Bacterial Stress CRN SDE Ensemble

Chemical Langevin ensemble of the four-species generalised bacterial stress response network from Utkarsh et al., Comput. Methods Appl. Mech. Eng. 428 (2024) 117109 (arXiv:2304.06835; artifacts, credit Torkel Loman for the Catalyst network). The paper converted a Catalyst SDEProblem through ModelingToolkit into an out-of-place StaticArray form. This notebook uses that CLE directly so the benchmark does not depend on Catalyst / MTK conversion.

Species $(\sigma, A_1, A_2, A_3)$, parameters $(S, D, \tau, v_0, n, \eta)$. Production / degradation pairs:

• \[\emptyset \leftrightarrow \sigma\]

  at $v_0 + \frac{(S\sigma)^n}{(S\sigma)^n + (DA_3)^n + 1}$ and $\sigma$

• \[\emptyset \leftrightarrow A_1\]

  at $\sigma/\tau$ and $A_1/\tau$

• \[\emptyset \leftrightarrow A_2\]

  at $A_1/\tau$ and $A_2/\tau$

• \[\emptyset \leftrightarrow A_3\]

  at $A_2/\tau$ and $A_3/\tau$

Diffusion columns are $\eta\,\nu_j\sqrt{r_j}$. $t \in [0, 1000]$, EM / GPUEM with $dt = 0.1$. Parameter grids are the paper's; $N$ is the number of $S$ and $D$ log-spaced samples (the runner used $N = 2, 4$).

using Printf
using CUDA
using DiffEqGPU, StochasticDiffEq, StaticArrays
using Plots

@assert CUDA.functional() "This benchmark requires a functional CUDA GPU"
println("GPU: ", CUDA.name(CUDA.device()))

const BACKEND = CUDA.CUDABackend()
const KERNEL = EnsembleGPUKernel(BACKEND, 0.0)

gr()

GPU: Tesla V100-PCIE-32GB
Plots.GRBackend()

function crn_f(u, p, t)
    T = eltype(u)
    σ, A1, A2, A3 = u
    S, D, τ, v0, n, η = p
    hill_num = (S * σ)^n
    hill = hill_num / (hill_num + (D * A3)^n + one(T))
    dσ = v0 + hill - σ
    dA1 = (σ - A1) / τ
    dA2 = (A1 - A2) / τ
    dA3 = (A2 - A3) / τ
    return SVector{4, T}(dσ, dA1, dA2, dA3)
end

function crn_g(u, p, t)
    T = eltype(u)
    σ, A1, A2, A3 = u
    S, D, τ, v0, n, η = p
    z = zero(T)
    hill_num = (S * σ)^n
    hill = hill_num / (hill_num + (D * A3)^n + one(T))
    s1 = η * sqrt(max(v0 + hill, z))
    s2 = η * sqrt(max(σ, z))
    s3 = η * sqrt(max(σ / τ, z))
    s4 = η * sqrt(max(A1 / τ, z))
    s5 = η * sqrt(max(A1 / τ, z))
    s6 = η * sqrt(max(A2 / τ, z))
    s7 = η * sqrt(max(A2 / τ, z))
    s8 = η * sqrt(max(A3 / τ, z))
    return SMatrix{4, 8, T}(
        s1, z, z, z,
        -s2, z, z, z,
        z, s3, z, z,
        z, -s4, z, z,
        z, z, s5, z,
        z, z, -s6, z,
        z, z, z, s7,
        z, z, z, -s8
    )
end

function crn_parameters(N; T::Type = Float32)
    S_grid = T.(10 .^ range(T(-1), stop = T(2), length = max(N, 2))[1:N])
    D_grid = T.(10 .^ range(T(-1), stop = T(2), length = max(N, 2))[1:N])
    τ_grid = T[0.1, 0.15, 0.20, 0.30, 0.50, 0.75, 1.0, 1.5, 2.0, 3.0, 5.0, 7.50,
               10.0, 15.0, 20.0, 30.0, 50.0, 75.0, 100.0][1:2:19]
    v0_grid = T[0.01, 0.02, 0.03, 0.05, 0.075, 0.1, 0.15, 0.20]
    n_grid = T[2.0, 3.0, 4.0]
    η_grid = T[0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1]
    return collect(Iterators.product(S_grid, D_grid, τ_grid, v0_grid, n_grid, η_grid))
end

function make_crn_ensemble(parameters; T::Type = Float32)
    u0 = SVector{4, T}(T(0.1), T(0.1), T(0.1), T(0.1))
    tspan = (zero(T), T(1000))
    p0 = SVector{6, T}(T(2.3), T(5), T(10), T(0.1), T(3), T(0.1))
    g0 = zeros(SMatrix{4, 8, T})
    prob = SDEProblem{false}(crn_f, crn_g, u0, tspan, p0; noise_rate_prototype = g0)
    function prob_func(prob, ctx)
        pi = parameters[ctx.sim_id]
        remake(prob;
            p = SVector{6, T}(pi[1], pi[2], pi[3], pi[4], pi[5], pi[6]),
            u0 = SVector{4, T}(pi[4], pi[4], pi[4], pi[4]))
    end
    return EnsembleProblem(prob, prob_func = prob_func, safetycopy = false)
end

function min_seconds(fn; warmup = 1, samples = 3)
    for _ in 1:warmup
        fn()
    end
    ts = Vector{Float64}(undef, samples)
    for i in 1:samples
        ts[i] = @elapsed fn()
    end
    return minimum(ts)
end

min_seconds (generic function with 1 method)

let
    ps = crn_parameters(2)
    ens = make_crn_ensemble(ps)
    sol = solve(ens, GPUEM(), KERNEL; trajectories = 2, save_everystep = false,
        adaptive = false, dt = 0.1f0)
    println("CRN smoke: 2 trajectories, u[1][end] = ", sol.u[1].u[end])
    @assert all(i -> length(sol.u[i].u[end]) == 4, 1:2)
end

CRN smoke: 2 trajectories, u[1][end] = Float32[0.009919406, 0.010128644, 0.
010087123, 0.010349157]

const NS = [2, 4]
ntraj = Int[]
t_gpu = Float64[]
t_cpu = Float64[]

for N in NS
    ps = crn_parameters(N)
    n = length(ps)
    push!(ntraj, n)
    @info "crn gpu" N n
    ens = make_crn_ensemble(ps)
    push!(t_gpu,
        min_seconds(() -> (CUDA.@sync solve(ens, GPUEM(), KERNEL; trajectories = n,
            save_everystep = false, adaptive = false, dt = 0.1f0); nothing)))
    @info "crn cpu" N n
    ens64 = make_crn_ensemble(crn_parameters(N; T = Float64); T = Float64)
    push!(t_cpu,
        min_seconds(() -> (solve(ens64, EM(), EnsembleThreads(); trajectories = n,
            save_everystep = false, adaptive = false, dt = 0.1); nothing);
            samples = 2))
end

p = plot(ntraj, t_gpu .* 1e3; xscale = :log10, yscale = :log10,
    xlabel = "trajectories", ylabel = "time (ms)",
    label = "GPUEM + EnsembleGPUKernel", marker = :circle, legend = :topleft,
    title = "CRN SDE ensemble, EM dt = 0.1")
plot!(p, ntraj, t_cpu .* 1e3; label = "EM + EnsembleThreads", marker = :utriangle)
p

println("CRN SDE (ms)")
@printf("%6s %10s %12s %12s\n", "N", "traj", "GPU", "CPU")
for (i, N) in enumerate(NS)
    @printf("%6d %10d %12.3f %12.3f\n", N, ntraj[i], t_gpu[i] * 1e3, t_cpu[i] * 1e3)
end

CRN SDE (ms)
     N       traj          GPU          CPU
     2       6720      117.296      378.649
     4      26880      364.196     1191.786

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/DiffEqGPU","crn_sde.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 9354 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver4)
Threads: 58 default, 0 interactive, 29 GC (on 58 virtual cores)
Environment:
  JULIA_CPU_THREADS = 58
  JULIA_NUM_PRECOMPILE_TASKS = 58
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/DiffEqGPU/Project.toml`
⌃ [052768ef] CUDA v6.2.2
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  [789caeaf] StochasticDiffEq v7.2.0
  [de0858da] Printf v1.11.0
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

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  [e98f9f5b] NVTX_jll v3.2.2+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [458c3c95] OpenSSL_jll v3.5.8+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
  [1e29f10c] demumble_jll v1.3.0+0
  [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
  [f8abcde7] micromamba_jll v2.3.1+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+0
  [4d7b5844] pixi_jll v0.76.2+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 v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf 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 v1.11.0
  [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 v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
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`