Linear SDE Ensemble — GPUEM vs EnsembleThreads

Linear geometric Brownian motion ensemble from Utkarsh et al., Comput. Methods Appl. Mech. Eng. 428 (2024) 117109 (arXiv:2304.06835; artifacts).

$

dX = p\, X\, dt + q\, X\, dW, \qquad X_0 = (0.1, 0.1, 0.1),\quad p = 1.5,\quad q = 0.01,\quad t \in [0, 1] $

Fixed-step Euler–Maruyama with $dt = 2^{-8}$. GPU uses GPUEM + EnsembleGPUKernel in Float32; CPU uses EM + EnsembleThreads in Float64, matching the paper scripts.

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()

f(u, p, t) = p[1] * u
g(u, p, t) = p[2] * u

function make_sde_ensemble(; T::Type = Float32)
    u0 = SVector{3, T}(T(0.1), T(0.1), T(0.1))
    tspan = (zero(T), one(T))
    p = SVector{2, T}(T(1.5), T(0.01))
    prob = SDEProblem{false}(f, g, u0, tspan, p; seed = 1234)
    return EnsembleProblem(prob, safetycopy = false)
end

function min_seconds(fn; warmup = 1, samples = 5)
    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

function time_sde(n, ensemblealg, alg; T = Float32)
    ens = make_sde_ensemble(; T)
    dt = T(1 // 2^8)
    run = if ensemblealg isa EnsembleThreads
        () -> (solve(ens, alg, ensemblealg; trajectories = n, save_everystep = false,
            adaptive = false, dt = dt); nothing)
    else
        () -> (CUDA.@sync solve(ens, alg, ensemblealg; trajectories = n,
            save_everystep = false, adaptive = false, dt = dt); nothing)
    end
    return min_seconds(run)
end

time_sde (generic function with 1 method)

let
    ens = make_sde_ensemble()
    sol = solve(ens, GPUEM(), KERNEL; trajectories = 4, save_everystep = false,
        adaptive = false, dt = Float32(1 // 2^8))
    @assert length(sol.u) == 4
    println("GPUEM smoke: 4 trajectories, u[end] = ", sol.u[1].u[end])
end

GPUEM smoke: 4 trajectories, u[end] = Float32[0.43873417, 0.45378098, 0.441
85403]

const TRAJ_GPU = [8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608]
const TRAJ_CPU = [8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152]

t_gpu = Float64[]
t_cpu = Float64[]
for n in TRAJ_GPU
    @info "sde gpu" n
    push!(t_gpu, time_sde(n, KERNEL, GPUEM()))
end
for n in TRAJ_CPU
    @info "sde cpu" n
    push!(t_cpu, time_sde(n, EnsembleThreads(), EM(); T = Float64))
end

p = plot(TRAJ_GPU, t_gpu .* 1e3; xscale = :log10, yscale = :log10,
    xlabel = "trajectories", ylabel = "time (ms)", label = "GPUEM + EnsembleGPUKernel",
    marker = :circle, legend = :topleft, title = "Linear SDE ensemble, EM dt = 2^{-8}")
plot!(p, TRAJ_CPU, t_cpu .* 1e3; label = "EM + EnsembleThreads", marker = :utriangle)
p

println("Linear SDE (ms)")
@printf("%10s %12s %12s\n", "N", "GPU", "CPU")
for n in TRAJ_GPU
    tg = t_gpu[findfirst(==(n), TRAJ_GPU)] * 1e3
    tc = (i = findfirst(==(n), TRAJ_CPU); i === nothing ? NaN : t_cpu[i] * 1e3)
    @printf("%10d %12.3f %12.3f\n", n, tg, tc)
end

Linear SDE (ms)
         N          GPU          CPU
         8        0.317        0.461
        32        0.334        0.426
       128        0.409        0.473
       512        0.712        0.699
      2048        2.006        1.972
      8192        8.146        7.184
     32768       31.943       39.958
    131072      158.628      122.606
    524288      684.913     1362.222
   2097152     2826.687     5152.480
   8388608    11403.889          NaN

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","linear_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`
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  [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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  [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`