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"""Print elapsed sampling time, allocated bytes, and the Turing chain."""
function display_ess_per_sec(chain, timed)
println("Elapsed time: $(round(timed.time; digits=2)) seconds")
println("Allocated: $(round(timed.bytes / 2^30; digits=3)) GiB ($(timed.bytes) bytes)")
display(chain)
end
# Distributions requires a vector mean; sol[:, i] can be a scalar after ODE 7.
_mvn(u, σ) = MvNormal(
let n = length(σ)
μ = u isa AbstractVector ? collect(float.(u)) : [float(u)]
length(μ) == n ? μ : fill(μ[1], n)
end,
Diagonal(σ .^ 2),
)
"""Extract and display Stan's internal timing from its CSV output files."""
function display_stan_timing(stan_result)
sample_files = stan_result.model.sample_file
for (chain_idx, f) in enumerate(sample_files)
isfile(f) || continue
lines = readlines(f)
println("Chain $chain_idx timing (from Stan CSV):")
for line in lines
if startswith(line, "#") && occursin("Elapsed Time", line)
println(" ", strip(line[2:end]))
elseif startswith(line, "#") && occursin("seconds", line)
println(" ", strip(line[2:end]))
end
end
end
endMain.var"##WeaveSandBox#277".display_stan_timinggr(fmt = :png)Plots.GRBackend()Initializing the problem
g1 = @ode_def LorenzExample begin
dx = σ*(y-x)
dy = x*(ρ-z) - y
dz = x*y - β*z
end σ ρ βMain.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, ModelingToolkitBase.System}(
Main.var"##WeaveSandBox#277".var"##ParameterizedDiffEqFunction#279", Linear
Algebra.UniformScaling{Bool}(true), nothing, Main.var"##WeaveSandBox#277".v
ar"##ParameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"##Pa
rameterizedJacobianFunction#281", nothing, nothing, nothing, nothing, nothi
ng, 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
2.66prob = 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}:
-9.69271 -8.02846 -8.40063 -9.39594 … 16.5353 -4.45959 -13.5884
-9.58049 -8.91238 -7.65649 -11.2989 23.1859 -4.72553 -14.5461
28.4096 25.3779 28.2925 26.6266 30.1962 22.253 30.6405Plots 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 (slightly above Stan's default of 0.8) consistently across the Stan and Turing 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)))14359.102935 seconds (4.89 M allocations: 239.199 MiB, 0.00% gc time, 0.03%
compilation time)
14388.879378 seconds (33.98 M allocations: 1.663 GiB, 0.00% gc time, 0.11%
compilation time: 2% of which was recompilation)
1000×6 DataFrame
Row │ sigma1.1 sigma1.2 sigma1.3 theta_1 theta_2 theta_3
│ Float64 Float64 Float64 Float64 Float64 Float64
──────┼─────────────────────────────────────────────────────────
1 │ 6.22845 11.3498 7.93824 7.62997 23.6666 2.03611
2 │ 6.14876 11.1648 8.01161 7.61103 23.7056 2.03618
3 │ 6.22808 11.8151 8.1129 7.70348 23.6005 2.03676
4 │ 6.37299 11.428 8.07464 7.74501 23.6322 2.0398
5 │ 6.21385 12.0494 8.09854 7.70777 23.582 2.036
6 │ 6.47193 12.3461 7.87264 7.71413 23.664 2.04083
7 │ 6.57056 12.2662 8.01327 7.67235 23.6508 2.03829
8 │ 6.73797 12.468 8.2567 7.73241 23.6403 2.04027
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
994 │ 6.15482 11.0841 8.16703 7.12751 24.7464 1.9131
995 │ 6.08791 11.2688 8.11591 7.17329 24.6934 1.90428
996 │ 6.22838 11.2009 7.80523 7.17969 24.6423 1.89836
997 │ 6.12579 11.4956 7.64484 7.24835 24.5844 1.89578
998 │ 5.80987 11.9855 7.58341 7.3127 24.5185 1.89522
999 │ 5.73001 11.7633 7.67404 7.33637 24.5018 1.89603
1000 │ 5.76831 11.3105 7.62109 7.30038 24.5155 1.89393
985 rows omittedStan's internal timing (excluding data serialization and CSV parsing):
display_stan_timing(bayesian_result_stan)Chain 1 timing (from Stan CSV):
Elapsed Time: 4308.66 seconds (Warm-up)
10046.7 seconds (Sampling)
14355.3 seconds (Total)Direct Turing.jl
We use per-dimension noise parameters (matching Stan) with InverseGamma(2, 3) priors on each σ. The model solves the same static-array problem sprob with Vern9 at reltol = abstol = 1e-8 that the DiffEqBayes Turing backend below is given, so the two runs differ only in how the Turing model is written.
@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, reltol = 1e-8, abstol = 1e-8)
# Observations.
for i in axes(data, 2)
data[:, i] ~ _mvn(predicted(t[i]), σ)
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)
timed_turing_direct = @timed chain = sample(model, Turing.NUTS(0.85), 10_000; progress = false)
chain╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter = 1001:11000
│
│ → chain = 1:1
│
│
│
│ Parameters (4) ── AbstractPPL.VarName
│
│ Vector{Float64} σ (3,)
│
│ Float64 σ_param, ρ, β
│
│
│
│ Extras (14)
│
│ Int64 n_steps, tree_depth
│
│ Bool is_accept, numerical_error
│
│ Float64 acceptance_rate, log_density, hamiltonian_energy,
│
│ hamiltonian_energy_error, max_hamiltonian_energy_error, step_si
ze, │
│ nom_step_size, logprior, loglikelihood, logjoint
│
╰──────────────────────────────────────────────────────────────────────────
────╯display_ess_per_sec(chain, timed_turing_direct)Elapsed time: 3069.24 seconds
Allocated: 421.822 GiB (452927759664 bytes)
╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter = 1001:11000
│
│ → chain = 1:1
│
│
│
│ Parameters (4) ── AbstractPPL.VarName
│
│ Vector{Float64} σ (3,)
│
│ Float64 σ_param, ρ, β
│
│
│
│ Extras (14)
│
│ Int64 n_steps, tree_depth
│
│ Bool is_accept, numerical_error
│
│ Float64 acceptance_rate, log_density, hamiltonian_energy,
│
│ hamiltonian_energy_error, max_hamiltonian_energy_error, step_si
ze, │
│ nom_step_size, logprior, loglikelihood, logjoint
│
╰──────────────────────────────────────────────────────────────────────────
────╯Using Turing.jl backend
# DiffEqBayes captures `likelihood` inside its Turing model, so both calls below
# must pass the same callable for the warmup to compile the timed specialization.
lorenz_likelihood(u, p, t, σ) = _mvn(u, σ)
# Warmup run to compile all code paths before timing, matching the direct model above
turing_inference(
sprob, Vern9(), t, data, priors;
sample_args = (sampler = Turing.NUTS(0.85), num_samples = 10),
solve_kwargs = Dict(:reltol => 1e-8, :abstol => 1e-8),
likelihood = lorenz_likelihood,
likelihood_dist_priors = [InverseGamma(2, 3), InverseGamma(2, 3), InverseGamma(2, 3)])
timed_turing_backend = @timed bayesian_result_turing = turing_inference(
sprob, 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 = lorenz_likelihood,
likelihood_dist_priors = [InverseGamma(2, 3), InverseGamma(2, 3), InverseGamma(2, 3)])
bayesian_result_turing╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter = 1001:11000
│
│ → chain = 1:1
│
│
│
│ Parameters (4) ── AbstractPPL.VarName
│
│ Float64 theta[1], theta[2], theta[3]
│
│ Vector{Float64} σ (3,)
│
│
│
│ Extras (14)
│
│ Int64 n_steps, tree_depth
│
│ Bool is_accept, numerical_error
│
│ Float64 acceptance_rate, log_density, hamiltonian_energy,
│
│ hamiltonian_energy_error, max_hamiltonian_energy_error, step_si
ze, │
│ nom_step_size, logprior, loglikelihood, logjoint
│
╰──────────────────────────────────────────────────────────────────────────
────╯The direct model above and this backend solve the same problem with the same sampler settings, and each timed run follows a 10-draw warmup that compiles its sampling path. The divergent-transition counts printed below show whether the two samplers took similar paths through the posterior; on this chaotic problem the counts can differ by orders of magnitude, in which case the wall times are not like-for-like measurements of the same trajectory:
println("Direct Turing model: $(round(timed_turing_direct.time; digits=2)) s, ",
"$(round(timed_turing_direct.bytes / 2^30; digits=3)) GiB allocated, ",
sum(chain[:numerical_error]), " divergent transitions")
println("DiffEqBayes backend: $(round(timed_turing_backend.time; digits=2)) s, ",
"$(round(timed_turing_backend.bytes / 2^30; digits=3)) GiB allocated, ",
sum(bayesian_result_turing[:numerical_error]), " divergent transitions")
println("Backend / direct: $(round(timed_turing_backend.time / timed_turing_direct.time; sigdigits=2))x time, ",
"$(round(timed_turing_backend.bytes / timed_turing_direct.bytes; sigdigits=2))x allocated")Direct Turing model: 3069.24 s, 421.822 GiB allocated, 857 divergent transi
tions
DiffEqBayes backend: 8074.22 s, 1162.961 GiB allocated, 11 divergent transi
tions
Backend / direct: 2.6x time, 2.8x allocatedUsing DynamicHMC.jl backend
@time bayesian_result_dynamichmc = dynamichmc_inference(
prob, Tsit5(), t, data, priors; solve_kwargs = (reltol = 1e-8, abstol = 1e-8))552.760865 seconds (192.04 M allocations: 18.854 GiB, 2.71% gc time, 1.97%
compilation time)
(posterior = [(parameters = [4.697752598649291, 21.965158185371006, 1.66854
5496743354], σ = [7.38342337831187, 7.279218639528952, 7.49043016730585]),
(parameters = [4.8447217940002, 21.851684269236817, 1.677614674986058], σ =
[7.515119161835431, 7.666763868641636, 7.235578776691011]), (parameters =
[4.72844711597368, 22.053166846467914, 1.6844596774003886], σ = [6.69153896
8484936, 8.50920271469932, 7.4019417612551415]), (parameters = [4.610673872
6984915, 21.980426578719815, 1.6616745831549438], σ = [6.741548287598121, 8
.082835562510041, 7.477372144722086]), (parameters = [4.641712498550828, 22
.059198236208545, 1.6698259274183866], σ = [6.77946654910855, 8.25430418789
069, 7.517424979054671]), (parameters = [4.591387004759307, 22.249994628645
176, 1.679098084064125], σ = [6.900853318784801, 8.427929976493846, 7.84045
5141984519]), (parameters = [4.588523224915479, 22.228312023175818, 1.68067
68538429633], σ = [6.893433931215722, 8.403488968344663, 7.888710187858041]
), (parameters = [4.595355824016725, 22.225184440023494, 1.6797182621140858
], σ = [7.3484658606253825, 8.120965914693516, 7.544803977165014]), (parame
ters = [4.569380963431411, 22.268867046232373, 1.6789094793840016], σ = [7.
658753199201719, 8.790551128001965, 6.767066768762526]), (parameters = [4.5
20059073011995, 22.26273733771215, 1.6654758328504073], σ = [7.640641475205
707, 8.716136564355931, 6.641912643824628]) … (parameters = [7.0343373785
40341, 24.678074566822954, 1.9266968352946447], σ = [6.138547976416683, 10.
013303060268338, 7.7116240557213676]), (parameters = [7.034337378540341, 24
.678074566822954, 1.9266968352946447], σ = [6.138547976416683, 10.013303060
268338, 7.7116240557213676]), (parameters = [7.034337378540341, 24.67807456
6822954, 1.9266968352946447], σ = [6.138547976416683, 10.013303060268338, 7
.7116240557213676]), (parameters = [7.034337378540341, 24.678074566822954,
1.9266968352946447], σ = [6.138547976416683, 10.013303060268338, 7.71162405
57213676]), (parameters = [7.034337378540341, 24.678074566822954, 1.9266968
352946447], σ = [6.138547976416683, 10.013303060268338, 7.7116240557213676]
), (parameters = [7.034337378540341, 24.678074566822954, 1.9266968352946447
], σ = [6.138547976416683, 10.013303060268338, 7.7116240557213676]), (param
eters = [7.034337378540341, 24.678074566822954, 1.9266968352946447], σ = [6
.138547976416683, 10.013303060268338, 7.7116240557213676]), (parameters = [
7.034337378540341, 24.678074566822954, 1.9266968352946447], σ = [6.13854797
6416683, 10.013303060268338, 7.7116240557213676]), (parameters = [7.0343373
78540341, 24.678074566822954, 1.9266968352946447], σ = [6.138547976416683,
10.013303060268338, 7.7116240557213676]), (parameters = [7.034337378540341,
24.678074566822954, 1.9266968352946447], σ = [6.138547976416683, 10.013303
060268338, 7.7116240557213676])], posterior_matrix = [1.5470842238558913 1.
5778898224350772 … 1.9508034968680537 1.9508034968680537; 3.089457479103199
7 3.0842780018311036 … 3.2059151800627874 3.2059151800627874; … ; 1.9850235
267133123 2.0368946056991426 … 2.3039145149477886 2.3039145149477886; 2.013
626228097031 1.979010353662177 … 2.042728808158583 2.042728808158583], tree
_statistics = DynamicHMC.TreeStatisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-
331.4606859960974, 8, turning at positions -29:226, 0.5641280623838211, 255
, DynamicHMC.Directions(0x92a8a9e2)), DynamicHMC.TreeStatisticsNUTS(-332.69
184820577, 7, turning at positions -109:18, 0.23081770983679023, 127, Dynam
icHMC.Directions(0x2799a112)), DynamicHMC.TreeStatisticsNUTS(-334.188516985
37367, 8, turning at positions -219:36, 0.23865014167908932, 255, DynamicHM
C.Directions(0xfff96c24)), DynamicHMC.TreeStatisticsNUTS(-331.5473127600112
, 5, turning at positions 11:26, 0.4735696149187773, 47, DynamicHMC.Directi
ons(0x509710aa)), DynamicHMC.TreeStatisticsNUTS(-331.41884477347503, 4, tur
ning at positions -16:-19, 0.9001467977374731, 27, DynamicHMC.Directions(0x
e4fb7ee8)), DynamicHMC.TreeStatisticsNUTS(-328.8472797151834, 7, turning at
positions -65:62, 0.8273723615433192, 127, DynamicHMC.Directions(0x9157bbb
e)), DynamicHMC.TreeStatisticsNUTS(-329.6475831664968, 6, turning at positi
ons 52:59, 0.06871672573780037, 119, DynamicHMC.Directions(0x8c1394c3)), Dy
namicHMC.TreeStatisticsNUTS(-331.4501137898888, 5, turning at positions -5:
-12, 0.8348336469361366, 39, DynamicHMC.Directions(0xaeae589b)), DynamicHMC
.TreeStatisticsNUTS(-332.790278943632, 6, turning at positions 99:114, 0.21
671540769815023, 127, DynamicHMC.Directions(0x81db13f2)), DynamicHMC.TreeSt
atisticsNUTS(-333.2970478949459, 5, turning at positions -24:-31, 0.5649023
47787276, 55, DynamicHMC.Directions(0x67b83b98)) … DynamicHMC.TreeStatist
icsNUTS(-320.5541735981091, 0, divergence at position 1, 0.0, 1, DynamicHMC
.Directions(0x6bcaa175)), DynamicHMC.TreeStatisticsNUTS(-321.7296532679743,
5, divergence at position 11, 2.539905238151943e-7, 33, DynamicHMC.Directi
ons(0x58641b69)), DynamicHMC.TreeStatisticsNUTS(-321.00408050630847, 5, tur
ning at positions 48:55, 1.5491165359956021e-9, 55, DynamicHMC.Directions(0
x5610ca7f)), DynamicHMC.TreeStatisticsNUTS(-320.67572271574824, 3, turning
at positions -3:4, 8.006592193007575e-9, 7, DynamicHMC.Directions(0x5bb6a91
4)), DynamicHMC.TreeStatisticsNUTS(-321.88608786821953, 1, divergence at po
sition -3, 3.553314745651557e-6, 3, DynamicHMC.Directions(0x7f4b6b34)), Dyn
amicHMC.TreeStatisticsNUTS(-322.48512912013155, 1, divergence at position 2
, 4.650629786703409e-225, 2, DynamicHMC.Directions(0x0eaf8e53)), DynamicHMC
.TreeStatisticsNUTS(-324.78246829459687, 5, divergence at position 8, 3.505
232012671796e-5, 36, DynamicHMC.Directions(0xa2699fa3)), DynamicHMC.TreeSta
tisticsNUTS(-321.9651183930931, 2, divergence at position 6, 4.963652115343
212e-5, 7, DynamicHMC.Directions(0x6657813e)), DynamicHMC.TreeStatisticsNUT
S(-322.46140374468706, 1, divergence at position 2, 4.935109771049365e-86,
2, DynamicHMC.Directions(0xb2a4700f)), DynamicHMC.TreeStatisticsNUTS(-325.9
797784737861, 3, divergence at position 12, 3.994328022746235e-8, 13, Dynam
icHMC.Directions(0x297cdf3e))], logdensities = [-328.2592029536249, -331.06
65774417257, -329.0949050284289, -330.5291256867349, -327.49121407317045, -
326.7123645127912, -328.2432218449867, -327.5508405484817, -327.94604513881
796, -329.37635112122155 … -319.7056650444975, -319.7056650444975, -319.7
056650444975, -319.7056650444975, -319.7056650444975, -319.7056650444975, -
319.7056650444975, -319.7056650444975, -319.7056650444975, -319.70566504449
75], κ = Gaussian kinetic energy (Diagonal), √diag(M⁻¹): [0.100977083630159
27, 0.03960323315132634, 0.03765180770706258, 0.12298383144547288, 0.150805
44558210104, 0.12709635400384847], ϵ = 0.016337880637118996)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 v3.16.2
⌃ [459566f4] DiffEqCallbacks v4.19.4
[31c24e10] Distributions v0.25.131
[bbc10e6e] DynamicHMC v3.6.1
[1dea7af3] OrdinaryDiffEq v7.8.1
[65888b18] ParameterizedFunctions v5.27.0
[91a5bcdd] Plots v1.41.7
⌃ [731186ca] RecursiveArrayTools v4.5.1
[31c91b34] SciMLBenchmarks v0.2.1 [loaded: `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/src/SciMLBenchmarks.jl` (v0.2.1) expected `/home/crackauc/.julia/packages/SciMLBenchmarks/ceJyd/src/SciMLBenchmarks.jl` (v0.2.1)]
[c1514b29] StanSample v7.10.3
⌃ [90137ffa] StaticArrays v1.9.20
⌃ [fce5fe82] Turing v0.46.1
[37e2e46d] LinearAlgebra v1.12.0
Info Packages marked with ⌃ have new versions available and may be upgradable.And the full manifest:
Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/BayesianInference/Manifest.toml`
[47edcb42] ADTypes v1.24.0
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⌃ [79e6a3ab] Adapt v4.7.0
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⌃ [aae01518] BandedMatrices v1.12.0
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⌅ [336ed68f] CSV v0.10.17
⌃ [d360d2e6] ChainRulesCore v1.26.1
[0ca39b1e] Chairmarks v1.3.1
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[c3611d14] ColorVectorSpace v0.11.0
⌃ [5ae59095] Colors v0.13.1
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[f70d9fcc] CommonWorldInvalidations v1.2.2
[34da2185] Compat v4.18.1
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[2e76f6c2] HarfBuzz_jll v100.14004.0+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.2.0+1
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.2.0+0
[1d63c593] LLVMOpenMP_jll v23.1.1+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
⌃ [4b2f31a3] Libmount_jll v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[856f044c] MKL_jll v2025.2.0+0
[e7412a2a] Ogg_jll v1.3.6+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.58.2+0
[30392449] Pixman_jll v0.46.4+0
[c0090381] Qt6Base_jll v6.10.2+2
[629bc702] Qt6Declarative_jll v6.10.2+2
[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[f50d1b31] Rmath_jll v0.5.2+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.4+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.13+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[a51aa0fd] Xorg_libXi_jll v1.8.4+0
[d1454406] Xorg_libXinerama_jll v1.1.7+0
[ec84b674] Xorg_libXrandr_jll v1.5.6+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.2.0+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
⌃ [35661453] Xorg_xkbcomp_jll v1.4.7+0
⌃ [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[3161d3a3] Zstd_jll v1.5.7+1
[35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
⌃ [a4ae2306] libaom_jll v3.14.1+0
[0ac62f75] libass_jll v0.17.5+0
[1183f4f0] libdecor_jll v0.2.2+0
[8e53e030] libdrm_jll v2.4.134+0
[2db6ffa8] libevdev_jll v1.13.4+0
[f638f0a6] libfdk_aac_jll v2.0.4+0
[36db933b] libinput_jll v1.28.1+0
⌃ [b53b4c65] libpng_jll v1.6.58+0
[9a156e7d] libva_jll v2.23.0+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
⌃ [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.13.0+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.12.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.12.1
[de0858da] Printf v1.11.0
[9abbd945] Profile v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.12.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.3.1+2
[deac9b47] LibCURL_jll v8.15.0+0
[e37daf67] LibGit2_jll v1.9.0+0
[29816b5a] LibSSH2_jll v1.11.3+1
[14a3606d] MozillaCACerts_jll v2025.11.4
[4536629a] OpenBLAS_jll v0.3.29+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.6+0
[efcefdf7] PCRE2_jll v10.44.0+1
[bea87d4a] SuiteSparse_jll v7.8.3+2
[83775a58] Zlib_jll v1.3.1+2
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.64.0+1
[3f19e933] p7zip_jll v17.7.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`