"Aayush Sabharwal, Bowen Zhu, Chris Rackauckas

The following benchmark is of 1122 ODEs with 24388 terms that describe a stiff chemical reaction network modeling the BCR signaling network from Barua et al.. We use ReactionNetworkImporters to load the BioNetGen model files as a Catalyst model, and then use ModelingToolkit to convert the Catalyst network model to ODEs.

The resultant large model is used to benchmark the time taken to compute a symbolic jacobian, generate a function to calculate it and call the function.

Jacobian construction uses the current Symbolics.sparsejacobian implementation, with derivative caches cleared before each sample. CSE is a build_function code-generation option, so only the code-generation and generated-function measurements compare CSE off and on.

using Catalyst, ReactionNetworkImporters,
    TimerOutputs, LinearAlgebra, ModelingToolkit, Chairmarks,
    LinearSolve, Symbolics, SymbolicUtils.Code, SparseArrays, CairoMakie,
    PrettyTables
using SymbolicIndexingInterface: default_values

datadir  = joinpath(dirname(pathof(ReactionNetworkImporters)),"../data/bcr")
const to = TimerOutput()
tf       = 100000.0

# generate ModelingToolkit ODEs
rn_raw = loadrxnetwork(BNGNetwork(), joinpath(datadir, "bcr.net"))
show(to)
rn    = complete(rn_raw; split = false)
obs = [eq.lhs for eq in observed(rn)]
osys = Catalyst.ode_model(rn)

rhs = [eq.rhs for eq in full_equations(osys)]
vars = unknowns(osys)
pars = parameters(osys)

Scanning blocks...done
Parsing parameters...done
Creating parameters...done
Parsing species...done
Creating variables...done
Setting up expression bindings...done
Parsing groups...done
Parsing functions...done
Parsing and adding reactions...done
────────────────────────────────────────────────────────────────────
                           Time                    Allocations      
                  ───────────────────────   ────────────────────────
Tot / % measured:      26.3s /   0.0%           1.55GiB /   0.0%    

Section   ncalls     time    %tot     avg     alloc    %tot      avg
────────────────────────────────────────────────────────────────────
────────────────────────────────────────────────────────────────────128-ele
ment Vector{SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{
SymbolicUtils.SymReal}}:
 p1
 p2
 p3
 p4
 p5
 p6
 p7
 p8
 p9
 p10
 ⋮
 _rateLaw2
 _rateLaw3
 _rateLaw4
 _rateLaw5
 _rateLaw6
 _rateLaw7
 _rateLaw8
 _rateLaw9
 _rateLaw10

Symbolics.clear_derivative_caches!()
@timeit to "Calculate symbolic jacobian" jac = Symbolics.sparsejacobian(rhs, vars);
args = (vars, pars, ModelingToolkit.get_iv(osys))
# out of place versions run into an error saying the expression is too large
# due to the `SymbolicUtils.Code.create_array` call. `iip_config` prevents it
# from trying to build the function.
kwargs = (; iip_config = (false, true), expression = Val{true})
@timeit to "Build jacobian - no CSE" _, jac_nocse_iip = build_function(jac, args...; cse = false, kwargs...);
@timeit to "Build jacobian - CSE" _, jac_cse_iip = build_function(jac, args...; cse = true, kwargs...);

jac_nocse_iip = eval(jac_nocse_iip)
jac_cse_iip = eval(jac_cse_iip)

defs = default_values(osys)
u = Float64[Symbolics.value(Symbolics.fixpoint_sub(var, defs)) for var in vars]
buffer_cse = similar(jac, Float64)
buffer_nocse = similar(jac, Float64)
p = Float64[Symbolics.value(Symbolics.fixpoint_sub(par, defs)) for par in pars]
tt = 0.0

@timeit to "Compile jacobian - CSE" jac_cse_iip(buffer_cse, u, p, tt)
@timeit to "Compute jacobian - CSE" jac_cse_iip(buffer_cse, u, p, tt)

@timeit to "Compile jacobian - no CSE" jac_nocse_iip(buffer_nocse, u, p, tt)
@timeit to "Compute jacobian - no CSE" jac_nocse_iip(buffer_nocse, u, p, tt)

@assert isapprox(buffer_cse, buffer_nocse, rtol = 1e-10)

show(to)

───────────────────────────────────────────────────────────────────────────
─────────────
                                               Time                    Allo
cations      
                                      ───────────────────────   ───────────
─────────────
          Tot / % measured:                 384s /  79.5%           17.3GiB
 /  72.8%    

Section                       ncalls     time    %tot     avg     alloc    
%tot      avg
───────────────────────────────────────────────────────────────────────────
─────────────
Compile jacobian - no CSE          1     171s   55.9%    171s   6.60GiB   5
2.3%  6.60GiB
Compile jacobian - CSE             1    94.4s   30.9%   94.4s   1.88GiB   1
4.9%  1.88GiB
Calculate symbolic jacobian        1    28.3s    9.3%   28.3s   2.93GiB   2
3.2%  2.93GiB
Build jacobian - no CSE            1    11.1s    3.6%   11.1s   1.09GiB    
8.6%  1.09GiB
Build jacobian - CSE               1    654ms    0.2%   654ms    125MiB    
1.0%   125MiB
Compute jacobian - no CSE          1    183μs    0.0%   183μs      176B    
0.0%     176B
Compute jacobian - CSE             1   80.5μs    0.0%  80.5μs      176B    
0.0%     176B
───────────────────────────────────────────────────────────────────────────
─────────────

We'll also measure scaling.

function run_and_time_construct!(rhs, vars, pars, iv, N, i, jac_times, jac_allocs, build_times, functions)
    outputs = rhs[1:N]
    jac_result = @be (Symbolics.clear_derivative_caches!(); Symbolics.sparsejacobian(outputs, vars))
    jac_times[i] = minimum(x -> x.time, jac_result.samples)
    jac_allocs[i] = minimum(x -> x.bytes, jac_result.samples)

    Symbolics.clear_derivative_caches!()
    jac = Symbolics.sparsejacobian(outputs, vars)
    args = (vars, pars, iv)
    kwargs = (; iip_config = (false, true), expression = Val{true})
    
    build_result = @be build_function(jac, args...; cse = false, kwargs...);
    build_times[1][i] = minimum(x -> x.time, build_result.samples)
    jacfn_nocse = eval(build_function(jac, args...; cse = false, kwargs...)[2])

    build_result = @be build_function(jac, args...; cse = true, kwargs...);
    build_times[2][i] = minimum(x -> x.time, build_result.samples)
    jacfn_cse = eval(build_function(jac, args...; cse = true, kwargs...)[2])

    functions[1][i] = let buffer = similar(jac, Float64), fn = jacfn_nocse
        function nocse(u, p, t)
            fn(buffer, u, p, t)
            buffer
        end
    end
    functions[2][i] = let buffer = similar(jac, Float64), fn = jacfn_cse
        function cse(u, p, t)
            fn(buffer, u, p, t)
            buffer
        end
    end

    return nothing
end

function run_and_time_call!(i, u, p, tt, functions, first_call_times, second_call_times)
    jacfn_nocse = functions[1][i]
    jacfn_cse = functions[2][i]

    call_result = @timed jacfn_nocse(u, p, tt)
    first_call_times[1][i] = call_result.time
    call_result = @timed jacfn_cse(u, p, tt)
    first_call_times[2][i] = call_result.time

    call_result = @be jacfn_nocse(u, p, tt)
    second_call_times[1][i] = minimum(x -> x.time, call_result.samples)
    call_result = @be jacfn_cse(u, p, tt)
    second_call_times[2][i] = minimum(x -> x.time, call_result.samples)
end

run_and_time_call! (generic function with 1 method)

Run benchmark

Chairmarks.DEFAULTS.seconds = 15.0
N = [10, 20, 40, 80, 160, 320]
jacobian_times = zeros(Float64, length(N))
jacobian_allocs = similar(jacobian_times)
functions = [Vector{Any}(undef, length(N)), Vector{Any}(undef, length(N))]
# [without_cse_times, with_cse_times]
build_times = [similar(jacobian_times), similar(jacobian_times)]
first_call_times = copy.(build_times)
second_call_times = copy.(build_times)

iv = ModelingToolkit.get_iv(osys)
run_and_time_construct!(rhs, vars, pars, iv, 10, 1, jacobian_times, jacobian_allocs, build_times, functions)
run_and_time_call!(1, u, p, tt, functions, first_call_times, second_call_times)
for (i, n) in enumerate(N)
    @info i n
    run_and_time_construct!(rhs, vars, pars, iv, n, i, jacobian_times, jacobian_allocs, build_times, functions)
end
for (i, n) in enumerate(N)
    @info i n
    run_and_time_call!(i, u, p, tt, functions, first_call_times, second_call_times)
end

Plot figures

tabledata = hcat(N, jacobian_times, jacobian_allocs, build_times..., first_call_times..., second_call_times...)
header = ["N", "Jacobian time", "Jacobian allocated memory (B)", "`build_function` time (no CSE)", "`build_function` time (CSE)", "First call time (no CSE)", "First call time (CSE)", "Second call time (no CSE)", "Second call time (CSE)"]
pretty_table(tabledata; column_labels = header, backend = :html)

<table> <thead> <tr class = "columnLabelRow"> <th style = "font-weight: bold; text-align: right;">N</th> <th style = "font-weight: bold; text-align: right;">Jacobian time</th> <th style = "font-weight: bold; text-align: right;">Jacobian allocated memory (B)</th> <th style = "font-weight: bold; text-align: right;">build_function time (no CSE)</th> <th style = "font-weight: bold; text-align: right;">build_function time (CSE)</th> <th style = "font-weight: bold; text-align: right;">First call time (no CSE)</th> <th style = "font-weight: bold; text-align: right;">First call time (CSE)</th> <th style = "font-weight: bold; text-align: right;">Second call time (no CSE)</th> <th style = "font-weight: bold; text-align: right;">Second call time (CSE)</th> </tr> </thead> <tbody> <tr class = "dataRow"> <td style = "text-align: right;">10.0</td> <td style = "text-align: right;">3.18885</td> <td style = "text-align: right;">4.2047e8</td> <td style = "text-align: right;">0.0245617</td> <td style = "text-align: right;">0.0311531</td> <td style = "text-align: right;">9.65231</td> <td style = "text-align: right;">6.34166</td> <td style = "text-align: right;">3.55363e-6</td> <td style = "text-align: right;">2.9439e-6</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">20.0</td> <td style = "text-align: right;">4.28213</td> <td style = "text-align: right;">5.47698e8</td> <td style = "text-align: right;">0.0366498</td> <td style = "text-align: right;">0.0473515</td> <td style = "text-align: right;">14.9811</td> <td style = "text-align: right;">8.93726</td> <td style = "text-align: right;">7.11975e-6</td> <td style = "text-align: right;">4.49667e-6</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">40.0</td> <td style = "text-align: right;">5.64052</td> <td style = "text-align: right;">7.19497e8</td> <td style = "text-align: right;">0.0591457</td> <td style = "text-align: right;">0.0687831</td> <td style = "text-align: right;">25.1312</td> <td style = "text-align: right;">14.1411</td> <td style = "text-align: right;">1.29245e-5</td> <td style = "text-align: right;">6.955e-6</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">80.0</td> <td style = "text-align: right;">9.30943</td> <td style = "text-align: right;">1.10733e9</td> <td style = "text-align: right;">0.108699</td> <td style = "text-align: right;">0.105643</td> <td style = "text-align: right;">47.6296</td> <td style = "text-align: right;">22.9037</td> <td style = "text-align: right;">2.546e-5</td> <td style = "text-align: right;">1.16395e-5</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">160.0</td> <td style = "text-align: right;">10.9206</td> <td style = "text-align: right;">1.2665e9</td> <td style = "text-align: right;">0.141835</td> <td style = "text-align: right;">0.143031</td> <td style = "text-align: right;">62.6152</td> <td style = "text-align: right;">29.6462</td> <td style = "text-align: right;">3.7879e-5</td> <td style = "text-align: right;">1.5319e-5</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">320.0</td> <td style = "text-align: right;">12.6784</td> <td style = "text-align: right;">1.43459e9</td> <td style = "text-align: right;">0.188567</td> <td style = "text-align: right;">0.187348</td> <td style = "text-align: right;">79.3665</td> <td style = "text-align: right;">41.6618</td> <td style = "text-align: right;">4.107e-5</td> <td style = "text-align: right;">2.163e-5</td> </tr> </tbody> </table>

f = Figure(size = (750, 400))
titles = [
    "Jacobian symbolic computation", "Jacobian symbolic computation", "Code generation",
    "Numerical function compilation", "Numerical function evaluation"]
labels = ["Time (seconds)", "Allocated memory (bytes)",
    "Time (seconds)", "Time (seconds)", "Time (seconds)"]
times = [jacobian_times, jacobian_allocs, build_times, first_call_times, second_call_times]
axes = Axis[]
for i in 1:2
    label = labels[i]
    data = times[i]
    ax = Axis(f[1, i], xscale = log10, yscale = log10, xlabel = "model size",
        xlabelsize = 10, ylabel = label, ylabelsize = 10, xticks = N,
        title = titles[i], titlesize = 12, xticklabelsize = 10, yticklabelsize = 10)
    push!(axes, ax)
    scatterlines!(ax, N, data)
end
axes2 = Axis[]
# make equal y-axis unit length
mn3, mx3 = extrema(reduce(vcat, times[3]))
xn3 = log10(mx3 / mn3)
mn4, mx4 = extrema(reduce(vcat, times[4]))
xn4 = log10(mx4 / mn4)
mn5, mx5 = extrema(reduce(vcat, times[5]))
xn5 = log10(mx5 / mn5)
xn = max(xn3, xn4, xn5)
xn += 0.2
hxn = xn / 2
hxn3 = (log10(mx3) + log10(mn3)) / 2
hxn4 = (log10(mx4) + log10(mn4)) / 2
hxn5 = (log10(mx5) + log10(mn5)) / 2
ylims = [(exp10(hxn3 - hxn), exp10(hxn3 + hxn)), (exp10(hxn4 - hxn), exp10(hxn4 + hxn)),
    (exp10(hxn5 - hxn), exp10(hxn5 + hxn))]
for i in 1:3
    ir = i + 2
    label = labels[ir]
    data = times[ir]
    ax = Axis(f[2, i], xscale = log10, yscale = log10, xlabel = "model size",
        xlabelsize = 10, ylabel = label, ylabelsize = 10, xticks = N,
        title = titles[ir], titlesize = 12, xticklabelsize = 10, yticklabelsize = 10)
    ylims!(ax, ylims[i]...)
    push!(axes2, ax)
    scatterlines!(ax, N, data[1], label = "without CSE")
    scatterlines!(ax, N, data[2], label = "with CSE")
end
Legend(f[1, 3], axes2[1], "Code generation", tellwidth = false, labelsize = 12, titlesize = 15)
save("bcr.pdf", f)
f

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/Symbolics","BCR.jmd")

Computer Information:

Julia Version 1.13.0
Commit d1c37793dd2 (2026-09-09 19:00 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-20.1.8 (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 `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Symbolics/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
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  [479239e8] Catalyst v16.4.3
  [0ca39b1e] Chairmarks v1.3.1
⌃ [992eb4ea] CondaPkg v0.2.33
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⌃ [7ed4a6bd] LinearSolve v5.17.3
⌃ [961ee093] ModelingToolkit v11.43.1
⌅ [bac558e1] OrderedCollections v1.8.2 [loaded: v2.0.1]
  [1dea7af3] OrdinaryDiffEq v7.8.1
  [91a5bcdd] Plots v1.41.7
  [f27b6e38] Polynomials v4.1.3
  [08abe8d2] PrettyTables v3.4.8
⌃ [6099a3de] PythonCall v0.9.35
  [b4db0fb7] ReactionNetworkImporters v1.5.0
  [31c91b34] SciMLBenchmarks v0.2.1 [loaded: `/home/crackauc/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/src/SciMLBenchmarks.jl` (v0.2.1) expected `/home/crackauc/.julia/packages/SciMLBenchmarks/ceJyd/src/SciMLBenchmarks.jl` (v0.2.1)]
  [10745b16] Statistics v1.11.5
  [123dc426] SymEngine v0.13.2
  [2efcf032] SymbolicIndexingInterface v0.3.55
⌃ [d1185830] SymbolicUtils v4.46.6
⌃ [0c5d862f] Symbolics v7.39.2
⌅ [a759f4b9] TimerOutputs v0.5.29
  [95ff35a0] XSteam v0.3.0
  [37e2e46d] LinearAlgebra v1.13.0
  [9a3f8284] Random v1.11.0
  [2f01184e] SparseArrays v1.13.0
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`

And the full manifest:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Symbolics/Manifest.toml`
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⌃ [4fba245c] ArrayInterface v7.30.1
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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`
Info Packages marked with [deprecated] are no longer maintained. Use `status --deprecated -m` to see more information.