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, 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
end

Main.var"##WeaveSandBox#277".display_stan_timing

gr(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#277".LorenzExample{Main.var"##WeaveSandBox#277".var
"###ParameterizedDiffEqFunction#279", Main.var"##WeaveSandBox#277".var"###P
arameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"###Paramet
erizedJacobianFunction#281", Nothing, Nothing, ModelingToolkit.System}(Main
.var"##WeaveSandBox#277".var"##ParameterizedDiffEqFunction#279", LinearAlge
bra.UniformScaling{Bool}(true), nothing, Main.var"##WeaveSandBox#277".var"#
#ParameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"##Parame
terizedJacobianFunction#281", nothing, nothing, nothing, nothing, nothing, 
nothing, nothing, [:x, :y, :z], :t, nothing, Model ##Parameterized#278:
Equations (3):
  3 standard: see equations(##Parameterized#278)
Unknowns (3): see unknowns(##Parameterized#278)
  x(t)
  y(t)
  z(t)
Parameters (3): see parameters(##Parameterized#278)
  σ
  ρ
  β, nothing, nothing)

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
  2.66

prob = ODEProblem(g1, r0, tspan, p)
sol = solve(prob, Tsit5())

retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 355-element Vector{Float64}:
  0.0
  3.5678604836301404e-5
  0.0012320942519312387
  0.00556932465348067
  0.012366524722957406
  0.021160814850103675
  0.033296251971569796
  0.04900115478366526
  0.06936892249678134
  0.09517047537763844
  ⋮
 29.383016799571845
 29.460494075076628
 29.540306930513268
 29.625471076888655
 29.723135438241798
 29.805484193360808
 29.910086909807717
 29.99928302640111
 30.0
u: 355-element Vector{Vector{Float64}}:
 [1.0, 0.0, 0.0]
 [0.9996434557625105, 0.0009988049817849054, 1.7814349300524496e-8]
 [0.9879653604576064, 0.03426824324278913, 2.0964175442932688e-5]
 [0.9500089426818266, 0.1514709435028955, 0.00040920017142123774]
 [0.9033877560315204, 0.32596836188271083, 0.001892216392693151]
 [0.8639680997970994, 0.5392700815038988, 0.005168731067070007]
 [0.8433235120785849, 0.8199540219704273, 0.0119181078390778]
 [0.8665761894886946, 1.178060982280894, 0.024523932292864075]
 [0.9699424017971061, 1.6684270372332624, 0.04902023715188545]
 [1.2120261198364632, 2.3949595105940595, 0.10071541720535906]
 ⋮
 [-11.953731623976742, -8.724911172152208, 34.75350663343156]
 [-8.414218124478245, -3.3997394618081014, 32.42494675534079]
 [-5.095020000229003, -2.072653537110552, 27.39746410268826]
 [-3.578767653386599, -2.916342234602008, 22.594599695010295]
 [-3.9132937007923942, -5.183232333060949, 18.662230583885023]
 [-5.62219113490472, -8.549611643866369, 17.379536103765123]
 [-9.796979599919887, -14.53093015590858, 21.494882238678578]
 [-13.15432518607142, -14.732570431860434, 31.42586687218547]
 [-13.165446107264613, -14.689326774867084, 31.504660801755886]

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: 355-element Vector{Float64}:
  0.0
  3.5678604836301404e-5
  0.0012320942519312387
  0.00556932465348067
  0.012366524722957406
  0.021160814850103675
  0.033296251971569796
  0.04900115478366526
  0.06936892249678134
  0.09517047537763844
  ⋮
 29.383016896164897
 29.460494173696
 29.540307024777988
 29.62547117109964
 29.72313561053084
 29.805484348246978
 29.91008712510907
 29.99928320804487
 30.0
u: 355-element Vector{StaticArraysCore.SVector{3, Float64}}:
 [1.0, 0.0, 0.0]
 [0.9996434557625105, 0.0009988049817849054, 1.7814349300524496e-8]
 [0.9879653604576064, 0.03426824324278913, 2.0964175442932688e-5]
 [0.9500089426818266, 0.1514709435028955, 0.00040920017142123774]
 [0.9033877560315204, 0.32596836188271083, 0.001892216392693151]
 [0.8639680997970994, 0.5392700815038988, 0.005168731067070007]
 [0.8433235120785849, 0.8199540219704273, 0.0119181078390778]
 [0.8665761894886946, 1.178060982280894, 0.024523932292864075]
 [0.9699424017971061, 1.6684270372332624, 0.04902023715188545]
 [1.2120261198364632, 2.3949595105940595, 0.10071541720535906]
 ⋮
 [-11.953728707099234, -8.724903559414773, 34.75350730511037]
 [-8.414213772728967, -3.399736204270524, 32.42494152319427]
 [-5.0950176680667125, -2.0726539133804285, 27.39745893707636]
 [-3.5787673480301443, -2.9163438983419585, 22.594595712232938]
 [-3.9132960668681473, -5.183237769500962, 18.66222616103544]
 [-5.622195731286272, -8.549619474241986, 17.379536947095986]
 [-9.796989656834683, -14.530940358320164, 21.4949007487875]
 [-13.154327787595237, -14.732559461298061, 31.42588636022459]
 [-13.165445843155602, -14.689326671922553, 31.50466026912453]

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}:
 -9.93527  -8.67702  -8.04024   -8.6215  …  15.7979  -4.00063  -13.0561
 -8.90803  -8.67557  -6.79671  -10.3481     23.2883  -4.70514  -15.0132
 28.0059   25.0552   27.6012    26.1561     29.5471  20.9232    31.2951

Plots 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.

We use adapt_delta = 0.85 (Stan's default) consistently across all backends for a fair comparison. Stan infers a separate noise parameter per data dimension (3 for the 3D Lorenz system).

@time bayesian_result_stan = stan_inference(
    prob, :rk45, t, data, priors;
    solve_kwargs = Dict(:reltol => 1e-8, :abstol => 1e-8),
    sample_kwargs = Dict(:delta => 0.85),
    vars = (DiffEqBayes.StanODEData(), InverseGamma(2, 3)))

29720.056207 seconds (4.85 M allocations: 237.477 MiB, 0.00% gc time, 0.01%
 compilation time)
29744.262861 seconds (15.97 M allocations: 790.756 MiB, 0.00% gc time, 0.04
% compilation time: <1% 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    3.9941    0.0000    0.0000     2.7491     6.4611    1.7034 
    ⋯
    sigma1.2    0.5652    0.0000    0.0000     2.7453    10.0118    1.7206 
    ⋯
    sigma1.3    1.0002    0.0000    0.0000    49.8133        NaN    1.0285 
    ⋯
     theta_1    5.7287    0.0000    0.0000     6.6433    17.6814    1.2756 
    ⋯
     theta_2   39.5220    0.0000    0.0000    11.6979        NaN    1.1269 
    ⋯
     theta_3    1.3846    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    3.9941    3.9941    3.9941    3.9941    3.9941
    sigma1.2    0.5652    0.5652    0.5652    0.5652    0.5652
    sigma1.3    1.0002    1.0002    1.0002    1.0002    1.0002
     theta_1    5.7287    5.7287    5.7287    5.7287    5.7287
     theta_2   39.5220   39.5220   39.5220   39.5220   39.5221
     theta_3    1.3846    1.3846    1.3846    1.3846    1.3846

Stan's internal timing (excluding data serialization and CSV parsing):

display_stan_timing(bayesian_result_stan)

Chain 1 timing (from Stan CSV):
  Elapsed Time: 14339.3 seconds (Warm-up)
  15376.2 seconds (Sampling)
  29715.5 seconds (Total)

Direct Turing.jl

We use per-dimension noise parameters (matching Stan) with InverseGamma(2, 3) priors on each σ.

@model function fitlorenz(data, prob)
    # Prior distributions.
    σ ~ filldist(InverseGamma(2, 3), 3)
    σ_param ~ truncated(Normal(10, 2), 1, 15)
    ρ ~ truncated(Normal(30, 5), 1, 45)
    β ~ truncated(Normal(2.5, 0.5), 1, 4)

    # Simulate Lorenz model.
    p = SA[σ_param, 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], Diagonal(σ .^ 2))
    end

    return nothing
end

model = fitlorenz(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

Chains MCMC chain (10000×20×1 Array{Float64, 3}):

Iterations        = 1001:1:11000
Number of chains  = 1
Samples per chain = 10000
Wall duration     = 6195.95 seconds
Compute duration  = 6195.95 seconds
parameters        = σ[1], σ[2], σ[3], σ_param, ρ, β
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      rhat 
  e ⋯
      Symbol   Float64   Float64   Float64    Float64    Float64   Float64 
    ⋯

        σ[1]    0.3821    0.0000    0.0000    21.9965    30.5146    1.5109 
    ⋯
        σ[2]    0.6126    0.0000    0.0000    20.9415    34.3309    1.9951 
    ⋯
        σ[3]    0.3698    0.0000    0.0000    20.8387    30.6869    2.1021 
    ⋯
     σ_param   12.8795    0.0000    0.0000        NaN        NaN       NaN 
    ⋯
           ρ   34.3810    0.0000    0.0000        NaN        NaN       NaN 
    ⋯
           β    1.5667    0.0000    0.0000    22.7338        NaN    1.3273 
    ⋯
                                                                1 column om
itted

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

        σ[1]    0.3821    0.3821    0.3821    0.3821    0.3821
        σ[2]    0.6126    0.6126    0.6126    0.6126    0.6126
        σ[3]    0.3698    0.3698    0.3698    0.3698    0.3698
     σ_param   12.8795   12.8795   12.8795   12.8795   12.8795
           ρ   34.3810   34.3810   34.3810   34.3810   34.3810
           β    1.5667    1.5667    1.5667    1.5667    1.5667

display_ess_per_sec(chain, elapsed_turing_direct)

Elapsed time: 6196.35 seconds

ESS/s (effective samples per second, bulk):
  σ[1]: 0.0
  σ[2]: 0.0
  σ[3]: 0.0
  σ_param: NaN
  ρ: NaN
  β: 0.0

Minimum ESS/s (bulk): NaN

Using Turing.jl backend

@time bayesian_result_turing = turing_inference(
    prob, Vern9(), t, data, priors;
    sample_args = (sampler = Turing.NUTS(0.85), num_samples = 10_000),
    solve_kwargs = Dict(: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)])

18529.567688 seconds (19.99 G allocations: 1.643 TiB, 8.29% gc time, 0.18% 
compilation time)
Chains MCMC chain (10000×20×1 Array{Float64, 3}):

Iterations        = 1001:1:11000
Number of chains  = 1
Samples per chain = 10000
Wall duration     = 18515.9 seconds
Compute duration  = 18515.9 seconds
parameters        = theta[1], theta[2], theta[3], σ[1], σ[2], σ[3]
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      rhat 
  e ⋯
      Symbol   Float64   Float64   Float64    Float64    Float64   Float64 
    ⋯

    theta[1]   10.1859    0.0000    0.0000        NaN        NaN       NaN 
    ⋯
    theta[2]   38.2072    0.0000    0.0000        NaN        NaN       NaN 
    ⋯
    theta[3]    2.7341    0.0000    0.0000        NaN        NaN       NaN 
    ⋯
        σ[1]    3.4307    0.0000    0.0000        NaN        NaN       NaN 
    ⋯
        σ[2]    0.7834    0.0000    0.0000    21.5739    24.5684    1.5942 
    ⋯
        σ[3]    1.2429    0.0000    0.0000    23.5517        NaN    1.1862 
    ⋯
                                                                1 column om
itted

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

    theta[1]   10.1859   10.1859   10.1859   10.1859   10.1859
    theta[2]   38.2072   38.2072   38.2072   38.2072   38.2072
    theta[3]    2.7341    2.7341    2.7341    2.7341    2.7341
        σ[1]    3.4307    3.4307    3.4307    3.4307    3.4307
        σ[2]    0.7834    0.7834    0.7834    0.7834    0.7834
        σ[3]    1.2429    1.2429    1.2429    1.2429    1.2429

Using DynamicHMC.jl backend

@time bayesian_result_dynamichmc = dynamichmc_inference(
    prob, Tsit5(), t, data, priors; solve_kwargs = (reltol = 1e-8, abstol = 1e-8))

118.464064 seconds (111.72 M allocations: 9.652 GiB, 7.35% gc time, 7.24% c
ompilation time)
(posterior = [(parameters = [1.8760312729317854, 25.37896546752704, 1.68280
29896978622], σ = [10.66157578848245, 13.410745124891944, 7.647804791086839
]), (parameters = [1.2176112555886092, 26.321736968587377, 1.63823997682052
7], σ = [16.044193322076016, 13.997339133179825, 5.763035562689531]), (para
meters = [1.8470336401876193, 25.274599973369657, 1.785911077912919], σ = [
12.91936297632157, 10.313898132155376, 6.162203282108906]), (parameters = [
2.095700630853876, 25.173714837500505, 1.9051737396567519], σ = [12.9154649
64212807, 13.130968958505958, 6.814272905732188]), (parameters = [1.5418148
459044465, 25.93222739289616, 1.646418956122406], σ = [12.652102091335065, 
10.733178728737316, 6.792574875463415]), (parameters = [1.4599614131950025,
 26.203863959708027, 1.6870494008621635], σ = [13.392346874955017, 12.35819
144645353, 5.305854515657605]), (parameters = [1.4530588035139267, 25.09364
530863547, 1.6806582970456676], σ = [12.407663163171874, 13.211997231328375
, 5.452833557278067]), (parameters = [1.4767453116551985, 26.40666596558259
7, 1.5575657452139864], σ = [11.00558108655802, 10.59682323491853, 7.440246
99288743]), (parameters = [1.5505273428301227, 25.10858024672293, 1.7572750
475027112], σ = [12.29775639335089, 10.937482265410273, 5.959763697648363])
, (parameters = [1.9568484769734167, 26.178426339988416, 1.852182954682399]
, σ = [13.146684232025676, 13.716098114563875, 8.015340827193961])  …  (par
ameters = [4.338470033802378, 23.919258965435265, 1.8456270353301087], σ = 
[6.935949019458352, 10.601640613366557, 9.161025100642156]), (parameters = 
[4.338470033802378, 23.919258965435265, 1.8456270353301087], σ = [6.9359490
19458352, 10.601640613366557, 9.161025100642156]), (parameters = [4.3384700
33802378, 23.919258965435265, 1.8456270353301087], σ = [6.935949019458352, 
10.601640613366557, 9.161025100642156]), (parameters = [4.338470033802378, 
23.919258965435265, 1.8456270353301087], σ = [6.935949019458352, 10.6016406
13366557, 9.161025100642156]), (parameters = [4.338470033802378, 23.9192589
65435265, 1.8456270353301087], σ = [6.935949019458352, 10.601640613366557, 
9.161025100642156]), (parameters = [4.338470033802378, 23.919258965435265, 
1.8456270353301087], σ = [6.935949019458352, 10.601640613366557, 9.16102510
0642156]), (parameters = [4.279518978333829, 24.52414806855977, 1.846161004
4359422], σ = [6.880804453395815, 10.604176969632706, 8.952497966802962]), 
(parameters = [4.279518978333829, 24.52414806855977, 1.8461610044359422], σ
 = [6.880804453395815, 10.604176969632706, 8.952497966802962]), (parameters
 = [4.279518978333829, 24.52414806855977, 1.8461610044359422], σ = [6.88080
4453395815, 10.604176969632706, 8.952497966802962]), (parameters = [4.29183
7237968101, 24.306740884954337, 1.832521334744501], σ = [6.925785458327831,
 10.59184219681094, 8.96156362129023])], posterior_matrix = [0.629158520451
3721 0.19689095215993097 … 1.4538406150379144 1.456714901861154; 3.23392069
9734697 3.2703950985268326 … 3.1996582675824174 3.1907537145505134; … ; 2.5
960562606188167 2.638867249635477 … 2.3612479772381714 2.3600841007463353; 
2.034418651245377 1.7514643432995414 … 2.19193259578101 2.192944723066226],
 tree_statistics = DynamicHMC.TreeStatisticsNUTS[DynamicHMC.TreeStatisticsN
UTS(-364.6000131756984, 5, turning at positions -3:28, 0.9471286991929042, 
31, DynamicHMC.Directions(0x92fa531c)), DynamicHMC.TreeStatisticsNUTS(-366.
3005221587259, 5, turning at positions 28:59, 0.9815225557940406, 63, Dynam
icHMC.Directions(0x465c1a7b)), DynamicHMC.TreeStatisticsNUTS(-370.336323057
9123, 6, turning at positions -25:-56, 0.7346595056737527, 95, DynamicHMC.D
irections(0x449f9627)), DynamicHMC.TreeStatisticsNUTS(-371.59117806496437, 
5, turning at positions -18:-49, 0.9967708979863921, 63, DynamicHMC.Directi
ons(0x0c4fb3ce)), DynamicHMC.TreeStatisticsNUTS(-363.9875752302064, 6, turn
ing at positions -10:53, 0.9846960882134211, 63, DynamicHMC.Directions(0xb7
723c75)), DynamicHMC.TreeStatisticsNUTS(-365.950603518262, 5, turning at po
sitions -7:-22, 0.9578688006802725, 47, DynamicHMC.Directions(0x38fe5019)),
 DynamicHMC.TreeStatisticsNUTS(-365.6685521537958, 5, turning at positions 
-18:-49, 0.8496907916182462, 63, DynamicHMC.Directions(0x69290f8e)), Dynami
cHMC.TreeStatisticsNUTS(-367.7412737128681, 5, turning at positions 32:63, 
0.9879942338870209, 63, DynamicHMC.Directions(0xb50bfd3f)), DynamicHMC.Tree
StatisticsNUTS(-365.81739814867973, 5, turning at positions -27:4, 0.997495
2377869817, 31, DynamicHMC.Directions(0xcb5ab9a4)), DynamicHMC.TreeStatisti
csNUTS(-365.1182186540352, 6, turning at positions 48:111, 0.95685393756142
98, 127, DynamicHMC.Directions(0xe70f276f))  …  DynamicHMC.TreeStatisticsNU
TS(-334.7690565099726, 0, divergence at position -1, 0.0, 1, DynamicHMC.Dir
ections(0xf2b80278)), DynamicHMC.TreeStatisticsNUTS(-336.2469123474379, 1, 
divergence at position 3, 0.0488563116708394, 3, DynamicHMC.Directions(0xe8
5985f3)), DynamicHMC.TreeStatisticsNUTS(-335.7780919364989, 2, divergence a
t position 1, 3.0599441328414395e-5, 4, DynamicHMC.Directions(0xa7747a7c)),
 DynamicHMC.TreeStatisticsNUTS(-338.32404466693987, 0, divergence at positi
on 1, 0.0, 1, DynamicHMC.Directions(0x1111427f)), DynamicHMC.TreeStatistics
NUTS(-334.5594177292511, 1, turning at positions 0:1, 1.7867730820404535e-1
7, 1, DynamicHMC.Directions(0x9c79ec79)), DynamicHMC.TreeStatisticsNUTS(-33
7.92459744639683, 0, divergence at position -1, 0.0, 1, DynamicHMC.Directio
ns(0x6e6ec9c8)), DynamicHMC.TreeStatisticsNUTS(-335.1513242980172, 1, diver
gence at position 2, 0.5, 2, DynamicHMC.Directions(0xa8289c6f)), DynamicHMC
.TreeStatisticsNUTS(-336.2306311119405, 0, divergence at position -1, 0.0, 
1, DynamicHMC.Directions(0xda0c1f26)), DynamicHMC.TreeStatisticsNUTS(-336.3
1879822998553, 0, divergence at position 1, 0.0, 1, DynamicHMC.Directions(0
x66178eed)), DynamicHMC.TreeStatisticsNUTS(-334.331673651966, 2, turning at
 positions -3:0, 0.06871860032419076, 3, DynamicHMC.Directions(0x1a164234))
], logdensities = [-363.3551358261815, -364.9095175125483, -363.31266781562
42, -361.8165922996491, -362.0726794042541, -363.186479980284, -363.0250053
4314885, -363.99083807823223, -363.08523212573044, -362.76021281954485  …  
-332.9465002978295, -332.9465002978295, -332.9465002978295, -332.9465002978
295, -332.9465002978295, -332.9465002978295, -332.1949281704365, -332.19492
81704365, -332.1949281704365, -333.1068622868298], κ = Gaussian kinetic ene
rgy (Diagonal), √diag(M⁻¹): [0.29185408547381164, 1.1588620418539393, 0.275
05818957859696, 0.33152559856542224, 0.1456632503965011, 0.3955213864644185
], ϵ = 0.03753482599514242)

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 demonstrates chaotic behavior and exhibits instability for different tolerance values. We use 1e-8 as the tolerance as it makes its uncertainty small enough to be trusted in the (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 1.12.7
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:

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  [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
  [e7412a2a] Ogg_jll v1.3.6+0
  [9bd350c2] OpenSSH_jll v10.5.1+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 v5.8.3+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
  [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
  [a9144af2] libsodium_jll v1.0.21+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
  [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
  [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`