Brusselator sparse AD benchmarks

using ADTypes
using LinearAlgebra, SparseArrays
using BenchmarkTools
import DifferentiationInterface as DI
import ForwardDiff
using Plots
using SparseConnectivityTracer: TracerSparsityDetector
using SparseMatrixColorings: GreedyColoringAlgorithm
using Symbolics: SymbolicsSparsityDetector
using Test

Definitions

brusselator_f(x, y, t) = (((x - 0.3)^2 + (y - 0.6)^2) <= 0.1^2) * (t >= 1.1) * 5.0

limit(a, N) =
    if a == N + 1
        1
    elseif a == 0
        N
    else
        a
    end;

function brusselator_2d!(du, u)
    t = 0.0
    N = size(u, 1)
    xyd = range(0; stop = 1, length = N)
    p = (3.4, 1.0, 10.0, step(xyd))
    A, B, alpha, dx = p
    alpha = alpha / dx^2

    @inbounds for I in CartesianIndices((N, N))
        i, j = Tuple(I)
        x, y = xyd[I[1]], xyd[I[2]]
        ip1, im1, jp1,
        jm1 = limit(i + 1, N),
        limit(i - 1, N), limit(j + 1, N),
        limit(j - 1, N)
        du[i, j, 1] = alpha *
                      (u[im1, j, 1] + u[ip1, j, 1] + u[i, jp1, 1] + u[i, jm1, 1] -
                       4u[i, j, 1]) +
                      B +
                      u[i, j, 1]^2 * u[i, j, 2] - (A + 1) * u[i, j, 1] +
                      brusselator_f(x, y, t)
        du[i, j, 2] = alpha *
                      (u[im1, j, 2] + u[ip1, j, 2] + u[i, jp1, 2] + u[i, jm1, 2] -
                       4u[i, j, 2]) +
                      A * u[i, j, 1] - u[i, j, 1]^2 * u[i, j, 2]
    end
end;

function init_brusselator_2d(N::Integer)
    xyd = range(0; stop = 1, length = N)
    N = length(xyd)
    u = zeros(N, N, 2)
    for I in CartesianIndices((N, N))
        x = xyd[I[1]]
        y = xyd[I[2]]
        u[I, 1] = 22 * (y * (1 - y))^(3 / 2)
        u[I, 2] = 27 * (x * (1 - x))^(3 / 2)
    end
    return u
end;

Correctness

x0_32 = init_brusselator_2d(32);

Sparsity detection

S1 = ADTypes.jacobian_sparsity(
    brusselator_2d!, similar(x0_32), x0_32, TracerSparsityDetector()
)
S2 = ADTypes.jacobian_sparsity(
    brusselator_2d!, similar(x0_32), x0_32, SymbolicsSparsityDetector()
)
@test S1 == S2

Test Passed

Coloring

c1 = ADTypes.column_coloring(S1, GreedyColoringAlgorithm())
@test length(unique(c1)) <= size(S1, 2)

Test Passed

Differentiation

backend = AutoSparse(
    AutoForwardDiff();
    sparsity_detector = TracerSparsityDetector(),
    coloring_algorithm = GreedyColoringAlgorithm()
);

prep = DI.prepare_jacobian(brusselator_2d!, similar(x0_32), backend, x0_32);
J1 = DI.jacobian!(
    brusselator_2d!, similar(x0_32), similar(S1, eltype(x0_32)), prep, backend, x0_32
)

@test nnz(J1) > 0

Test Passed

Benchmarks

N_values = 2 .^ (2:8)

7-element Vector{Int64}:
   4
   8
  16
  32
  64
 128
 256

Sparsity detection

td1, td2 = zeros(length(N_values)), zeros(length(N_values))
for (i, N) in enumerate(N_values)
    @info "Benchmarking sparsity detection: N=$N"
    x0 = init_brusselator_2d(N)
    td1[i] = @belapsed ADTypes.jacobian_sparsity(
        $brusselator_2d!, $(similar(x0)), $x0, TracerSparsityDetector()
    )
    td2[i] = @belapsed ADTypes.jacobian_sparsity(
        $brusselator_2d!, $(similar(x0)), $x0, SymbolicsSparsityDetector()
    )
end

let
    pld = plot(;
        title = "Sparsity detection on the Brusselator",
        xlabel = "Input size N",
        ylabel = "Runtime [s]"
    )
    plot!(
        pld,
        N_values,
        td1;
        lw = 2,
        linestyle = :auto,
        markershape = :auto,
        label = "SparseConnectivityTracer"
    )
    plot!(pld, N_values, td2; lw = 2, linestyle = :auto,
        markershape = :auto, label = "Symbolics")
    plot!(pld; xscale = :log10, yscale = :log10, legend = :topleft, minorgrid = true)
    pld
end

Coloring

tc1 = zeros(length(N_values))
for (i, N) in enumerate(N_values)
    @info "Benchmarking coloring: N=$N"
    x0 = init_brusselator_2d(N)
    S = ADTypes.jacobian_sparsity(
        brusselator_2d!, similar(x0), x0, TracerSparsityDetector()
    )
    tc1[i] = @belapsed ADTypes.column_coloring($S, GreedyColoringAlgorithm())
end

let
    plc = plot(;
        title = "Coloring on the Brusselator", xlabel = "Input size N", ylabel = "Runtime [s]"
    )
    plot!(
        plc,
        N_values,
        tc1;
        lw = 2,
        linestyle = :auto,
        markershape = :auto,
        label = "SparseMatrixColorings"
    )
    plot!(plc; xscale = :log10, yscale = :log10, legend = :topleft, minorgrid = true)
    plc
end

Differentiation

tj1 = zeros(length(N_values))
for (i, N) in enumerate(N_values)
    @info "Benchmarking differentiation: N=$N"
    x0 = init_brusselator_2d(N)
    S = ADTypes.jacobian_sparsity(
        brusselator_2d!, similar(x0), x0, TracerSparsityDetector()
    )
    J = similar(S, eltype(x0))

    tj1[i] = @belapsed DI.jacobian!($brusselator_2d!, _y, _J, _prep, $backend, $x0) setup=(
        _y = similar($x0);
        _J = similar($J);
        _prep = DI.prepare_jacobian($brusselator_2d!, similar($x0), $backend, $x0)
    ) evals=1
end

let
    plj = plot(;
        title = "Sparse Jacobian on the Brusselator", xlabel = "Input size N", ylabel = "Runtime [s]"
    )
    plot!(
        plj,
        N_values,
        tj1;
        lw = 2,
        linestyle = :auto,
        markershape = :auto,
        label = "DifferentiationInterface"
    )
    plot!(plj; xscale = :log10, yscale = :log10, legend = :topleft, minorgrid = true)
    plj
end

Summary

let
    pl = plot(;
        title = "Sparse AD pipeline on the Brusselator",
        xlabel = "Input size N",
        ylabel = "Runtime [s]"
    )
    plot!(
        pl,
        N_values,
        td1;
        lw = 2,
        linestyle = :dot,
        markershape = :utriangle,
        label = "sparsity detection (SCT)"
    )
    plot!(
        pl,
        N_values,
        td2;
        lw = 2,
        linestyle = :dot,
        markershape = :dtriangle,
        label = "sparsity detection (Symbolics)"
    )
    plot!(
        pl,
        N_values,
        tc1;
        lw = 2,
        linestyle = :dashdot,
        markershape = :diamond,
        label = "coloring (SparseMatrixColorings)"
    )
    plot!(
        pl,
        N_values,
        tj1;
        lw = 2,
        linestyle = :dash,
        markershape = :pentagon,
        label = "differentiation (DI + ForwardDiff)"
    )
    plot!(pl; xscale = :log10, yscale = :log10, minorgrid = true, legend = :topleft)
    pl
end

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/AutomaticDifferentiationSparse","BrusselatorSparseAD.jmd")

Computer Information:

Julia Version 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 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
  LLVM: libLLVM-18.1.7 (ORCJIT, znver2)
  GC: Built with stock GC
Threads: 128 default, 1 interactive, 128 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/AutomaticDifferentiationSparse/Project.toml`
  [47edcb42] ADTypes v1.24.0
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  [a0c0ee7d] DifferentiationInterface v0.7.21
  [a82114a7] DifferentiationInterfaceTest v0.11.0
⌃ [7da242da] Enzyme v0.13.199
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  [37e2e46d] LinearAlgebra v1.12.0
  [2f01184e] SparseArrays v1.12.0
  [8dfed614] Test v1.11.0
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/AutomaticDifferentiationSparse/Manifest.toml`
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  [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
  [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
  [009596ad] mtdev_jll v1.1.7+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 v1.7.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.12.1
  [de0858da] Printf v1.11.0
  [9abbd945] Profile 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 v1.12.0
  [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 v1.3.1+2
  [deac9b47] LibCURL_jll v8.15.0+0
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.6+0
  [efcefdf7] PCRE2_jll v10.44.0+1
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.7.0+0
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`