Fitzhugh-Nagumo Bayesian Parameter Estimation Benchmarks

using DiffEqBayes, BenchmarkTools
using OrdinaryDiffEq, RecursiveArrayTools, Distributions, ParameterizedFunctions, StanSample, DynamicHMC
using Plots
gr(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 - v^3/3 -w + l
  dw = τinv*(v +  a - b*w)
end a b τinv l
(::Main.##WeaveSandBox#291.FitzhughNagumo{Main.##WeaveSandBox#291.var"###Pa
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}:
  0.0
  0.15079562872319327
  0.6663735500745417
  1.4549121831880751
  2.6341750328847153
  3.787286306555317
  5.149281980854215
  6.7648099160689386
  7.606020767982898
  8.324334020953115
  9.040772533539053
  9.552574456558219
  9.985205763666121
 10.0
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,:])
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}}:
 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

@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
itted

Quantiles
  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

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 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
Environment:
  JULIA_CPU_THREADS = 128
  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
  [ebbdde9d] DiffEqBayes v3.0.0
  [459566f4] DiffEqCallbacks v2.24.0
  [31c24e10] Distributions v0.25.67
  [bbc10e6e] DynamicHMC v3.1.2
  [1dea7af3] OrdinaryDiffEq v6.20.0
  [65888b18] ParameterizedFunctions v5.13.2
  [91a5bcdd] Plots v1.31.7
  [731186ca] RecursiveArrayTools v2.32.0
  [31c91b34] SciMLBenchmarks v0.1.1
  [c1514b29] StanSample v6.9.4
  [fce5fe82] Turing v0.21.10

And the full manifest:

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