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Fitzhugh-Nagumo Bayesian Parameter Estimation Benchmarks

using DiffEqBayes, BenchmarkTools
using OrdinaryDiffEq, RecursiveArrayTools, Distributions, ParameterizedFunctions, StanSample, DynamicHMC
using Plots, StaticArrays, Turing, LinearAlgebra
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 - 0.33*v^3 -w + l
  dw = τinv*(v +  a - b*w)
end a b τinv l
(::Main.var"##WeaveSandBox#225".FitzhughNagumo{Main.var"##WeaveSandBox#225"
.var"###ParameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".var"
###ParameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###Par
ameterizedJacobianFunction#229", Nothing, Nothing, ModelingToolkit.ODESyste
m}) (generic function 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: 13-element Vector{Float64}:
  0.0
  0.1502916178003539
  0.6611859977697417
  1.4391494636342572
  2.5894515498152293
  3.760237603808549
  5.101014094147208
  6.709997158618223
  7.604553280596642
  8.336547620442024
  9.03127910678863
  9.55639994619208
 10.0
u: 13-element Vector{Vector{Float64}}:
 [1.0, 1.0]
 [1.0247192356111163, 1.0109189409610948]
 [1.0944137320832108, 1.0492393331998289]
 [1.1525604499975908, 1.1092966016287371]
 [1.1446577644416096, 1.195273810878899]
 [1.0557695278493895, 1.2718985704582837]
 [0.865959919956831, 1.3388184704641362]
 [0.3675855933126252, 1.3735376027635644]
 [-0.359442563141841, 1.3493319765636338]
 [-1.3772888489033577, 1.2781711287398287]
 [-1.905699772833817, 1.1680024379627088]
 [-1.9707492682163554, 1.0777291974859278]
 [-1.965045343773361, 1.0031251493361766]
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
  0.6611859977697417
  1.4391494636342572
  2.5894515498152293
  3.760237603808549
  5.101014094147208
  6.709997158618223
  7.604553280596642
  8.336547620442024
  9.03127910678863
  9.55639994619208
 10.0
u: 13-element Vector{StaticArraysCore.SVector{2, Float64}}:
 [1.0, 1.0]
 [1.0247192356111163, 1.0109189409610948]
 [1.0944137320832108, 1.0492393331998289]
 [1.1525604499975908, 1.1092966016287371]
 [1.1446577644416096, 1.195273810878899]
 [1.0557695278493895, 1.2718985704582837]
 [0.865959919956831, 1.3388184704641362]
 [0.3675855933126252, 1.3735376027635644]
 [-0.359442563141841, 1.3493319765636338]
 [-1.3772888489033577, 1.2781711287398287]
 [-1.905699772833817, 1.1680024379627088]
 [-1.9707492682163554, 1.0777291974859278]
 [-1.965045343773361, 1.0031251493361766]

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}:
 0.6625   1.1411   0.809948  0.834286  …  -0.888931  -1.95485  -2.24427
 1.16987  1.01141  1.17188   1.34561       1.2748     1.10598   0.988366

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, 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

@time bayesian_result_stan = stan_inference(prob_ode_fitzhughnagumo,t,data,priors; delta = 0.65, num_samples = 10_000, print_summary=false, vars=(DiffEqBayes.StanODEData(), InverseGamma(2, 3)))
23.122573 seconds (2.13 M allocations: 145.103 MiB, 0.30% gc time, 6.91% c
ompilation time)
 46.149503 seconds (6.77 M allocations: 470.665 MiB, 0.38% gc time, 9.51% c
ompilation time: <1% of which was recompilation)
Chains MCMC chain (10000×6×1 Array{Float64, 3}):

Iterations        = 1:1:10000
Number of chains  = 1
Samples per chain = 10000
parameters        = sigma1.1, sigma1.2, theta_1, theta_2, theta_3, theta_4
internals         = 

Summary Statistics
  parameters      mean       std      mcse    ess_bulk    ess_tail      rha
t   ⋯
      Symbol   Float64   Float64   Float64     Float64     Float64   Float6
4   ⋯

    sigma1.1    0.4465    0.1473    0.0111    101.0569     33.7685    1.015
9   ⋯
    sigma1.2    0.3606    0.1099    0.0036    986.5857   1476.4564    1.001
1   ⋯
     theta_1    0.9020    0.3293    0.0196    335.7560   1564.9912    1.004
0   ⋯
     theta_2    0.9067    0.2847    0.0049   3280.5830   4112.6736    1.002
5   ⋯
     theta_3    0.0957    0.0439    0.0010   1764.0137   1399.3717    1.000
8   ⋯
     theta_4    0.5454    0.1005    0.0052    321.1343    672.9208    1.002
0   ⋯
                                                                1 column om
itted

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

    sigma1.1    0.2243    0.3509    0.4235    0.5195    0.7903
    sigma1.2    0.2041    0.2825    0.3440    0.4181    0.6257
     theta_1    0.2291    0.6583    0.9218    1.1626    1.4504
     theta_2    0.2907    0.7322    0.9175    1.1104    1.4133
     theta_3    0.0285    0.0657    0.0896    0.1185    0.1991
     theta_4    0.3821    0.4692    0.5372    0.6073    0.7661

Direct Turing.jl

@model function fitlv(data, prob)

    # Prior distributions.
    σ ~ InverseGamma(2, 3)
    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 Lotka-Volterra model. 
    p = SA[a,b,τinv,l]
    _prob = remake(prob, p = p)
    predicted = solve(_prob, Tsit5(); saveat=t)

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

    return nothing
end

model = fitlv(data, sprob_ode_fitzhughnagumo)

@time chain = sample(model, Turing.NUTS(0.65), 10000; progress=false)
71.711285 seconds (245.10 M allocations: 42.342 GiB, 6.61% gc time, 30.05%
 compilation time)
Chains MCMC chain (10000×17×1 Array{Float64, 3}):

Iterations        = 1001:1:11000
Number of chains  = 1
Samples per chain = 10000
Wall duration     = 63.86 seconds
Compute duration  = 63.86 seconds
parameters        = σ, a, b, τinv, l
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      mcse    ess_bulk    ess_tail      rha
t   ⋯
      Symbol   Float64   Float64   Float64     Float64     Float64   Float6
4   ⋯

           σ    0.3027    0.0602    0.0009   4271.7108   5107.9954    1.000
3   ⋯
           a    0.9475    0.3241    0.0047   4487.1248   3640.7400    1.000
0   ⋯
           b    0.8993    0.2977    0.0043   4509.0980   4239.4574    0.999
9   ⋯
        τinv    0.0815    0.0349    0.0006   3351.7321   3974.2185    1.000
4   ⋯
           l    0.5219    0.0812    0.0013   3865.9606   3931.8080    1.000
2   ⋯
                                                                1 column om
itted

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

           σ    0.2118    0.2611    0.2938    0.3350    0.4430
           a    0.2469    0.7258    0.9783    1.1989    1.4627
           b    0.2746    0.6969    0.9181    1.1213    1.4139
        τinv    0.0272    0.0562    0.0772    0.1021    0.1605
           l    0.3812    0.4653    0.5154    0.5715    0.6997

Turing.jl backend

@time bayesian_result_turing = turing_inference(prob_ode_fitzhughnagumo,Tsit5(),t,data,priors;num_samples = 10_000)
51.189120 seconds (245.93 M allocations: 38.558 GiB, 7.17% gc time, 20.06%
 compilation time)
Chains MCMC chain (10000×17×1 Array{Float64, 3}):

Iterations        = 1001:1:11000
Number of chains  = 1
Samples per chain = 10000
Wall duration     = 48.72 seconds
Compute duration  = 48.72 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      mcse    ess_bulk    ess_tail      rha
t   ⋯
      Symbol   Float64   Float64   Float64     Float64     Float64   Float6
4   ⋯

    theta[1]    0.9492    0.3167    0.0045   4738.3528   4198.0564    1.000
8   ⋯
    theta[2]    0.8994    0.2970    0.0050   3661.5575   4061.6882    0.999
9   ⋯
    theta[3]    0.0815    0.0358    0.0006   2892.5687   3574.1292    1.000
1   ⋯
    theta[4]    0.5205    0.0810    0.0013   3789.2942   4373.2390    0.999
9   ⋯
        σ[1]    0.3010    0.0590    0.0008   5127.0438   5637.4514    1.000
1   ⋯
                                                                1 column om
itted

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

    theta[1]    0.2723    0.7323    0.9807    1.2004    1.4513
    theta[2]    0.2727    0.7002    0.9216    1.1227    1.4075
    theta[3]    0.0264    0.0552    0.0772    0.1020    0.1621
    theta[4]    0.3789    0.4644    0.5140    0.5683    0.6999
        σ[1]    0.2101    0.2589    0.2927    0.3334    0.4387

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 1.10.5
Commit 6f3fdf7b362 (2024-08-27 14:19 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`
⌃ [6e4b80f9] BenchmarkTools v1.3.2
⌃ [ebbdde9d] DiffEqBayes v3.6.0
⌅ [459566f4] DiffEqCallbacks v2.29.1
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⌃ [65888b18] ParameterizedFunctions v5.15.0
⌃ [91a5bcdd] Plots v1.39.0
⌅ [731186ca] RecursiveArrayTools v2.38.7
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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`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.

And the full manifest:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-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`
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