Fitzhugh-Nagumo Bayesian Parameter Estimation Benchmarks

using DiffEqBayes, BenchmarkTools
using OrdinaryDiffEq, RecursiveArrayTools, Distributions, ParameterizedFunctions, StanSample, DynamicHMC
using Plots

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 - v^3/3 -w + l
  dw = τinv*(v +  a - b*w)
end a b τinv l
rameterizedDiffEqFunction#474", Main.##WeaveSandBox#291.var"###Parameterize
dTGradFunction#475", Main.##WeaveSandBox#291.var"###ParameterizedJacobianFu
nction#476", Nothing, Nothing, ModelingToolkit.ODESystem}) (generic functio
n with 1 method)
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: 14-element Vector{Float64}:
u: 14-element Vector{Vector{Float64}}:
 [1.0, 1.0]
 [1.0242787914016627, 1.0109527801835287]
 [1.092538282536039, 1.0495725586393927]
 [1.147894455050522, 1.1102123746508352]
 [1.134543879398286, 1.1975474699280848]
 [1.043276210840273, 1.271868870768599]
 [0.8446920608722805, 1.3381007150055888]
 [0.31354431600484334, 1.3689380066234902]
 [-0.40983461431568685, 1.342759553988513]
 [-1.4082542877988236, 1.2706202676829645]
 [-1.9097832365284388, 1.1563319270772914]
 [-1.9618464462657994, 1.068871311583418]
 [-1.9544223952589859, 0.9966726802665666]
 [-1.953862989878742, 0.9942458207861209]

Data is genereated by adding noise to the solution obtained above.

t = collect(range(1,stop=10,length=10))
sig = 0.20
data = convert(Array, VectorOfArray([(sol(t[i]) + sig*randn(2)) for i in 1:length(t)]))
2×10 Matrix{Float64}:
 0.925003  1.30003   1.3525   0.771624  …  -1.02135  -2.10478   -2.27248
 1.15794   0.945443  1.41933  1.52572       1.1178    0.863596   1.25334

Plot of the data and the solution.

scatter(t, data[1,:])
scatter!(t, data[2,:])

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}}:
 Truncated(Distributions.Normal{Float64}(μ=1.0, σ=0.5); lower=0.0, upper=1.
 Truncated(Distributions.Normal{Float64}(μ=1.0, σ=0.5); lower=0.0, upper=1.
 Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0, upper=0.
 Truncated(Distributions.Normal{Float64}(μ=0.5, σ=0.5); lower=0.0, upper=1.


Stan.jl backend

@btime bayesian_result_stan = stan_inference(prob_ode_fitzhughnagumo,t,data,priors; print_summary=false, vars=(DiffEqBayes.StanODEData(), InverseGamma(2, 3)))
Error: IOError: cd(""): no such file or directory (ENOENT)

Turing.jl backend

@btime bayesian_result_turing = turing_inference(prob_ode_fitzhughnagumo,Tsit5(),t,data,priors;num_samples = 10_000)
45.500 s (374329729 allocations: 40.83 GiB)
Chains MCMC chain (10000×17×1 Array{Float64, 3}):

Iterations        = 1001:1:11000
Number of chains  = 1
Samples per chain = 10000
Wall duration     = 45.38 seconds
Compute duration  = 45.38 seconds
parameters        = theta[1], theta[2], theta[3], theta[4], σ[1]
internals         = lp, n_steps, is_accept, acceptance_rate, log_density, h
amiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, 
tree_depth, numerical_error, step_size, nom_step_size

Summary Statistics
  parameters      mean       std   naive_se      mcse         ess      rhat
      Symbol   Float64   Float64    Float64   Float64     Float64   Float64

    theta[1]    0.9568    0.3165     0.0032    0.0048   3954.4668    1.0003
    theta[2]    0.8886    0.3162     0.0032    0.0051   3987.4750    0.9999
    theta[3]    0.0626    0.0336     0.0003    0.0006   2872.4762    1.0002
    theta[4]    0.4575    0.0755     0.0008    0.0012   3202.4666    1.0000
        σ[1]    0.3125    0.0633     0.0006    0.0010   3514.1320    0.9999
                                                                1 column om

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

    theta[1]    0.2920    0.7433    0.9861    1.2035    1.4614
    theta[2]    0.2227    0.6729    0.9079    1.1297    1.4301
    theta[3]    0.0148    0.0381    0.0567    0.0804    0.1452
    theta[4]    0.3291    0.4049    0.4499    0.5022    0.6271
        σ[1]    0.2168    0.2675    0.3026    0.3467    0.4623


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.


These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization

To locally run this benchmark, do the following commands:

using SciMLBenchmarks

Computer Information:

Julia Version 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
  BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

      Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/BayesianInference/Project.toml`
  [6e4b80f9] BenchmarkTools v1.3.1
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  [31c24e10] Distributions v0.25.67
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  [65888b18] ParameterizedFunctions v5.13.2
  [91a5bcdd] Plots v1.31.7
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  [31c91b34] SciMLBenchmarks v0.1.1
  [c1514b29] StanSample v6.9.4
  [fce5fe82] Turing v0.21.10

And the full manifest:

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  [aacddb02] JpegTurbo_jll v2.1.2+0
  [c1c5ebd0] LAME_jll v3.100.1+0
  [88015f11] LERC_jll v3.0.0+1
  [dd4b983a] LZO_jll v2.10.1+0
  [e9f186c6] Libffi_jll v3.2.2+1
  [d4300ac3] Libgcrypt_jll v1.8.7+0
  [7e76a0d4] Libglvnd_jll v1.3.0+3
  [7add5ba3] Libgpg_error_jll v1.42.0+0
  [94ce4f54] Libiconv_jll v1.16.1+1
  [4b2f31a3] Libmount_jll v2.35.0+0
  [89763e89] Libtiff_jll v4.4.0+0
  [38a345b3] Libuuid_jll v2.36.0+0
  [856f044c] MKL_jll v2022.0.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
  [458c3c95] OpenSSL_jll v1.1.17+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [2f80f16e] PCRE_jll v8.44.0+0
  [30392449] Pixman_jll v0.40.1+0
  [ea2cea3b] Qt5Base_jll v5.15.3+1
  [f50d1b31] Rmath_jll v0.3.0+0
  [a2964d1f] Wayland_jll v1.19.0+0
  [2381bf8a] Wayland_protocols_jll v1.25.0+0
  [02c8fc9c] XML2_jll v2.9.14+0
  [aed1982a] XSLT_jll v1.1.34+0
  [4f6342f7] Xorg_libX11_jll v1.6.9+4
  [0c0b7dd1] Xorg_libXau_jll v1.0.9+4
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.3+4
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
  [c7cfdc94] Xorg_libxcb_jll v1.13.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.0+4
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.2+4
  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
  [c5fb5394] Xorg_xtrans_jll v1.4.0+3
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  [3161d3a3] Zstd_jll v1.5.2+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [b53b4c65] libpng_jll v1.6.38+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
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  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.8.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
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  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v0.5.2+0
  [deac9b47] LibCURL_jll v7.81.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.0+0
  [14a3606d] MozillaCACerts_jll v2022.2.1
  [4536629a] OpenBLAS_jll v0.3.20+0
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.40.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+0
  [83775a58] Zlib_jll v1.2.12+3
  [8e850b90] libblastrampoline_jll v5.1.0+0
  [8e850ede] nghttp2_jll v1.41.0+1
  [3f19e933] p7zip_jll v17.4.0+0