Fitzhugh-Nagumo Bayesian Parameter Estimation Benchmarks
using DiffEqBayes, BenchmarkToolsusing OrdinaryDiffEq, RecursiveArrayTools, Distributions, ParameterizedFunctions,
StanSample, DynamicHMC
using Plots, StaticArrays, Turing, LinearAlgebra"""Print elapsed sampling time and the Turing chain."""
function display_ess_per_sec(chain, elapsed)
println("Elapsed time: $(round(elapsed; digits=2)) seconds")
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()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, ModelingToolkitBase.System}
(Main.var"##WeaveSandBox#277".var"##ParameterizedDiffEqFunction#279", Linea
rAlgebra.UniformScaling{Bool}(true), nothing, Main.var"##WeaveSandBox#277".
var"##ParameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"##P
arameterizedJacobianFunction#281", nothing, nothing, nothing, nothing, noth
ing, 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]
[1.1525604472298032, 1.1092965960073389]
[1.1446577625483758, 1.1952738138449215]
[1.0557695077719014, 1.2718985818139574]
[0.8659598744812583, 1.3388184800875969]
[0.3675854021172527, 1.3735376018319738]
[-0.3594427955481846, 1.3493319650351674]
[-1.3772889489189293, 1.2781711184359077]
[-1.905699839713037, 1.1680023987534751]
[-1.9707492736430974, 1.0777291565175875]
[-1.9650453438870346, 1.0031251492628281]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]
[1.1525604472298032, 1.1092965960073389]
[1.1446577625483758, 1.1952738138449215]
[1.0557695077719014, 1.2718985818139574]
[0.8659598744812583, 1.3388184800875969]
[0.3675854021172527, 1.3735376018319738]
[-0.3594427955481846, 1.3493319650351674]
[-1.3772889489189293, 1.2781711184359077]
[-1.905699839713037, 1.1680023987534751]
[-1.9707492736430974, 1.0777291565175875]
[-1.9650453438870346, 1.0031251492628281]Data is generated by adding noise to the solution obtained above.
t = collect(range(1, stop = 10, length = 10))
sig = 0.20
data = convert(Array, VectorOfArray([(sol(t[i]) + sig*randn(2)) for i in 1:length(t)]))2×10 Matrix{Float64}:
1.14221 1.21014 1.16264 0.753106 … -1.39607 -2.19142 -1.96639
0.884597 1.27066 1.4347 0.992987 1.22151 1.48364 1.22042Plot 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)))77.813261 seconds (4.89 M allocations: 239.380 MiB, 0.25% gc time, 4.85% c
ompilation time)
112.749641 seconds (34.54 M allocations: 1.705 GiB, 0.45% gc time, 14.56% c
ompilation time: 2% of which was recompilation)
10000×6 DataFrame
Row │ sigma1.1 sigma1.2 theta_1 theta_2 theta_3 theta_4
│ Float64 Float64 Float64 Float64 Float64 Float64
───────┼─────────────────────────────────────────────────────────────
1 │ 0.365038 0.380852 1.29242 0.308144 0.034126 0.494373
2 │ 0.86269 0.326737 1.04728 0.964062 0.0469634 0.421378
3 │ 0.347589 0.293769 0.861474 0.371166 0.0608809 0.590995
4 │ 0.416714 0.324897 0.332552 0.423464 0.0986979 0.58639
5 │ 0.321029 0.384109 0.901959 0.732185 0.0506977 0.45285
6 │ 0.349643 0.314845 1.39298 0.889437 0.0639647 0.584447
7 │ 0.377891 0.393565 1.3832 0.973481 0.0932183 0.726972
8 │ 0.39721 0.353124 0.509067 0.555115 0.0476631 0.411749
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
9994 │ 0.264968 0.300227 0.969477 0.724397 0.0677749 0.535409
9995 │ 0.454213 0.475924 0.795519 0.706114 0.0834425 0.592522
9996 │ 0.418423 0.512727 0.933985 0.913035 0.0725448 0.473902
9997 │ 0.426785 0.332422 0.846243 0.585637 0.060041 0.530267
9998 │ 0.467653 0.321215 0.745938 0.73412 0.0715218 0.500027
9999 │ 0.411892 0.258825 1.21189 0.663628 0.0661827 0.602318
10000 │ 0.558332 0.359402 1.17441 1.21245 0.0746187 0.481378
9985 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: 5.736 seconds (Warm-up)
68.293 seconds (Sampling)
74.029 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 axes(data, 2)
data[:, i] ~ _mvn(predicted(t[i]), σ)
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)
chain╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter = 1001:11000
│
│ → chain = 1:1
│
│
│
│ Parameters (5) ── AbstractPPL.VarName
│
│ Vector{Float64} σ (2,)
│
│ Float64 a, b, τinv, l
│
│
│
│ 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, elapsed_turing_direct)Elapsed time: 101.39 seconds
╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter = 1001:11000
│
│ → chain = 1:1
│
│
│
│ Parameters (5) ── AbstractPPL.VarName
│
│ Vector{Float64} σ (2,)
│
│ Float64 a, b, τinv, l
│
│
│
│ 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
│
╰──────────────────────────────────────────────────────────────────────────
────╯Turing.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) -> _mvn(u, σ),
likelihood_dist_priors = [InverseGamma(2, 3), InverseGamma(2, 3)])145.064 s (740383443 allocations: 44.52 GiB)
╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter = 1001:11000
│
│ → chain = 1:1
│
│
│
│ Parameters (5) ── AbstractPPL.VarName
│
│ Float64 theta[1], theta[2], theta[3], theta[4]
│
│ Vector{Float64} σ (2,)
│
│
│
│ 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
│
╰──────────────────────────────────────────────────────────────────────────
────╯Conclusion
FitzHugh-Ngumo is a standard problem for parameter estimation studies. In the FitzHugh-Nagumo model the parameters to be estimated were [0.7,0.8,0.08,0.5]. dynamichmc_inference has issues with the model and hence was excluded from this benchmark.
Appendix
These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.
To locally run this benchmark, do the following commands:
using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/BayesianInference","DiffEqBayesFitzHughNagumo.jmd")Computer Information:
Julia Version 1.12.7
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 v3.16.1
[459566f4] DiffEqCallbacks v4.19.3
[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 but compatibility constraints restrict them 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`
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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`