Fitzhugh-Nagumo Bayesian Parameter Estimation Benchmarks
using DiffEqBayes, BenchmarkToolsusing OrdinaryDiffEq, RecursiveArrayTools, Distributions, ParameterizedFunctions,
StanSample, DynamicHMC
using Plots, StaticArrays, Turing, LinearAlgebra"""Display ESS/s (effective samples per second) from a Turing chain."""
function display_ess_per_sec(chain, elapsed)
stats = summarystats(chain)
ess_bulk = stats[:, :ess_bulk]
println("Elapsed time: $(round(elapsed; digits=2)) seconds\n")
println("ESS/s (effective samples per second, bulk):")
for (i, param) in enumerate(stats[:, :parameters])
println(" $param: $(round(ess_bulk[i] / elapsed; digits=1))")
end
println("\nMinimum ESS/s (bulk): $(round(minimum(ess_bulk) / elapsed; digits=1))")
end
"""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()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 lMain.var"##WeaveSandBox#277".FitzhughNagumo{Main.var"##WeaveSandBox#277".va
r"###ParameterizedDiffEqFunction#279", Main.var"##WeaveSandBox#277".var"###
ParameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"###Parame
terizedJacobianFunction#281", Nothing, Nothing, ModelingToolkit.System}(Mai
n.var"##WeaveSandBox#277".var"##ParameterizedDiffEqFunction#279", LinearAlg
ebra.UniformScaling{Bool}(true), nothing, Main.var"##WeaveSandBox#277".var"
##ParameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"##Param
eterizedJacobianFunction#281", nothing, nothing, nothing, nothing, nothing,
nothing, nothing, [:v, :w], :t, nothing, Model ##Parameterized#278:
Equations (2):
2 standard: see equations(##Parameterized#278)
Unknowns (2): see unknowns(##Parameterized#278)
v(t)
w(t)
Parameters (4): see parameters(##Parameterized#278)
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]
[1.1446577625483758, 1.1952738138449215]
[1.0557695077719014, 1.2718985818139574]
[0.8659598744812584, 1.3388184800875969]
[0.367585402117253, 1.373537601831974]
[-0.359442795548185, 1.3493319650351676]
[-1.3772889489189262, 1.2781711184359077]
[-1.905699839713036, 1.1680023987534751]
[-1.9707492736430972, 1.0777291565175877]
[-1.9650453438870348, 1.0031251492628284]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]
[1.1446577625483758, 1.1952738138449215]
[1.0557695077719014, 1.2718985818139574]
[0.8659598744812584, 1.3388184800875969]
[0.367585402117253, 1.373537601831974]
[-0.359442795548185, 1.3493319650351676]
[-1.3772889489189262, 1.2781711184359077]
[-1.905699839713036, 1.1680023987534751]
[-1.9707492736430972, 1.0777291565175877]
[-1.9650453438870348, 1.0031251492628284]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}:
0.986173 1.43124 0.83495 0.912354 … -0.966216 -1.90734 -2.3265
1.14278 1.09743 1.1091 1.28319 1.40656 0.995218 0.874711Plot 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
We use adapt_delta = 0.85 (Stan's default) consistently across all backends for a fair comparison.
bayesian_result_stan = @time stan_inference(
prob_ode_fitzhughnagumo, :rk45, t, data, priors;
print_summary = false,
sample_kwargs = Dict(:delta => 0.85, :num_samples => 10_000),
vars = (DiffEqBayes.StanODEData(), InverseGamma(2, 3)))52.004302 seconds (4.85 M allocations: 237.371 MiB, 0.14% gc time, 7.38% c
ompilation time)
81.350151 seconds (15.98 M allocations: 808.437 MiB, 0.18% gc time, 13.51%
compilation time: <1% of which was recompilation)
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 0.4463 0.1292 0.0017 6543.6143 5302.6721 1.000
1 ⋯
sigma1.2 0.3506 0.1056 0.0014 6099.9390 5487.2198 1.000
1 ⋯
theta_1 0.9173 0.3219 0.0045 4757.7852 4228.7728 1.000
2 ⋯
theta_2 0.9460 0.2868 0.0041 4652.6520 3984.2736 1.000
2 ⋯
theta_3 0.0981 0.0452 0.0007 3709.4382 3368.9362 1.000
4 ⋯
theta_4 0.5275 0.0959 0.0016 4020.7463 3701.4248 1.000
8 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
sigma1.1 0.2675 0.3549 0.4226 0.5104 0.7650
sigma1.2 0.2001 0.2748 0.3323 0.4067 0.6073
theta_1 0.2417 0.6968 0.9432 1.1667 1.4432
theta_2 0.3338 0.7574 0.9684 1.1644 1.4247
theta_3 0.0284 0.0658 0.0928 0.1234 0.2019
theta_4 0.3641 0.4611 0.5202 0.5839 0.7432Stan's internal timing (excluding data serialization and CSV parsing):
display_stan_timing(bayesian_result_stan)Chain 1 timing (from Stan CSV):
Elapsed Time: 4.706 seconds (Warm-up)
43.453 seconds (Sampling)
48.159 seconds (Total)Direct Turing.jl
We use per-dimension noise parameters (matching Stan) with InverseGamma(2, 3) priors on each σ.
@model function fitfhn(data, prob)
# Prior distributions.
σ ~ filldist(InverseGamma(2, 3), 2)
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 FitzHugh-Nagumo 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], Diagonal(σ .^ 2))
end
return nothing
end
model = fitfhn(data, sprob_ode_fitzhughnagumo)
# 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)
chainChains MCMC chain (10000×20×1 Array{Float64, 3}):
Iterations = 1001:1:11000
Number of chains = 1
Samples per chain = 10000
Wall duration = 99.72 seconds
Compute duration = 99.72 seconds
parameters = σ[1], σ[2], 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[1] 0.4474 0.1321 0.0020 5280.7854 4955.8395 1.000
3 ⋯
σ[2] 0.3494 0.1060 0.0014 5975.9299 5634.9272 1.000
3 ⋯
a 0.9179 0.3247 0.0049 4240.8767 4314.7052 1.000
1 ⋯
b 0.9434 0.2927 0.0041 4937.1615 4960.3285 1.000
5 ⋯
τinv 0.0970 0.0458 0.0008 3246.0404 3315.4761 1.000
7 ⋯
l 0.5257 0.0973 0.0017 3370.5208 3508.6678 1.000
6 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ[1] 0.2688 0.3563 0.4220 0.5089 0.7708
σ[2] 0.1998 0.2744 0.3311 0.4041 0.6023
a 0.2222 0.6999 0.9483 1.1672 1.4463
b 0.3092 0.7461 0.9675 1.1653 1.4283
τinv 0.0252 0.0653 0.0917 0.1224 0.2003
l 0.3504 0.4606 0.5181 0.5836 0.7464display_ess_per_sec(chain, elapsed_turing_direct)Elapsed time: 100.15 seconds
ESS/s (effective samples per second, bulk):
σ[1]: 52.7
σ[2]: 59.7
a: 42.3
b: 49.3
τinv: 32.4
l: 33.7
Minimum ESS/s (bulk): 32.4Turing.jl backend
@btime bayesian_result_turing = turing_inference(
prob_ode_fitzhughnagumo, Tsit5(), t, data, priors;
sample_args = (sampler = Turing.NUTS(0.85), num_samples = 10_000),
likelihood = (u, p, t, σ) -> MvNormal(u, Diagonal(σ .^ 2)),
likelihood_dist_priors = [InverseGamma(2, 3), InverseGamma(2, 3)])84.596 s (328217861 allocations: 21.55 GiB)
Chains MCMC chain (10000×20×1 Array{Float64, 3}):
Iterations = 1001:1:11000
Number of chains = 1
Samples per chain = 10000
Wall duration = 89.37 seconds
Compute duration = 89.37 seconds
parameters = theta[1], theta[2], theta[3], theta[4], σ[1], σ[2]
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 ⋯
theta[1] 0.9240 0.3211 0.0045 4923.7021 4575.9903 1.000
1 ⋯
theta[2] 0.9490 0.2890 0.0041 4666.8215 3381.5229 1.000
4 ⋯
theta[3] 0.0970 0.0444 0.0007 4254.9109 4134.7980 1.000
4 ⋯
theta[4] 0.5251 0.0963 0.0015 4247.9960 4143.7748 1.000
6 ⋯
σ[1] 0.4460 0.1324 0.0018 5915.7216 5211.3424 1.000
2 ⋯
σ[2] 0.3502 0.1064 0.0013 6758.2917 5913.3633 1.000
2 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
theta[1] 0.2521 0.7001 0.9444 1.1711 1.4494
theta[2] 0.3294 0.7571 0.9666 1.1672 1.4358
theta[3] 0.0256 0.0653 0.0922 0.1232 0.1967
theta[4] 0.3557 0.4583 0.5179 0.5827 0.7386
σ[1] 0.2651 0.3546 0.4215 0.5076 0.7690
σ[2] 0.2008 0.2734 0.3317 0.4055 0.6118Conclusion
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 1.12.7
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 v3.13.0
⌃ [459566f4] DiffEqCallbacks v4.19.2
⌃ [31c24e10] Distributions v0.25.127
[bbc10e6e] DynamicHMC v3.6.1
⌅ [1dea7af3] OrdinaryDiffEq v6.111.0
⌃ [65888b18] ParameterizedFunctions v5.19.0
[91a5bcdd] Plots v1.41.7
⌅ [731186ca] RecursiveArrayTools v3.54.0
⌃ [31c91b34] SciMLBenchmarks v0.1.3 [loaded: v0.2.0]
[c1514b29] StanSample v7.10.3
[90137ffa] StaticArrays v1.9.19
⌅ [fce5fe82] Turing v0.42.9
[37e2e46d] LinearAlgebra v1.12.0
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`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 v0.5.3
[621f4979] AbstractFFTs v1.5.0
[80f14c24] AbstractMCMC v5.16.0
⌅ [7a57a42e] AbstractPPL v0.13.6
[1520ce14] AbstractTrees v0.4.5
[7d9f7c33] Accessors v0.1.45
[79e6a3ab] Adapt v4.7.0
[0bf59076] AdvancedHMC v0.8.6
[5b7e9947] AdvancedMH v0.8.10
⌅ [576499cb] AdvancedPS v0.7.2
⌅ [b5ca4192] AdvancedVI v0.6.2
[66dad0bd] AliasTables v1.1.3
[dce04be8] ArgCheck v2.5.0
[ec485272] ArnoldiMethod v0.4.0
[4fba245c] ArrayInterface v7.30.0
[4c555306] ArrayLayouts v1.12.2
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[39de3d68] AxisArrays v0.4.8
[198e06fe] BangBang v0.4.9
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[e2ed5e7c] Bijections v0.2.2
⌅ [76274a88] Bijectors v0.15.16
[62783981] BitTwiddlingConvenienceFunctions v0.1.6
[8e7c35d0] BlockArrays v1.10.0
⌃ [70df07ce] BracketingNonlinearSolve v1.12.1
[2a0fbf3d] CPUSummary v0.2.7
[336ed68f] CSV v0.10.17
[082447d4] ChainRules v1.73.0
[d360d2e6] ChainRulesCore v1.26.1
[0ca39b1e] Chairmarks v1.3.1
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[a80b9123] CommonMark v1.0.4
[38540f10] CommonSolve v0.2.14
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[f70d9fcc] CommonWorldInvalidations v1.2.0
[34da2185] Compat v4.18.1
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[bbc10e6e] DynamicHMC v3.6.1
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[615f187c] IfElse v0.1.1
⌅ [3263718b] ImplicitDiscreteSolve v1.10.0
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[22cec73e] InitialValues v0.3.1
[842dd82b] InlineStrings v1.4.5
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[41ab1584] InvertedIndices v1.3.1
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[1019f520] JLFzf v0.1.11
[692b3bcd] JLLWrappers v1.8.0
⌅ [682c06a0] JSON v0.21.4
[ae98c720] Jieko v0.2.1
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⌃ [ccbc3e58] JumpProcesses v9.29.0
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[ba0b0d4f] Krylov v0.10.9
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[23fbe1c1] Latexify v0.16.12
[10f19ff3] LayoutPointers v0.1.17
[1fad7336] LazyStack v0.1.3
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⌃ [87fe0de2] LineSearch v0.1.14
⌃ [d3d80556] LineSearches v7.5.1
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⌅ [2ab3a3ac] LogExpFunctions v0.3.29
[e6f89c97] LoggingExtras v1.2.0
⌃ [c7f686f2] MCMCChains v6.0.7
[be115224] MCMCDiagnosticTools v0.3.19
[e80e1ace] MLJModelInterface v1.12.1
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[dbb5928d] MappedArrays v0.4.3
[a3b82374] MatrixFactorizations v3.1.3
[bb5d69b7] MaybeInplace v0.1.8
[442fdcdd] Measures v0.3.3
[e1d29d7a] Missings v1.2.0
[dbe65cb8] MistyClosures v2.1.0
⌅ [961ee093] ModelingToolkit v10.32.1
[2e0e35c7] Moshi v0.3.12
[46d2c3a1] MuladdMacro v0.2.7
[102ac46a] MultivariatePolynomials v0.5.19
[ffc61752] Mustache v1.0.21
[d8a4904e] MutableArithmetics v1.8.0
⌅ [d41bc354] NLSolversBase v7.10.0
[77ba4419] NaNMath v1.1.4
[86f7a689] NamedArrays v0.10.5
[d9ec5142] NamedTupleTools v0.14.3
[c020b1a1] NaturalSort v1.0.0
⌃ [8913a72c] NonlinearSolve v4.16.0
⌃ [be0214bd] NonlinearSolveBase v2.11.2
⌃ [5959db7a] NonlinearSolveFirstOrder v2.0.0
⌃ [9a2c21bd] NonlinearSolveQuasiNewton v1.12.0
⌃ [26075421] NonlinearSolveSpectralMethods v1.6.0
[6fe1bfb0] OffsetArrays v1.17.0
⌅ [429524aa] Optim v1.13.3
[3bd65402] Optimisers v0.4.9
⌃ [7f7a1694] Optimization v5.4.0
⌅ [bca83a33] OptimizationBase v4.2.0
⌃ [36348300] OptimizationOptimJL v0.4.8
⌅ [bac558e1] OrderedCollections v1.8.2 [loaded: v2.0.1]
⌅ [1dea7af3] OrdinaryDiffEq v6.111.0
⌅ [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.11.0
⌅ [6ad6398a] OrdinaryDiffEqBDF v1.26.0
⌅ [bbf590c4] OrdinaryDiffEqCore v3.28.0
⌅ [50262376] OrdinaryDiffEqDefault v1.14.0
⌅ [4302a76b] OrdinaryDiffEqDifferentiation v2.7.0
⌅ [9286f039] OrdinaryDiffEqExplicitRK v1.12.0
⌅ [e0540318] OrdinaryDiffEqExponentialRK v1.15.0
⌅ [becaefa8] OrdinaryDiffEqExtrapolation v1.18.0
⌅ [5960d6e9] OrdinaryDiffEqFIRK v1.26.0
⌅ [101fe9f7] OrdinaryDiffEqFeagin v1.10.0
⌅ [d3585ca7] OrdinaryDiffEqFunctionMap v1.11.0
⌅ [d28bc4f8] OrdinaryDiffEqHighOrderRK v1.12.0
⌅ [9f002381] OrdinaryDiffEqIMEXMultistep v1.14.0
⌅ [521117fe] OrdinaryDiffEqLinear v1.12.0
⌅ [1344f307] OrdinaryDiffEqLowOrderRK v1.13.0
⌅ [b0944070] OrdinaryDiffEqLowStorageRK v1.15.0
⌅ [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.28.0
⌅ [c9986a66] OrdinaryDiffEqNordsieck v1.11.0
⌅ [5dd0a6cf] OrdinaryDiffEqPDIRK v1.14.0
⌅ [5b33eab2] OrdinaryDiffEqPRK v1.10.0
⌅ [04162be5] OrdinaryDiffEqQPRK v1.10.0
⌅ [af6ede74] OrdinaryDiffEqRKN v1.12.0
⌅ [43230ef6] OrdinaryDiffEqRosenbrock v1.29.0
⌅ [2d112036] OrdinaryDiffEqSDIRK v1.14.0
⌅ [669c94d9] OrdinaryDiffEqSSPRK v1.14.0
⌅ [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.14.0
⌅ [358294b1] OrdinaryDiffEqStabilizedRK v1.11.1
⌅ [fa646aed] OrdinaryDiffEqSymplecticRK v1.13.0
⌅ [b1df2697] OrdinaryDiffEqTsit5 v1.12.0
⌅ [79d7bb75] OrdinaryDiffEqVerner v1.14.0
[90014a1f] PDMats v0.11.41
⌃ [65888b18] ParameterizedFunctions v5.19.0
⌅ [d96e819e] Parameters v0.12.3
⌅ [69de0a69] Parsers v2.8.7
[ccf2f8ad] PlotThemes v3.3.0
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[91a5bcdd] Plots v1.41.7
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⌃ [d236fae5] PreallocationTools v0.4.34
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⌅ [08abe8d2] PrettyTables v2.4.0
[27ebfcd6] Primes v0.5.7
[33c8b6b6] ProgressLogging v0.1.6
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⌅ [731186ca] RecursiveArrayTools v3.54.0
[189a3867] Reexport v1.2.2
[05181044] RelocatableFolders v1.0.1
[ae029012] Requires v1.3.1
[ae5879a3] ResettableStacks v1.4.0
[79098fc4] Rmath v0.9.0
⌅ [f2b01f46] Roots v2.3.0
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⌃ [9dfe8606] SCCNonlinearSolve v1.13.0
[94e857df] SIMDTypes v0.1.0
[26aad666] SSMProblems v0.6.1
⌅ [0bca4576] SciMLBase v2.153.1
⌃ [31c91b34] SciMLBenchmarks v0.1.3 [loaded: v0.2.0]
⌃ [19f34311] SciMLJacobianOperators v0.1.17
⌅ [a6db7da4] SciMLLogging v1.10.1
[c0aeaf25] SciMLOperators v1.30.0
[431bcebd] SciMLPublic v1.3.0
[53ae85a6] SciMLStructures v1.10.5
[30f210dd] ScientificTypesBase v3.1.0
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[efcf1570] Setfield v1.1.2
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⌃ [727e6d20] SimpleNonlinearSolve v2.11.0
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[a57abbd0] SparseColumnPivotedQR v2.1.7
[9f842d2f] SparseConnectivityTracer v1.2.3
[dc90abb0] SparseInverseSubset v0.1.3
[0a514795] SparseMatrixColorings v0.4.27
[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
[aedffcd0] Static v1.4.6
[0d7ed370] StaticArrayInterface v1.10.0
[90137ffa] StaticArrays v1.9.19
[1e83bf80] StaticArraysCore v1.4.4
[64bff920] StatisticalTraits v3.5.0
[10745b16] Statistics v1.11.4
[82ae8749] StatsAPI v1.8.0
[2913bbd2] StatsBase v0.34.13
⌅ [4c63d2b9] StatsFuns v1.5.2
[7792a7ef] StrideArraysCore v0.5.9
[5e0ebb24] Strided v2.6.4
[4db3bf67] StridedViews v0.5.2
[69024149] StringEncodings v0.3.7
⌅ [892a3eda] StringManipulation v0.4.7
[09ab397b] StructArrays v0.7.3
⌃ [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 v1.14.0
[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
[1c621080] TestItems v1.1.0
[8290d209] ThreadingUtilities v0.5.6
⌅ [a759f4b9] TimerOutputs v0.5.29
[3bb67fe8] TranscodingStreams v0.11.3
[84d833dd] TransformVariables v0.8.26
[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 v0.42.9
[5c2747f8] URIs v1.7.0
[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 v1.4.3
[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 v0.2.8
[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 v2.8.3+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
[020c3dae] Git_LFS_jll v3.7.1+0
[f8c6e375] Git_jll v2.55.0+0
[7746bdde] Glib_jll v2.88.3+0
[3b182d85] Graphite2_jll v1.3.16+0
[2e76f6c2] HarfBuzz_jll v100.14003.0+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 v4.1.0+0
[1d63c593] LLVMOpenMP_jll v22.1.7+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
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[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
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[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 v5.8.3+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
[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 v3.14.1+0
[0ac62f75] libass_jll v0.17.5+0
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[f638f0a6] libfdk_aac_jll v2.0.4+0
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[f27f6e37] libvorbis_jll v1.3.8+0
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[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.13.0+0
[0dad84c5] ArgTools v1.1.2
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[2a0f44e3] Base64 v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
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[9fa8497b] Future v1.11.0
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[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
[1a1011a3] SharedArrays 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
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[4536629a] OpenBLAS_jll v0.3.29+0
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[efcefdf7] PCRE2_jll v10.44.0+1
[bea87d4a] SuiteSparse_jll v7.8.3+2
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`