Fitzhugh-Nagumo Bayesian Parameter Estimation Benchmarks

using DiffEqBayes, BenchmarkTools

using OrdinaryDiffEq, RecursiveArrayTools, Distributions, ParameterizedFunctions,
      StanSample, DynamicHMC
using Plots, StaticArrays, Turing, LinearAlgebra

gr(fmt = :png)

Plots.GRBackend()

Defining the problem.

The FitzHugh-Nagumo model is a simplified version of Hodgkin-Huxley model and is used to describe an excitable system (e.g. neuron).

fitz = @ode_def FitzhughNagumo begin
    dv = v - 0.33*v^3 - w + l
    dw = τinv*(v + a - b*w)
end a b τinv l

Main.var"##WeaveSandBox#225".FitzhughNagumo{Main.var"##WeaveSandBox#225".va
r"###ParameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".var"###
ParameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###Parame
terizedJacobianFunction#229", Nothing, Nothing, ModelingToolkit.System}(Mai
n.var"##WeaveSandBox#225".var"##ParameterizedDiffEqFunction#227", LinearAlg
ebra.UniformScaling{Bool}(true), nothing, Main.var"##WeaveSandBox#225".var"
##ParameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"##Param
eterizedJacobianFunction#229", nothing, nothing, nothing, nothing, nothing,
 nothing, nothing, [:v, :w], :t, nothing, Model ##Parameterized#226:
Equations (2):
  2 standard: see equations(##Parameterized#226)
Unknowns (2): see unknowns(##Parameterized#226)
  v(t)
  w(t)
Parameters (4): see parameters(##Parameterized#226)
  a
  b
  τinv
  l, nothing, nothing)

prob_ode_fitzhughnagumo = ODEProblem(fitz, [1.0, 1.0], (0.0, 10.0), [0.7, 0.8, 1/12.5, 0.5])
sol = solve(prob_ode_fitzhughnagumo, Tsit5())

retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 13-element Vector{Float64}:
  0.0
  0.1502916178003539
  0.6611860158920579
  1.4391493908273403
  2.589451591547814
  3.7602377960785525
  5.101014337183989
  6.709997524274457
  7.604553475030161
  8.336547696252527
  9.031279335406245
  9.556400185811816
 10.0
u: 13-element Vector{Vector{Float64}}:
 [1.0, 1.0]
 [1.0247192356111163, 1.0109189409610948]
 [1.0944137341238236, 1.049239334584406]
 [1.1525604472298034, 1.1092965960073389]
 [1.1446577625483763, 1.1952738138449215]
 [1.0557695077719018, 1.2718985818139574]
 [0.8659598744812588, 1.3388184800875969]
 [0.36758540211725504, 1.3735376018319743]
 [-0.3594427955481789, 1.3493319650351683]
 [-1.3772889489189244, 1.278171118435909]
 [-1.9056998397130338, 1.1680023987534764]
 [-1.9707492736430954, 1.077729156517589]
 [-1.9650453438870357, 1.0031251492628297]

sprob_ode_fitzhughnagumo = ODEProblem{false, SciMLBase.FullSpecialize}(
    fitz, SA[1.0, 1.0], (0.0, 10.0), SA[0.7, 0.8, 1 / 12.5, 0.5])
sol = solve(sprob_ode_fitzhughnagumo, Tsit5())

retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 13-element Vector{Float64}:
  0.0
  0.1502916178003539
  0.6611860158920579
  1.4391493908273403
  2.589451591547814
  3.7602377960785525
  5.101014337183989
  6.709997524274457
  7.604553475030161
  8.336547696252527
  9.031279335406245
  9.556400185811816
 10.0
u: 13-element Vector{StaticArraysCore.SVector{2, Float64}}:
 [1.0, 1.0]
 [1.0247192356111163, 1.0109189409610948]
 [1.0944137341238236, 1.049239334584406]
 [1.1525604472298034, 1.1092965960073389]
 [1.1446577625483763, 1.1952738138449215]
 [1.0557695077719018, 1.2718985818139574]
 [0.8659598744812588, 1.3388184800875969]
 [0.36758540211725504, 1.3735376018319743]
 [-0.3594427955481789, 1.3493319650351683]
 [-1.3772889489189244, 1.278171118435909]
 [-1.9056998397130338, 1.1680023987534764]
 [-1.9707492736430954, 1.077729156517589]
 [-1.9650453438870357, 1.0031251492628297]

Data is generated by adding noise to the solution obtained above.

t = collect(range(1, stop = 10, length = 10))
sig = 0.20
data = convert(Array, VectorOfArray([(sol(t[i]) + sig*randn(2)) for i in 1:length(t)]))

2×10 Matrix{Float64}:
 1.0724    1.0841   1.31174  0.863462  …  -0.741135  -1.92516  -2.14103
 0.989612  1.35783  1.45561  1.48824       1.16961    1.29289   1.06208

Plot of the data and the solution.

scatter(t, data[1, :])
scatter!(t, data[2, :])
plot!(sol)

Priors for the parameters which will be passed for the Bayesian Inference

priors = [truncated(Normal(1.0, 0.5), 0, 1.5), truncated(Normal(1.0, 0.5), 0, 1.5),
    truncated(Normal(0.0, 0.5), 0.0, 0.5), truncated(Normal(0.5, 0.5), 0, 1)]

4-element Vector{Distributions.Truncated{Distributions.Normal{Float64}, Dis
tributions.Continuous, Float64, Float64, Float64}}:
 Truncated(Distributions.Normal{Float64}(μ=1.0, σ=0.5); lower=0.0, upper=1.
5)
 Truncated(Distributions.Normal{Float64}(μ=1.0, σ=0.5); lower=0.0, upper=1.
5)
 Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0, upper=0.
5)
 Truncated(Distributions.Normal{Float64}(μ=0.5, σ=0.5); lower=0.0, upper=1.
0)

Benchmarks

Stan.jl backend

@time bayesian_result_stan = stan_inference(
    prob_ode_fitzhughnagumo, t, data, priors; delta = 0.65, num_samples = 10_000,
    print_summary = false, vars = (DiffEqBayes.StanODEData(), InverseGamma(2, 3)))

Error: MethodError: no method matching stan_inference(::SciMLBase.ODEProble
m{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, Main.var
"##WeaveSandBox#225".FitzhughNagumo{Main.var"##WeaveSandBox#225".var"###Par
ameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".var"###Paramete
rizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###ParameterizedJ
acobianFunction#229", Nothing, Nothing, ModelingToolkit.System}, Base.Pairs
{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}, :
:Vector{Float64}, ::Matrix{Float64}, ::Vector{Distributions.Truncated{Distr
ibutions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float
64}}, ::Nothing; delta::Float64, num_samples::Int64, print_summary::Bool, v
ars::Tuple{DiffEqBayes.StanODEData, Distributions.InverseGamma{Float64}})

Closest candidates are:
  stan_inference(::SciMLBase.AbstractSciMLProblem, ::Any, ::Any, ::Any, ::A
ny; stanmodel, likelihood, vars, sample_u0, solve_kwargs, diffeq_string, sa
mple_kwargs, output_format, print_summary, tmpdir) got unsupported keyword 
arguments "delta", "num_samples"
   @ DiffEqBayes /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae
0a-f4d2d937f953/packages/DiffEqBayes/gFKkQ/src/stan_inference.jl:57
  stan_inference(::SciMLBase.AbstractSciMLProblem, ::Any, ::Any, ::Any; ...
)
   @ DiffEqBayes /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae
0a-f4d2d937f953/packages/DiffEqBayes/gFKkQ/src/stan_inference.jl:57

Direct Turing.jl

@model function fitlv(data, prob)

    # Prior distributions.
    σ ~ InverseGamma(2, 3)
    a ~ truncated(Normal(1.0, 0.5), 0, 1.5)
    b ~ truncated(Normal(1.0, 0.5), 0, 1.5)
    τinv ~ truncated(Normal(0.0, 0.5), 0.0, 0.5)
    l ~ truncated(Normal(0.5, 0.5), 0, 1)

    # Simulate Lotka-Volterra model. 
    p = SA[a, b, τinv, l]
    _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_ode_fitzhughnagumo)

@time chain = sample(model, Turing.NUTS(0.65), 10000; progress = false)

75.807597 seconds (292.42 M allocations: 22.407 GiB, 4.93% gc time, 33.08%
 compilation time: <1% of which was recompilation)
Chains MCMC chain (10000×19×1 Array{Float64, 3}):

Iterations        = 1001:1:11000
Number of chains  = 1
Samples per chain = 10000
Wall duration     = 63.29 seconds
Compute duration  = 63.29 seconds
parameters        = σ, a, b, τinv, l
internals         = n_steps, is_accept, acceptance_rate, log_density, hamil
tonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree
_depth, numerical_error, step_size, nom_step_size, logprior, loglikelihood,
 logjoint

Summary Statistics
  parameters      mean       std      mcse    ess_bulk    ess_tail      rha
t   ⋯
      Symbol   Float64   Float64   Float64     Float64     Float64   Float6
4   ⋯

           σ    0.2558    0.0518    0.0009   3089.4986   3260.6092    1.000
7   ⋯
           a    0.9560    0.3164    0.0049   4245.0104   4361.2728    1.000
3   ⋯
           b    0.8996    0.3121    0.0051   3724.4158   4219.0973    1.001
5   ⋯
        τinv    0.0688    0.0311    0.0008   1773.1781   1595.3976    1.002
0   ⋯
           l    0.4815    0.0667    0.0015   2117.4782   2239.2364    1.002
2   ⋯
                                                                1 column om
itted

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

           σ    0.1789    0.2196    0.2481    0.2824    0.3801
           a    0.2751    0.7446    0.9884    1.2017    1.4534
           b    0.2387    0.6913    0.9243    1.1366    1.4190
        τinv    0.0217    0.0458    0.0645    0.0865    0.1421
           l    0.3666    0.4358    0.4755    0.5207    0.6342

Turing.jl backend

@time bayesian_result_turing = turing_inference(
    prob_ode_fitzhughnagumo, Tsit5(), t, data, priors; num_samples = 10_000)

Error: MethodError: no method matching turing_inference(::SciMLBase.ODEProb
lem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, Main.v
ar"##WeaveSandBox#225".FitzhughNagumo{Main.var"##WeaveSandBox#225".var"###P
arameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".var"###Parame
terizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###Parameterize
dJacobianFunction#229", Nothing, Nothing, ModelingToolkit.System}, Base.Pai
rs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem},
 ::OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), t
ypeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, ::Vector{Float64
}, ::Matrix{Float64}, ::Vector{Distributions.Truncated{Distributions.Normal
{Float64}, Distributions.Continuous, Float64, Float64, Float64}}; num_sampl
es::Int64)

Closest candidates are:
  turing_inference(::SciMLBase.AbstractSciMLProblem, ::Any, ::Any, ::Any, :
:Any; likelihood_dist_priors, likelihood, syms, sample_u0, progress, solve_
kwargs, sample_args, sample_kwargs) got unsupported keyword argument "num_s
amples"
   @ DiffEqBayes /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae
0a-f4d2d937f953/packages/DiffEqBayes/gFKkQ/src/turing_inference.jl:1

Conclusion

FitzHugh-Ngumo is a standard problem for parameter estimation studies. In the FitzHugh-Nagumo model the parameters to be estimated were [0.7,0.8,0.08,0.5]. dynamichmc_inference has issues with the model and hence was excluded from this benchmark.

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","DiffEqBayesFitzHughNagumo.jmd")

Computer Information:

Julia Version 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 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-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/BayesianInference/Project.toml`
  [6e4b80f9] BenchmarkTools v1.6.3
  [ebbdde9d] DiffEqBayes v3.11.0
  [459566f4] DiffEqCallbacks v4.12.0
  [31c24e10] Distributions v0.25.123
  [bbc10e6e] DynamicHMC v3.6.0
  [1dea7af3] OrdinaryDiffEq v6.108.0
⌃ [65888b18] ParameterizedFunctions v5.19.0
  [91a5bcdd] Plots v1.41.6
  [731186ca] RecursiveArrayTools v3.48.0
  [31c91b34] SciMLBenchmarks v0.1.3
  [c1514b29] StanSample v7.10.2
  [90137ffa] StaticArrays v1.9.17
  [fce5fe82] Turing v0.42.8
  [37e2e46d] LinearAlgebra
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/BayesianInference/Manifest.toml`
  [47edcb42] ADTypes v1.21.0
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  [d125e4d3] ManualMemory v0.1.8
  [dbb5928d] MappedArrays v0.4.3
  [bb5d69b7] MaybeInplace v0.1.4
  [739be429] MbedTLS v1.1.10
  [442fdcdd] Measures v0.3.3
  [128add7d] MicroCollections v0.2.0
  [e1d29d7a] Missings v1.2.0
  [dbe65cb8] MistyClosures v2.1.0
⌅ [961ee093] ModelingToolkit v10.32.1
  [2e0e35c7] Moshi v0.3.7
  [46d2c3a1] MuladdMacro v0.2.4
  [102ac46a] MultivariatePolynomials v0.5.13
  [ffc61752] Mustache v1.0.21
  [d8a4904e] MutableArithmetics v1.6.7
⌅ [d41bc354] NLSolversBase v7.10.0
  [77ba4419] NaNMath v1.1.3
  [86f7a689] NamedArrays v0.10.5
  [d9ec5142] NamedTupleTools v0.14.3
  [c020b1a1] NaturalSort v1.0.0
⌃ [8913a72c] NonlinearSolve v4.15.0
⌃ [be0214bd] NonlinearSolveBase v2.11.2
⌅ [5959db7a] NonlinearSolveFirstOrder v1.11.1
  [9a2c21bd] NonlinearSolveQuasiNewton v1.12.0
  [26075421] NonlinearSolveSpectralMethods v1.6.0
  [6fe1bfb0] OffsetArrays v1.17.0
  [4d8831e6] OpenSSL v1.6.1
⌅ [429524aa] Optim v1.13.3
  [3bd65402] Optimisers v0.4.7
⌃ [7f7a1694] Optimization v5.4.0
⌅ [bca83a33] OptimizationBase v4.2.0
⌃ [36348300] OptimizationOptimJL v0.4.8
  [bac558e1] OrderedCollections v1.8.1
  [1dea7af3] OrdinaryDiffEq v6.108.0
  [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.9.0
  [6ad6398a] OrdinaryDiffEqBDF v1.21.0
  [bbf590c4] OrdinaryDiffEqCore v3.9.0
  [50262376] OrdinaryDiffEqDefault v1.12.0
  [4302a76b] OrdinaryDiffEqDifferentiation v2.1.0
  [9286f039] OrdinaryDiffEqExplicitRK v1.9.0
  [e0540318] OrdinaryDiffEqExponentialRK v1.13.0
  [becaefa8] OrdinaryDiffEqExtrapolation v1.15.0
  [5960d6e9] OrdinaryDiffEqFIRK v1.23.0
  [101fe9f7] OrdinaryDiffEqFeagin v1.8.0
  [d3585ca7] OrdinaryDiffEqFunctionMap v1.9.0
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v1.9.0
  [9f002381] OrdinaryDiffEqIMEXMultistep v1.12.0
  [521117fe] OrdinaryDiffEqLinear v1.10.0
  [1344f307] OrdinaryDiffEqLowOrderRK v1.10.0
  [b0944070] OrdinaryDiffEqLowStorageRK v1.12.0
  [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.23.0
  [c9986a66] OrdinaryDiffEqNordsieck v1.9.0
  [5dd0a6cf] OrdinaryDiffEqPDIRK v1.11.0
  [5b33eab2] OrdinaryDiffEqPRK v1.8.0
  [04162be5] OrdinaryDiffEqQPRK v1.8.0
  [af6ede74] OrdinaryDiffEqRKN v1.10.0
  [43230ef6] OrdinaryDiffEqRosenbrock v1.25.0
  [2d112036] OrdinaryDiffEqSDIRK v1.12.0
  [669c94d9] OrdinaryDiffEqSSPRK v1.11.0
  [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.11.0
  [358294b1] OrdinaryDiffEqStabilizedRK v1.8.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v1.11.0
  [b1df2697] OrdinaryDiffEqTsit5 v1.9.0
  [79d7bb75] OrdinaryDiffEqVerner v1.11.0
  [90014a1f] PDMats v0.11.37
  [65ce6f38] PackageExtensionCompat v1.0.2
⌃ [65888b18] ParameterizedFunctions v5.19.0
  [d96e819e] Parameters v0.12.3
  [69de0a69] Parsers v2.8.3
  [ccf2f8ad] PlotThemes v3.3.0
  [995b91a9] PlotUtils v1.4.4
  [91a5bcdd] Plots v1.41.6
  [e409e4f3] PoissonRandom v0.4.7
  [f517fe37] Polyester v0.7.19
  [1d0040c9] PolyesterWeave v0.2.2
  [2dfb63ee] PooledArrays v1.4.3
  [85a6dd25] PositiveFactorizations v0.2.4
⌃ [d236fae5] PreallocationTools v0.4.34
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.5.1
⌅ [08abe8d2] PrettyTables v2.4.0
  [27ebfcd6] Primes v0.5.7
  [33c8b6b6] ProgressLogging v0.1.6
  [92933f4c] ProgressMeter v1.11.0
  [43287f4e] PtrArrays v1.4.0
  [1fd47b50] QuadGK v2.11.2
  [74087812] Random123 v1.7.1
  [e6cf234a] RandomNumbers v1.6.0
  [b3c3ace0] RangeArrays v0.3.2
  [c84ed2f1] Ratios v0.4.5
  [c1ae055f] RealDot v0.1.0
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [731186ca] RecursiveArrayTools v3.48.0
  [189a3867] Reexport v1.2.2
  [05181044] RelocatableFolders v1.0.1
  [ae029012] Requires v1.3.1
  [ae5879a3] ResettableStacks v1.2.0
  [79098fc4] Rmath v0.9.0
  [f2b01f46] Roots v2.2.12
  [7e49a35a] RuntimeGeneratedFunctions v0.5.17
  [9dfe8606] SCCNonlinearSolve v1.11.0
  [94e857df] SIMDTypes v0.1.0
  [26aad666] SSMProblems v0.6.1
  [0bca4576] SciMLBase v2.144.0
  [31c91b34] SciMLBenchmarks v0.1.3
  [19f34311] SciMLJacobianOperators v0.1.12
  [a6db7da4] SciMLLogging v1.9.1
  [c0aeaf25] SciMLOperators v1.15.1
  [431bcebd] SciMLPublic v1.0.1
  [53ae85a6] SciMLStructures v1.10.0
  [30f210dd] ScientificTypesBase v3.1.0
  [6c6a2e73] Scratch v1.3.0
  [91c51154] SentinelArrays v1.4.9
  [efcf1570] Setfield v1.1.2
  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.2.0
  [727e6d20] SimpleNonlinearSolve v2.11.0
  [699a6c99] SimpleTraits v0.9.5
  [a2af1166] SortingAlgorithms v1.2.2
  [9f842d2f] SparseConnectivityTracer v1.2.1
  [dc90abb0] SparseInverseSubset v0.1.2
  [0a514795] SparseMatrixColorings v0.4.23
  [276daf66] SpecialFunctions v2.7.1
  [171d559e] SplittablesBase v0.1.15
  [860ef19b] StableRNGs v1.0.4
  [d0ee94f6] StanBase v4.12.4
  [c1514b29] StanSample v7.10.2
  [aedffcd0] Static v1.3.1
  [0d7ed370] StaticArrayInterface v1.9.0
  [90137ffa] StaticArrays v1.9.17
  [1e83bf80] StaticArraysCore v1.4.4
  [64bff920] StatisticalTraits v3.5.0
  [82ae8749] StatsAPI v1.8.0
  [2913bbd2] StatsBase v0.34.10
  [4c63d2b9] StatsFuns v1.5.2
  [7792a7ef] StrideArraysCore v0.5.8
  [5e0ebb24] Strided v2.3.2
  [4db3bf67] StridedViews v0.4.3
  [69024149] StringEncodings v0.3.7
  [892a3eda] StringManipulation v0.4.2
  [09ab397b] StructArrays v0.7.2
⌃ [2efcf032] SymbolicIndexingInterface v0.3.44
⌅ [19f23fe9] SymbolicLimits v0.2.3
⌅ [d1185830] SymbolicUtils v3.32.0
⌅ [0c5d862f] Symbolics v6.58.0
  [ab02a1b2] TableOperations v1.2.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.1
  [ed4db957] TaskLocalValues v0.1.3
  [02d47bb6] TensorCast v0.4.9
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [5d786b92] TerminalLoggers v0.1.7
  [1c621080] TestItems v1.0.0
  [8290d209] ThreadingUtilities v0.5.5
  [a759f4b9] TimerOutputs v0.5.29
  [3bb67fe8] TranscodingStreams v0.11.3
  [28d57a85] Transducers v0.4.85
  [84d833dd] TransformVariables v0.8.19
  [f9bc47f6] TransformedLogDensities v1.1.1
  [24ddb15e] TransmuteDims v0.1.17
  [410a4b4d] Tricks v0.1.13
  [781d530d] TruncatedStacktraces v1.4.0
  [9d95972d] TupleTools v1.6.0
  [fce5fe82] Turing v0.42.8
  [5c2747f8] URIs v1.6.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.28.0
  [a7c27f48] Unityper v0.1.6
  [41fe7b60] Unzip v0.2.0
  [81def892] VersionParsing v1.3.0
  [ea10d353] WeakRefStrings v1.4.2
  [44d3d7a6] Weave v0.10.12
  [efce3f68] WoodburyMatrices v1.1.0
  [76eceee3] WorkerUtilities v1.6.1
  [ddb6d928] YAML v0.4.16
  [c2297ded] ZMQ v1.5.1
  [700de1a5] ZygoteRules v0.2.7
  [6e34b625] Bzip2_jll v1.0.9+0
  [83423d85] Cairo_jll v1.18.5+1
  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.7.3+0
  [b22a6f82] FFMPEG_jll v8.0.1+0
  [a3f928ae] Fontconfig_jll v2.17.1+0
  [d7e528f0] FreeType2_jll v2.13.4+0
  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.4.1+0
  [d2c73de3] GR_jll v0.73.22+0
  [b0724c58] GettextRuntime_jll v0.22.4+0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [020c3dae] Git_LFS_jll v3.7.0+0
  [f8c6e375] Git_jll v2.53.0+0
  [7746bdde] Glib_jll v2.86.3+0
  [3b182d85] Graphite2_jll v1.3.15+0
  [2e76f6c2] HarfBuzz_jll v8.5.1+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [aacddb02] JpegTurbo_jll v3.1.4+0
  [c1c5ebd0] LAME_jll v3.100.3+0
  [88015f11] LERC_jll v4.0.1+0
  [1d63c593] LLVMOpenMP_jll v18.1.8+0
  [dd4b983a] LZO_jll v2.10.3+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 v2.41.3+0
  [89763e89] Libtiff_jll v4.7.2+0
  [38a345b3] Libuuid_jll v2.41.3+0
  [856f044c] MKL_jll v2025.2.0+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [9bd350c2] OpenSSH_jll v10.2.1+0
  [458c3c95] OpenSSL_jll v3.5.5+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.57.0+0
⌅ [30392449] Pixman_jll v0.44.2+0
⌅ [c0090381] Qt6Base_jll v6.8.2+2
⌅ [629bc702] Qt6Declarative_jll v6.8.2+1
⌅ [ce943373] Qt6ShaderTools_jll v6.8.2+1
⌃ [e99dba38] Qt6Wayland_jll v6.8.2+2
  [f50d1b31] Rmath_jll v0.5.1+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.2+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 v1.8.3+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
  [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 v2.44.0+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [8f1865be] ZeroMQ_jll v4.3.6+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 v3.13.1+0
  [0ac62f75] libass_jll v0.17.4+0
  [1183f4f0] libdecor_jll v0.2.2+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 v1.6.55+0
  [a9144af2] libsodium_jll v1.0.21+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.0.0+1
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
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`