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

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#225".LorenzExample{Main.var"##WeaveSandBox#225".var
"###ParameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".var"###P
arameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###Paramet
erizedJacobianFunction#229", Nothing, Nothing, ModelingToolkit.System}(Main
.var"##WeaveSandBox#225".var"##ParameterizedDiffEqFunction#227", LinearAlge
bra.UniformScaling{Bool}(true), nothing, Main.var"##WeaveSandBox#225".var"#
#ParameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"##Parame
terizedJacobianFunction#229", nothing, nothing, nothing, nothing, nothing, 
nothing, nothing, [:x, :y, :z], :t, nothing, Model ##Parameterized#226:
Equations (3):
  3 standard: see equations(##Parameterized#226)
Unknowns (3): see unknowns(##Parameterized#226)
  x(t)
  y(t)
  z(t)
Parameters (3): see parameters(##Parameterized#226)
  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: 362-element Vector{Float64}:
  0.0
  3.5678604836301404e-5
  0.0003924646531993154
  0.0032623432492218762
  0.009057689436955101
  0.016955558915156328
  0.02768838704624741
  0.041853949478017696
  0.06023708074309082
  0.08368091762034398
  ⋮
 29.457488314242962
 29.53970487795357
 29.60813559643932
 29.6799710290514
 29.75613146300546
 29.825569653667173
 29.88686353386663
 29.942576497493015
 30.0
u: 362-element Vector{Vector{Float64}}:
 [1.0, 0.0, 0.0]
 [0.9996434557625105, 0.0009988049817849054, 1.7814349300524274e-8]
 [0.9961045497425811, 0.010965399721242273, 2.1469572398550344e-6]
 [0.9693597308012994, 0.08976886167146739, 0.00014379729511899872]
 [0.9242069950726427, 0.2422792247788865, 0.0010460983294834716]
 [0.8800496030937968, 0.4387144269574134, 0.00342404857466947]
 [0.8483334484926083, 0.6915266982936876, 0.008487275934120025]
 [0.8494997037566883, 1.0144879834027536, 0.018211867521223127]
 [0.9138893419489335, 1.4424795940711108, 0.03669462180658799]
 [1.0888204830087638, 2.0521988687179387, 0.07402932237242585]
 ⋮
 [13.32238887310434, 18.101945961886766, 27.660768754320422]
 [14.021077095360623, 10.175467083806073, 37.98703009395567]
 [9.58370521371149, 1.793229083577783, 35.90388452654971]
 [4.63425990594524, -0.806923586723377, 29.7403383405032]
 [1.7566567267091238, -0.5799042223012193, 24.12416718752542]
 [0.7258408400096722, -0.10101690190353449, 20.029682888955108]
 [0.4286568979804371, 0.20942636755125477, 17.017933027580412]
 [0.39300509505101944, 0.4602673436547749, 14.680801993969931]
 [0.49560717391903136, 0.7842730243755545, 12.615660658671283]

sr0 = SA[1.0; 0.0; 0.0]
tspan = (0.0, 30.0)
sp = SA[10.0, 28.0, 2.66]
sprob = ODEProblem{false, SciMLBase.FullSpecialize}(g1, sr0, tspan, sp)
sol = solve(sprob, Tsit5())

retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 362-element Vector{Float64}:
  0.0
  3.5678604836301404e-5
  0.0003924646531993154
  0.0032623432492218762
  0.009057689436955101
  0.016955558915156328
  0.02768838704624741
  0.041853949478017696
  0.06023708074309082
  0.08368091762034398
  ⋮
 29.457488315815564
 29.539704873805782
 29.608135591720277
 29.67997101938466
 29.75613144220241
 29.825569627514067
 29.886863506921383
 29.942576473358972
 30.0
u: 362-element Vector{StaticArraysCore.SVector{3, Float64}}:
 [1.0, 0.0, 0.0]
 [0.9996434557625105, 0.0009988049817849054, 1.7814349300524274e-8]
 [0.9961045497425811, 0.010965399721242273, 2.1469572398550344e-6]
 [0.9693597308012994, 0.08976886167146739, 0.00014379729511899872]
 [0.9242069950726427, 0.24227922477888653, 0.0010460983294834716]
 [0.8800496030937968, 0.4387144269574134, 0.00342404857466947]
 [0.8483334484926083, 0.6915266982936876, 0.008487275934120025]
 [0.8494997037566883, 1.0144879834027536, 0.018211867521223127]
 [0.9138893419489335, 1.4424795940711108, 0.03669462180658799]
 [1.0888204830087638, 2.0521988687179387, 0.07402932237242585]
 ⋮
 [13.322389118665507, 18.101945726587324, 27.660769845223655]
 [14.021076982689753, 10.175466941149782, 37.98702995896426]
 [9.583705192805605, 1.7932291983155246, 35.90388439984461]
 [4.63426023460834, -0.8069234124416628, 29.740338705086153]
 [1.7566572048708329, -0.5799041979297636, 24.12416822308383]
 [0.7258411488531001, -0.10101684594938895, 20.029684030726465]
 [0.42865711854860805, 0.20942652456951336, 17.01793403736309]
 [0.3930053157847396, 0.4602676268925582, 14.680802763491037]
 [0.4956075290257965, 0.7842736395679019, 12.615660535439243]

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}:
 -9.68234  -7.95789  -8.71281   -9.92148  …  11.3468   3.90121    0.416424
 -8.65435  -8.67145  -6.78846  -10.4081      14.8211   0.442432   0.714294
 28.7055   25.0599   28.183     27.0174      25.8459  26.281     12.1428

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.

@time bayesian_result_stan = stan_inference(
    prob, t, data, priors; delta = 0.65, reltol = 1e-8, abstol = 1e-8,
    vars = (DiffEqBayes.StanODEData(), InverseGamma(2, 3)))

Error: MethodError: no method matching stan_inference(::SciMLBase.ODEProble
m{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, Main.var
"##WeaveSandBox#225".LorenzExample{Main.var"##WeaveSandBox#225".var"###Para
meterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".var"###Parameter
izedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###ParameterizedJa
cobianFunction#229", Nothing, Nothing, ModelingToolkit.System}, Base.Pairs{
Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}, ::
Vector{Float64}, ::Matrix{Float64}, ::Vector{Distributions.Truncated{Distri
butions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float6
4}}, ::Nothing; delta::Float64, reltol::Float64, abstol::Float64, vars::Tup
le{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", "reltol", "abstol"
   @ 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.
    α ~ InverseGamma(2, 3)
    σ ~ truncated(Normal(10, 2), 1, 15)
    ρ ~ truncated(Normal(30, 5), 1, 45)
    β ~ truncated(Normal(2.5, 0.5), 1, 4)

    # Simulate Lotka-Volterra model. 
    p = SA[σ, ρ, β]
    _prob = remake(prob, p = p)
    predicted = solve(_prob, Vern9(); saveat = t)

    # Observations.
    for i in 1:length(predicted)
        data[:, i] ~ MvNormal(predicted[i], α^2 * I)
    end

    return nothing
end

model = fitlv(data, sprob)

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

4654.930920 seconds (9.27 G allocations: 765.118 GiB, 2.37% gc time, 0.55% 
compilation time: <1% of which was recompilation)
Chains MCMC chain (10000×18×1 Array{Float64, 3}):

Iterations        = 1001:1:11000
Number of chains  = 1
Samples per chain = 10000
Wall duration     = 4642.49 seconds
Compute duration  = 4642.49 seconds
parameters        = α, σ, 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 
    ⋯

           α    2.0786    0.0000    0.0000    21.7150    47.6944    1.7867 
    ⋯
           σ   11.9087    0.0000    0.0000        NaN        NaN       NaN 
    ⋯
           ρ   36.5425    0.0000    0.0000        NaN        NaN       NaN 
    ⋯
           β    1.4046    0.0000    0.0000   660.4927   451.4442    1.0052 
    ⋯
                                                                1 column om
itted

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

           α    2.0786    2.0786    2.0786    2.0786    2.0786
           σ   11.9087   11.9087   11.9087   11.9087   11.9087
           ρ   36.5425   36.5425   36.5425   36.5425   36.5425
           β    1.4046    1.4046    1.4046    1.4046    1.4046

Using Turing.jl backend

@time bayesian_result_turing = turing_inference(
    prob, Vern9(), t, data, priors; reltol = 1e-8, abstol = 1e-8,
    likelihood = (u, p, t, σ) -> MvNormal(u, Diagonal((σ) .^ 2 .* ones(length(u)))),
    likelihood_dist_priors = [InverseGamma(2, 3), InverseGamma(2, 3), InverseGamma(2, 3)])

Error: MethodError: no method matching turing_inference(::SciMLBase.ODEProb
lem{Vector{Float64}, Tuple{Float64, Float64}, true, Vector{Float64}, Main.v
ar"##WeaveSandBox#225".LorenzExample{Main.var"##WeaveSandBox#225".var"###Pa
rameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".var"###Paramet
erizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###Parameterized
JacobianFunction#229", Nothing, Nothing, ModelingToolkit.System}, Base.Pair
s{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}, 
::OrdinaryDiffEqVerner.Vern9{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}}; reltol::F
loat64, abstol::Float64, likelihood::Main.var"##WeaveSandBox#225".var"#4#5"
, likelihood_dist_priors::Vector{Distributions.InverseGamma{Float64}})

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 arguments "relt
ol", "abstol"
   @ DiffEqBayes /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae
0a-f4d2d937f953/packages/DiffEqBayes/gFKkQ/src/turing_inference.jl:1

Using DynamicHMC.jl backend

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

837.665058 seconds (150.05 M allocations: 26.278 GiB, 0.61% gc time, 0.89% 
compilation time)
(posterior = @NamedTuple{parameters::Vector{Float64}, σ::Vector{Float64}}[(
parameters = [11.93318774657433, 18.526382590151794, 1.516646359815506], σ 
= [8.528127147447154, 8.500725240241099, 8.942103960016837]), (parameters =
 [11.298029079905936, 17.939462820652867, 1.5167088885838909], σ = [8.67846
7165745442, 8.503710953184436, 8.318172112742046]), (parameters = [11.28976
0691534978, 18.292554844750534, 1.5304402394629808], σ = [5.836698971590648
, 8.039013556355668, 7.6930688596275]), (parameters = [14.060970705968582, 
18.537623429199108, 1.4775890989462381], σ = [5.722871102871797, 8.44743553
153042, 7.322000474290479]), (parameters = [13.559736173077694, 18.73999062
7167774, 1.4907415845156533], σ = [5.7528539153834135, 8.3463013929304, 7.3
34934900442837]), (parameters = [13.589326642449825, 18.880416527133356, 1.
49511785362385], σ = [5.758753363503688, 8.329766902007833, 7.3160026809054
15]), (parameters = [13.815097893800857, 18.97436422724487, 1.4950922640725
244], σ = [5.7611399080313035, 8.344925882603441, 7.2456027796248]), (param
eters = [13.627617650068581, 18.82662931001996, 1.492500315224656], σ = [5.
762079726844361, 8.364600400263374, 7.1443191589656925]), (parameters = [13
.577901532883999, 18.57020112585152, 1.4842924018038626], σ = [5.7746271345
28725, 8.366266198919522, 7.127212413744965]), (parameters = [13.5478610654
12171, 18.702550470208067, 1.4839249486578883], σ = [5.779916063829454, 8.3
62153266653133, 7.1595838544735])  …  (parameters = [11.284470896069678, 18
.572605579138667, 1.543549611589748], σ = [6.440108692900828, 9.85851165008
6726, 6.483099171565424]), (parameters = [11.445783311117182, 18.3897306462
4198, 1.5207389816964518], σ = [6.590491084837057, 10.337793594990536, 6.68
70365166617916]), (parameters = [11.421233706848618, 18.148664571558772, 1.
5209550005552752], σ = [6.5933804197486925, 10.3406716947717, 6.67955540521
9149]), (parameters = [13.189349000396877, 18.6640712012023, 1.493925232744
7112], σ = [6.595309478233922, 10.301078528287789, 7.034928697213989]), (pa
rameters = [13.004493804704875, 18.630228926676192, 1.4967799836344624], σ 
= [6.574667901865449, 10.314012204395736, 6.987030582152954]), (parameters 
= [13.019664279324699, 18.990793056115685, 1.517606307639064], σ = [6.58808
12037501455, 10.248890951858481, 6.944190081547463]), (parameters = [12.942
998652336206, 18.85791836059732, 1.5142711260579014], σ = [6.58434966777710
8, 10.231541694152618, 7.014509802218115]), (parameters = [12.5173684713677
07, 18.658990864568455, 1.509581014659423], σ = [6.596523555533407, 10.1902
79364474979, 7.217620525070529]), (parameters = [13.549548334702404, 19.257
043061094233, 1.525722561736379], σ = [6.578315277531544, 9.93201887744481,
 8.450020445614037]), (parameters = [13.545078409220269, 19.25741499467326,
 1.5267614685951345], σ = [6.580790046801167, 9.928742050905726, 8.44594236
1090456])], posterior_matrix = [2.4793234046563866 2.4246282927983183 … 2.6
063532135378504 2.6060232643147123; 2.9191958021435767 2.8870029130749253 …
 2.957876867122074 2.9578961810939073; … ; 2.140151482237829 2.140502650949
3417 … 2.295763768313144 2.2954337883510836; 2.190770903780534 2.1184425327
08244 … 2.1341688609654494 2.1336861320852516], tree_statistics = DynamicHM
C.TreeStatisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-325.238159254184, 10, r
eached maximum depth without divergence or turning, 0.996392258185004, 1023
, DynamicHMC.Directions(0x8266e3bc)), DynamicHMC.TreeStatisticsNUTS(-324.70
02430916413, 9, turning at positions -114:397, 0.992473239978731, 511, Dyna
micHMC.Directions(0x9c27958d)), DynamicHMC.TreeStatisticsNUTS(-326.02274178
079807, 10, reached maximum depth without divergence or turning, 0.99646821
12477935, 1023, DynamicHMC.Directions(0x5e34e08b)), DynamicHMC.TreeStatisti
csNUTS(-326.2593012651297, 8, turning at positions 414:445, 0.5777535138919
557, 479, DynamicHMC.Directions(0x65c0ffdd)), DynamicHMC.TreeStatisticsNUTS
(-325.75916363398574, 7, turning at positions -70:-73, 0.9216410295709615, 
163, DynamicHMC.Directions(0x4551715a)), DynamicHMC.TreeStatisticsNUTS(-325
.1556309561799, 6, divergence at position -52, 0.8094230189123023, 80, Dyna
micHMC.Directions(0x51c4359c)), DynamicHMC.TreeStatisticsNUTS(-322.90378742
368716, 6, turning at positions -56:-119, 0.714853402199095, 127, DynamicHM
C.Directions(0xa9e30088)), DynamicHMC.TreeStatisticsNUTS(-324.1088644619041
5, 6, divergence at position -51, 0.8656106295218999, 98, DynamicHMC.Direct
ions(0x3ae2622f)), DynamicHMC.TreeStatisticsNUTS(-324.86306783591516, 6, di
vergence at position 46, 0.8378278716360014, 88, DynamicHMC.Directions(0x35
c68155)), DynamicHMC.TreeStatisticsNUTS(-324.45656936726925, 6, turning at 
positions 87:94, 0.9032037366671766, 127, DynamicHMC.Directions(0xebd2105e)
)  …  DynamicHMC.TreeStatisticsNUTS(-331.93103608019356, 6, turning at posi
tions -36:27, 0.9924495112166255, 63, DynamicHMC.Directions(0xaeeab99b)), D
ynamicHMC.TreeStatisticsNUTS(-331.14647251216866, 8, divergence at position
 220, 0.684278638045537, 357, DynamicHMC.Directions(0x94ffb376)), DynamicHM
C.TreeStatisticsNUTS(-325.6055649001309, 5, turning at positions 18:49, 0.9
459253891517425, 63, DynamicHMC.Directions(0xcd1598b1)), DynamicHMC.TreeSta
tisticsNUTS(-327.4407068134445, 8, turning at positions -113:-128, 0.950322
6730063562, 335, DynamicHMC.Directions(0x017046cf)), DynamicHMC.TreeStatist
icsNUTS(-329.09217469894446, 7, divergence at position -108, 0.713971857114
4624, 143, DynamicHMC.Directions(0xe9f06b23)), DynamicHMC.TreeStatisticsNUT
S(-324.2323299326903, 5, turning at positions 40:47, 0.8497164004834769, 47
, DynamicHMC.Directions(0xf78ae2bf)), DynamicHMC.TreeStatisticsNUTS(-327.26
590589769546, 7, turning at positions -172:-179, 0.21251423381859058, 247, 
DynamicHMC.Directions(0x20267e44)), DynamicHMC.TreeStatisticsNUTS(-321.7998
3762777573, 7, turning at positions -42:-49, 0.9906718982340966, 135, Dynam
icHMC.Directions(0xd6e7be56)), DynamicHMC.TreeStatisticsNUTS(-322.221676618
40217, 8, turning at positions 316:319, 0.7535312172195258, 343, DynamicHMC
.Directions(0xc5df99e7)), DynamicHMC.TreeStatisticsNUTS(-323.2844232571443,
 4, divergence at position 8, 0.08280136014248485, 19, DynamicHMC.Direction
s(0xbcde7fb4))], logdensities = [-321.93232556940364, -323.47210781921217, 
-322.32386503142874, -324.47077114280904, -321.82835205544234, -321.2144715
094589, -321.8167854474556, -321.83348552421756, -322.8731560369897, -323.0
9061934089374  …  -326.8640849761591, -323.2091052832689, -323.875850443751
6, -322.38108458882954, -322.0175147690097, -320.7366539525423, -321.024873
0565529, -320.39847462301583, -320.4965178073353, -320.5955326928465], κ = 
Gaussian kinetic energy (Diagonal), √diag(M⁻¹): [0.48210764004498113, 0.954
7714457688214, 0.2276640210779706, 0.15929535072903372, 0.13996298929705298
, 0.3793891385945444], ϵ = 0.0011319048551928217)

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 1.10.10
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`
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⌃ [65888b18] ParameterizedFunctions v5.19.0
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  [37e2e46d] LinearAlgebra
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

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  [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`