHMC.TreeStatisticsNUTS(-23.716560207673588, 5, turning at positions 2
7:58, 0.9366795825068057, 63, DynamicHMC.Directions(0x4ca93fba)), DynamicHM
C.TreeStatisticsNUTS(-24.320670438951034, 5, turning at positions -5:26, 0.
6013023559228335, 31, DynamicHMC.Directions(0xd983807a)), DynamicHMC.TreeSt
atisticsNUTS(-24.77413671453625, 5, turning at positions 25:56, 0.832680162
3682447, 63, DynamicHMC.Directions(0xc209cfb8)), DynamicHMC.TreeStatisticsN
UTS(-24.89270391556704, 5, turning at positions 42:57, 0.991046817569021, 6
3, DynamicHMC.Directions(0x4569e4f9)), DynamicHMC.TreeStatisticsNUTS(-21.89
6581766924843, 4, turning at positions -13:2, 0.9045283974296666, 15, Dynam
icHMC.Directions(0xfbfbe022)) … DynamicHMC.TreeStatisticsNUTS(-23.0983022
19631634, 5, turning at positions -20:11, 0.9968353583943, 31, DynamicHMC.D
irections(0x0b49abeb)), DynamicHMC.TreeStatisticsNUTS(-23.66077270880653, 5
, turning at positions -6:25, 0.40961885728455083, 31, DynamicHMC.Direction
s(0x5dd6e639)), DynamicHMC.TreeStatisticsNUTS(-19.973625965605954, 2, turni
ng at positions 2:5, 0.9937557978178075, 7, DynamicHMC.Directions(0x5661126
5)), DynamicHMC.TreeStatisticsNUTS(-19.40994411437341, 5, turning at positi
ons -23:-54, 0.9863414839986061, 63, DynamicHMC.Directions(0x62520689)), Dy
namicHMC.TreeStatisticsNUTS(-20.57961612552247, 4, turning at positions 0:1
5, 0.9120304796951139, 15, DynamicHMC.Directions(0x0f2eb6ef)), DynamicHMC.T
reeStatisticsNUTS(-22.480136527248902, 4, turning at positions -7:8, 0.9397
423935892305, 15, DynamicHMC.Directions(0x19d7c388)), DynamicHMC.TreeStatis
ticsNUTS(-22.240193611593234, 4, turning at positions 0:15, 0.9914080068613
943, 15, DynamicHMC.Directions(0xeac5b91f)), DynamicHMC.TreeStatisticsNUTS(
-22.25958807194615, 5, turning at positions -1:30, 0.8088302581551928, 31,
DynamicHMC.Directions(0xfdc4e61e)), DynamicHMC.TreeStatisticsNUTS(-24.74079
583616427, 4, turning at positions -16:-31, 0.9826229500009657, 31, Dynamic
HMC.Directions(0x36992100)), DynamicHMC.TreeStatisticsNUTS(-26.544760860063
676, 5, turning at positions 25:40, 0.9905121917672413, 47, DynamicHMC.Dire
ctions(0x0e209ab8))], logdensities = [-19.12371707211037, -20.0408620847806
8, -18.9823129414525, -20.20382254749095, -20.648028287341575, -19.06454976
8453507, -20.625595040579668, -21.017598450659413, -20.092010177932064, -19
.24071141901307 … -20.243370093379195, -19.014693464810357, -18.494906106
133126, -18.68014243359701, -19.574302984033622, -21.07764860671045, -19.44
2138303623658, -21.19374761799483, -21.7694111242984, -23.88074586615191],
κ = Gaussian kinetic energy (Diagonal), √diag(M⁻¹): [0.06080785472159253, 0
.12160565423754041, 0.08342888709172874, 0.108706919971907, 0.2968755249795
436, 0.3263190779494731], ϵ = 0.09531685367063045)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 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 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
LLVM: libLLVM-18.1.7 (ORCJIT, znver2)
GC: Built with stock GC
Threads: 128 default, 1 interactive, 128 GC (on 128 virtual cores)
Environment:
JULIA_NUM_THREADS = auto
Package Information:
Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/BayesianInference/Project.toml`
[6e4b80f9] BenchmarkTools v1.8.0
[ebbdde9d] DiffEqBayes v3.16.1
[459566f4] DiffEqCallbacks v4.19.3
[31c24e10] Distributions v0.25.131
[bbc10e6e] DynamicHMC v3.6.1
[1dea7af3] OrdinaryDiffEq v7.8.1
[65888b18] ParameterizedFunctions v5.27.0
[91a5bcdd] Plots v1.41.7
[731186ca] RecursiveArrayTools v4.5.1
[31c91b34] SciMLBenchmarks v0.2.1 [loaded: `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/src/SciMLBenchmarks.jl` (v0.2.1) expected `/home/crackauc/.julia/packages/SciMLBenchmarks/ceJyd/src/SciMLBenchmarks.jl` (v0.2.1)]
[c1514b29] StanSample v7.10.3
[90137ffa] StaticArrays v1.9.20
⌅ [fce5fe82] Turing v0.46.1
[37e2e46d] LinearAlgebra v1.12.0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`And the full manifest:
Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/BayesianInference/Manifest.toml`
[47edcb42] ADTypes v1.24.0
[14f7f29c] AMD v0.5.4
[621f4979] AbstractFFTs v1.5.0
[80f14c24] AbstractMCMC v5.16.0
[7a57a42e] AbstractPPL v0.15.5
[6e696c72] AbstractPlutoDingetjes v1.4.1
[1520ce14] AbstractTrees v0.4.5
[7d9f7c33] Accessors v0.1.45
[79e6a3ab] Adapt v4.7.0
[0bf59076] AdvancedHMC v0.8.7
[5b7e9947] AdvancedMH v0.8.10
⌅ [576499cb] AdvancedPS v0.7.2
[b5ca4192] AdvancedVI v0.7.0
[66dad0bd] AliasTables v1.1.3
[dce04be8] ArgCheck v2.5.0
[ec485272] ArnoldiMethod v0.4.0
[4fba245c] ArrayInterface v7.30.1
[4c555306] ArrayLayouts v1.12.2
[aae01518] BandedMatrices v1.12.0
[198e06fe] BangBang v0.4.9
[6e4b80f9] BenchmarkTools v1.8.0
[e2ed5e7c] Bijections v0.2.2
[76274a88] Bijectors v0.16.3
[b2a6c25c] BinaryHeaps v1.1.0
[caf10ac8] BipartiteGraphs v0.1.14
[8e7c35d0] BlockArrays v1.10.0
[70df07ce] BracketingNonlinearSolve v1.12.7
[336ed68f] CSV v0.10.17
[d360d2e6] ChainRulesCore v1.26.1
[0ca39b1e] Chairmarks v1.3.1
[9e997f8a] ChangesOfVariables v0.1.11
[944b1d66] CodecZlib v0.7.9
[35d6a980] ColorSchemes v3.31.0
[3da002f7] ColorTypes v0.12.1
[c3611d14] ColorVectorSpace v0.11.0
[5ae59095] Colors v0.13.1
⌅ [861a8166] Combinatorics v1.0.2
[38540f10] CommonSolve v0.2.14
[bbf7d656] CommonSubexpressions v0.3.1
[f70d9fcc] CommonWorldInvalidations v1.2.2
[34da2185] Compat v4.18.1
[5224ae11] CompatHelperLocal v0.1.29
[b152e2b5] CompositeTypes v0.1.4
[a33af91c] CompositionsBase v0.1.2
[2569d6c7] ConcreteStructs v0.2.8
[88cd18e8] ConsoleProgressMonitor v0.1.2
[187b0558] ConstructionBase v1.6.0
[d38c429a] Contour v0.6.3
[a8cc5b0e] Crayons v4.2.0
[9a962f9c] DataAPI v1.16.0
[a93c6f00] DataFrames v1.8.2
[864edb3b] DataStructures v0.19.6
[e2d170a0] DataValueInterfaces v1.0.0
[8bb1440f] DelimitedFiles v1.9.1
[b429d917] DensityInterface v0.4.0
[2b5f629d] DiffEqBase v7.21.0
[ebbdde9d] DiffEqBayes v3.16.1
[459566f4] DiffEqCallbacks v4.19.3
[163ba53b] DiffResults v1.1.0
[b552c78f] DiffRules v1.16.0
[a0c0ee7d] DifferentiationInterface v0.7.21
[0703355e] DimensionalData v0.30.2
[b4f34e82] Distances v0.10.12
[31c24e10] Distributions v0.25.131
[ffbed154] DocStringExtensions v0.9.5
[5b8099bc] DomainSets v0.8.1
[bbc10e6e] DynamicHMC v3.6.1
[366bfd00] DynamicPPL v0.42.12
[7c1d4256] DynamicPolynomials v0.6.8
[cad2338a] EllipticalSliceSampling v2.0.0
[4e289a0a] EnumX v1.0.7
[f151be2c] EnzymeCore v0.8.21
[e2ba6199] ExprTools v0.1.11
[55351af7] ExproniconLite v0.10.14
[411431e0] Extents v0.1.6
[c87230d0] FFMPEG v0.4.5
[7034ab61] FastBroadcast v1.4.0
[9aa1b823] FastClosures v0.3.2
[a4df4552] FastPower v1.5.0
[48062228] FilePathsBase v0.9.24
[1a297f60] FillArrays v1.17.0
[64ca27bc] FindFirstFunctions v3.2.1
[6a86dc24] FiniteDiff v2.33.0
⌅ [53c48c17] FixedPointNumbers v0.8.6
[4a37a8b9] FlexiChains v0.6.38
[1fa38f19] Format v1.3.7
⌃ [f6369f11] ForwardDiff v1.4.5
[a85aefff] FunctionMaps v0.1.2
[069b7b12] FunctionWrappers v1.1.3
[77dc65aa] FunctionWrappersWrappers v1.13.0
[d9f16b24] Functors v0.5.3
[46192b85] GPUArraysCore v0.2.0
[28b8d3ca] GR v0.73.27
⌃ [a0844989] Gamma v1.1.0
[86223c79] Graphs v1.15.0
⌅ [eafb193a] Highlights v0.5.3
[34004b35] HypergeometricFunctions v0.3.30
[3263718b] ImplicitDiscreteSolve v2.3.0
[d25df0c9] Inflate v0.1.5
[22cec73e] InitialValues v0.3.1
⌅ [842dd82b] InlineStrings v1.4.6
[18e54dd8] IntegerMathUtils v0.1.4
[85a1e053] Interfaces v0.3.2
[8197267c] IntervalSets v0.7.14
[3587e190] InverseFunctions v0.1.17
[41ab1584] InvertedIndices v1.3.1
[92d709cd] IrrationalConstants v0.2.6
[82899510] IteratorInterfaceExtensions v1.0.0
[1019f520] JLFzf v0.1.11
[692b3bcd] JLLWrappers v1.8.0
⌅ [682c06a0] JSON v0.21.4
[ae98c720] Jieko v0.2.1
[ccbc3e58] JumpProcesses v9.32.3
[ba0b0d4f] Krylov v0.10.9
[2faa5264] LHLFactorization v2.2.2
[b964fa9f] LaTeXStrings v1.4.1
[23fbe1c1] Latexify v0.16.12
[1fad7336] LazyStack v0.1.3
[1d6d02ad] LeftChildRightSiblingTrees v0.3.0
[6f1fad26] Libtask v0.9.19
[87fe0de2] LineSearch v0.1.17
[d3d80556] LineSearches v7.8.1
⌃ [7ed4a6bd] LinearSolve v5.17.1
[6fdf6af0] LogDensityProblems v2.2.0
[996a588d] LogDensityProblemsAD v1.13.1
⌅ [2ab3a3ac] LogExpFunctions v0.3.29
[e6f89c97] LoggingExtras v1.2.0
[be115224] MCMCDiagnosticTools v0.3.19
[e80e1ace] MLJModelInterface v1.12.1
[1914dd2f] MacroTools v0.5.16
[dbb5928d] MappedArrays v0.4.3
[bb5d69b7] MaybeInplace v0.1.8
[442fdcdd] Measures v0.3.3
[e1d29d7a] Missings v1.2.0
[dbe65cb8] MistyClosures v2.1.0
[961ee093] ModelingToolkit v11.42.1
⌃ [7771a370] ModelingToolkitBase v1.69.1
[6bb917b9] ModelingToolkitTearing v1.20.6
⌅ [2e0e35c7] Moshi v0.3.9
[46d2c3a1] MuladdMacro v0.2.7
[102ac46a] MultivariatePolynomials v0.5.19
[ffc61752] Mustache v1.0.21
[d8a4904e] MutableArithmetics v1.8.0
[d41bc354] NLSolversBase v8.0.1
[77ba4419] NaNMath v1.1.4
[d9ec5142] NamedTupleTools v0.14.3
[8913a72c] NonlinearSolve v4.30.0
[be0214bd] NonlinearSolveBase v2.49.5
[5959db7a] NonlinearSolveFirstOrder v2.6.1
[9a2c21bd] NonlinearSolveQuasiNewton v1.15.3
[26075421] NonlinearSolveSpectralMethods v1.8.3
[6fe1bfb0] OffsetArrays v1.17.0
[429524aa] Optim v2.3.1
[3bd65402] Optimisers v0.4.9
[7f7a1694] Optimization v5.9.0
[bca83a33] OptimizationBase v5.5.3
[36348300] OptimizationOptimJL v0.4.21
⌅ [bac558e1] OrderedCollections v1.8.2 [loaded: v2.0.1]
[1dea7af3] OrdinaryDiffEq v7.8.1
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.7
⌃ [bbf590c4] OrdinaryDiffEqCore v4.17.0
⌃ [50262376] OrdinaryDiffEqDefault v2.6.0
[4302a76b] OrdinaryDiffEqDifferentiation v3.11.5
[127b3ac7] OrdinaryDiffEqNonlinearSolve v2.9.6
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.7.1
[b4bd8bb3] OrdinaryDiffEqRosenbrockTableaus v2.4.2
[2d112036] OrdinaryDiffEqSDIRK v2.9.2
[b1df2697] OrdinaryDiffEqTsit5 v2.1.4
[79d7bb75] OrdinaryDiffEqVerner v2.4.1
[90014a1f] PDMats v0.11.41
[65888b18] ParameterizedFunctions v5.27.0
⌅ [d96e819e] Parameters v0.12.3
⌅ [69de0a69] Parsers v2.8.8
[569bd051] PartitionedDistributions v0.1.0
[ccf2f8ad] PlotThemes v3.3.0
[995b91a9] PlotUtils v1.4.4
[91a5bcdd] Plots v1.41.7
[e409e4f3] PoissonRandom v0.4.13
[2dfb63ee] PooledArrays v1.4.3
[85a6dd25] PositiveFactorizations v0.2.4
[d236fae5] PreallocationTools v1.7.1
[aea7be01] PrecompileTools v1.3.4
[21216c6a] Preferences v1.5.2
[08abe8d2] PrettyTables v3.4.8
[27ebfcd6] Primes v0.5.7
[33c8b6b6] ProgressLogging v0.1.6
[92933f4c] ProgressMeter v1.11.0
[43287f4e] PtrArrays v1.4.0
[0c0d3e7f] PureKLU v1.4.2
[1fd47b50] QuadGK v2.11.3
[74087812] Random123 v1.7.1
[e6cf234a] RandomNumbers v1.6.0
[988b38a3] ReadOnlyArrays v0.2.0
[795d4caa] ReadOnlyDicts v1.0.1
[3cdcf5f2] RecipesBase v1.3.4
[01d81517] RecipesPipeline v0.6.12
[731186ca] RecursiveArrayTools v4.5.1
[189a3867] Reexport v1.2.2
[05181044] RelocatableFolders v1.0.1
[ae029012] Requires v1.3.1
[9fe22ead] RespecializeParams v1.3.0
[79098fc4] Rmath v0.9.0
[f2b01f46] Roots v3.0.8
[7e49a35a] RuntimeGeneratedFunctions v0.5.26
[9dfe8606] SCCNonlinearSolve v1.15.3
[26aad666] SSMProblems v0.6.1
⌃ [0bca4576] SciMLBase v3.53.2
[31c91b34] SciMLBenchmarks v0.2.1 [loaded: `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/src/SciMLBenchmarks.jl` (v0.2.1) expected `/home/crackauc/.julia/packages/SciMLBenchmarks/ceJyd/src/SciMLBenchmarks.jl` (v0.2.1)]
[19f34311] SciMLJacobianOperators v0.1.19
[a6db7da4] SciMLLogging v2.1.0
[c0aeaf25] SciMLOperators v1.30.0
[431bcebd] SciMLPublic v1.3.0
[53ae85a6] SciMLStructures v1.10.5
[30f210dd] ScientificTypesBase v3.1.0
[6c6a2e73] Scratch v1.3.0
[91c51154] SentinelArrays v1.4.10
[efcf1570] Setfield v1.1.2
[992d4aef] Showoff v1.1.1
[727e6d20] SimpleNonlinearSolve v2.14.3
[699a6c99] SimpleTraits v0.9.6
[a2af1166] SortingAlgorithms v1.2.3
[a57abbd0] SparseColumnPivotedQR v2.1.8
[9f842d2f] SparseConnectivityTracer v1.2.3
[0a514795] SparseMatrixColorings v0.4.28
[276daf66] SpecialFunctions v2.9.0
[860ef19b] StableRNGs v1.0.4
[d0ee94f6] StanBase v4.12.4
[c1514b29] StanSample v7.10.3
[0c0c59c1] StarAlgebras v0.3.0
[64909d44] StateSelection v1.11.1
[90137ffa] StaticArrays v1.9.20
[1e83bf80] StaticArraysCore v1.4.4
[64bff920] StatisticalTraits v3.5.0
[10745b16] Statistics v1.11.5
[82ae8749] StatsAPI v1.8.0
[2913bbd2] StatsBase v0.34.13
⌅ [4c63d2b9] StatsFuns v1.5.3
[5e0ebb24] Strided v2.6.4
[4db3bf67] StridedViews v0.5.2
[69024149] StringEncodings v0.3.7
⌅ [892a3eda] StringManipulation v0.5.0
[2efcf032] SymbolicIndexingInterface v0.3.55
[19f23fe9] SymbolicLimits v1.2.1
⌃ [d1185830] SymbolicUtils v4.46.4
[0c5d862f] Symbolics v7.39.2
[ab02a1b2] TableOperations v1.2.0
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.14.0
[ed4db957] TaskLocalValues v0.1.3
[02d47bb6] TensorCast v0.4.9
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v2.0.0
[5d786b92] TerminalLoggers v0.1.8
[a759f4b9] TimerOutputs v1.2.1
[3bb67fe8] TranscodingStreams v0.11.3
[84d833dd] TransformVariables v0.8.26
[f9bc47f6] TransformedLogDensities v1.1.1
[24ddb15e] TransmuteDims v0.1.17
[781d530d] TruncatedStacktraces v1.4.0
[9d95972d] TupleTools v1.6.0
⌅ [fce5fe82] Turing v0.46.1
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
[41fe7b60] Unzip v0.2.0
[d30d5f5c] WeakCacheSets v0.1.0
[ea10d353] WeakRefStrings v1.4.3
[44d3d7a6] Weave v0.10.12
[76eceee3] WorkerUtilities v1.6.1
[ddb6d928] YAML v0.4.16
[6e34b625] Bzip2_jll v1.0.9+0
[83423d85] Cairo_jll v1.18.7+0
[ee1fde0b] Dbus_jll v1.16.2+0
[2702e6a9] EpollShim_jll v0.0.20230411+1
[2e619515] Expat_jll v2.8.4+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
[a3f928ae] Fontconfig_jll v2.17.1+0
[d7e528f0] FreeType2_jll v2.14.3+1
[559328eb] FriBidi_jll v1.0.17+0
[0656b61e] GLFW_jll v3.5.1+0
[d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
[61579ee1] Ghostscript_jll v9.55.1+0
[7746bdde] Glib_jll v2.88.3+0
[3b182d85] Graphite2_jll v1.3.16+0
[2e76f6c2] HarfBuzz_jll v100.14004.0+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.2.0+1
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.2.0+0
[1d63c593] LLVMOpenMP_jll v23.1.1+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[856f044c] MKL_jll v2025.2.0+0
[e7412a2a] Ogg_jll v1.3.6+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.58.2+0
[30392449] Pixman_jll v0.46.4+0
[c0090381] Qt6Base_jll v6.10.2+2
[629bc702] Qt6Declarative_jll v6.10.2+2
[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[f50d1b31] Rmath_jll v0.5.2+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.4+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.13+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[a51aa0fd] Xorg_libXi_jll v1.8.4+0
[d1454406] Xorg_libXinerama_jll v1.1.7+0
[ec84b674] Xorg_libXrandr_jll v1.5.6+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.2.0+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
[35661453] Xorg_xkbcomp_jll v1.4.7+0
[33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[3161d3a3] Zstd_jll v1.5.7+1
[35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
[a4ae2306] libaom_jll v3.14.1+0
[0ac62f75] libass_jll v0.17.5+0
[1183f4f0] libdecor_jll v0.2.2+0
[8e53e030] libdrm_jll v2.4.134+0
[2db6ffa8] libevdev_jll v1.13.4+0
[f638f0a6] libfdk_aac_jll v2.0.4+0
[36db933b] libinput_jll v1.28.1+0
[b53b4c65] libpng_jll v1.6.58+0
[9a156e7d] libva_jll v2.23.0+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.13.0+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.12.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.12.1
[de0858da] Printf v1.11.0
[9abbd945] Profile v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.12.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.3.1+2
[deac9b47] LibCURL_jll v8.15.0+0
[e37daf67] LibGit2_jll v1.9.0+0
[29816b5a] LibSSH2_jll v1.11.3+1
[14a3606d] MozillaCACerts_jll v2025.11.4
[4536629a] OpenBLAS_jll v0.3.29+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.6+0
[efcefdf7] PCRE2_jll v10.44.0+1
[bea87d4a] SuiteSparse_jll v7.8.3+2
[83775a58] Zlib_jll v1.3.1+2
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.64.0+1
[3f19e933] p7zip_jll v17.7.0+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`