GitHub

ProbNumDiffEq.jl vs. various solver packages

Adapted from SciMLBenchmarks.jl multi-language wrapper benchmark.

Code:
# Imports
using LinearAlgebra, Statistics
using StaticArrays, DiffEqDevTools, ParameterizedFunctions, Plots, SciMLBase, OrdinaryDiffEq
using ODEInterface, ODEInterfaceDiffEq, Sundials, SciPyDiffEq, deSolveDiffEq, LSODA
using LoggingExtras
ODEInterface.loadODESolvers()

using ProbNumDiffEq

# Workaround: SciPyDiffEq's odeint retcode check compares fullout["tcur"] == saveat[end],
# but tcur is the internal step endpoint (e.g. 10.07) which overshoots the requested time.
# Fix: use >= comparison and index into the tcur vector.
include_string(SciPyDiffEq, """
function DiffEqBase.__solve(prob::DiffEqBase.AbstractODEProblem,
        alg::SciPyAlgorithm, timeseries = [], ts = [], ks = [];
        dense = true, dt = nothing,
        dtmax = abs(prob.tspan[2] - prob.tspan[1]),
        dtmin = eps(eltype(prob.tspan)), save_everystep = false,
        saveat = eltype(prob.tspan)[], timeseries_errors = true,
        reltol = 1e-3, abstol = 1e-6, maxiters = 10_000,
        kwargs...)
    p = prob.p
    tspan = prob.tspan
    u0 = prob.u0

    if DiffEqBase.isinplace(prob)
        f = function (t, u)
            du = similar(u)
            prob.f(du, u, p, t)
            du
        end
    else
        f = (t, u) -> prob.f(u, p, t)
    end

    _saveat = isempty(saveat) ? nothing : saveat
    if _saveat isa Array
        __saveat = _saveat
    elseif _saveat isa Number
        __saveat = Array(tspan[1]:_saveat:tspan[2])
    elseif _saveat isa Nothing
        if save_everystep
            __saveat = nothing
        else
            __saveat = [tspan[1], tspan[2]]
        end
    else
        __saveat = Array(_saveat)
    end

    if alg isa odeint
        __saveat === nothing && error("saveat is required for odeint!")
        sol,
        fullout = integrate.odeint(f, u0, __saveat,
            hmax = dtmax,
            rtol = reltol, atol = abstol,
            full_output = 1, tfirst = true,
            mxstep = maxiters)
        tcur = fullout["tcur"]
        retcode = first(tcur) >= __saveat[end] ? ReturnCode.Success : ReturnCode.Failure
        ts = __saveat
        y = sol

        if u0 isa AbstractArray
            timeseries = Vector{typeof(u0)}(undef, length(ts))
            for i in 1:length(ts)
                timeseries[i] = @view y[i, :]
            end
        else
            timeseries = y
        end

    else
        sol = integrate.solve_ivp(f, tspan, u0,
            first_step = dt,
            max_step = dtmax,
            rtol = reltol, atol = abstol,
            t_eval = __saveat,
            dense_output = dense,
            method = string(alg)[13:(end - 2)])
        ts = sol["t"]
        y = sol["y"]
        retcode = sol["success"] == false ? ReturnCode.Failure : ReturnCode.Success

        if u0 isa AbstractArray
            timeseries = Vector{typeof(u0)}(undef, length(ts))
            for i in 1:length(ts)
                timeseries[i] = @view y[:, i]
            end
        else
            timeseries = y
        end
    end

    if !(alg isa odeint) && dense
        _interp = PyInterpolation(sol["sol"])
    else
        _interp = DiffEqBase.LinearInterpolation(ts, timeseries)
    end

    DiffEqBase.build_solution(prob, alg, ts, timeseries,
        interp = _interp,
        dense = dense,
        retcode = retcode,
        timeseries_errors = timeseries_errors)
end
""")


Plots.theme(
    :dao;
    markerstrokewidth=0.5,
    legend=:outertopright,
    margin=5Plots.mm,
    xticks=10.0 .^ (-16:1:16),
    yticks=10.0 .^ (-6:1:5),
)

# Constants used throughout this benchmark as we only consider final values
const DENSE = false # used to decide if we smooth or not
const SAVE_EVERYSTEP = false;

# COLORS and a realted utility
COLORS = Dict(
    "Julia" => :LightGreen,
    "Julia (static)" => :DarkGreen,
    "Hairer" => :Red,
    "SciPy" => :Yellow,
    "deSolve" => :Blue,
    "Sundials" => :Purple,
    "liblsoda" => :Purple,
    "ProbNumDiffEq: EK0" => :Gray30,
    "ProbNumDiffEq: EK1" => :Gray60,
)
tocolor(n) = if split(n, '(')[1] in keys(COLORS)
    COLORS[split(n, '(')[1]]
else
    COLORS[split(n, ':')[1]]
end

# Do not show "deprecated warnings"
deprecated_filter(log_args) = !contains(log_args.message, "deprecated")
filtered_logger = ActiveFilteredLogger(deprecated_filter, global_logger());

Non-Stiff Problem 1: Lotka-Volterra

Code:
f = @ode_def LotkaVolterra begin
  dx = a*x - b*x*y
  dy = -c*y + d*x*y
end a b c d
p = [1.5, 1, 3, 1]
tspan = (0.0, 10.0)
u0 = [1.0, 1.0]
prob = ODEProblem{true,SciMLBase.FullSpecialize()}(f,u0,tspan,p)
staticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(f,SVector{2}(u0),tspan,SVector{4}(p))

sol = solve(prob,Vern7(),abstol=1/10^14,reltol=1/10^14,dense=false)
test_sol = sol
plot(sol, title="Lotka-Volterra Solution", legend=false, xticks=:auto, yticks=:auto)

Code:
_setups = [
  "Julia: DP5" => Dict(:alg=>DP5())
  "Julia: Tsit5" => Dict(:alg=>Tsit5())
  "Julia: Vern7" => Dict(:alg=>Vern7())
  "Hairer: dopri5" => Dict(:alg=>ODEInterfaceDiffEq.dopri5())
  "SciPy: RK45" => Dict(:alg=>SciPyDiffEq.RK45())
  "SciPy: LSODA" => Dict(:alg=>SciPyDiffEq.LSODA())
  "SciPy: odeint" => Dict(:alg=>SciPyDiffEq.odeint())
  "deSolve: lsoda" => Dict(:alg=>deSolveDiffEq.lsoda())
  "deSolve: ode45" => Dict(:alg=>deSolveDiffEq.ode45())
  "Sundials: Adams" => Dict(:alg=>Sundials.CVODE_Adams())
  "ProbNumDiffEq: EK0(3)" => Dict(:alg=>EK0(order=3, smooth=DENSE))
  "ProbNumDiffEq: EK0(5)" => Dict(:alg=>EK0(order=5, smooth=DENSE))
  "ProbNumDiffEq: EK1(3)" => Dict(:alg=>EK1(order=3, smooth=DENSE))
  "ProbNumDiffEq: EK1(5)" => Dict(:alg=>EK1(order=5, smooth=DENSE))
  "ProbNumDiffEq: EK1(8)" => Dict(:alg=>EK1(order=8, smooth=DENSE))
]

labels = first.(_setups)
setups = last.(_setups)
colors = tocolor.(labels) |> permutedims

abstols = 1.0 ./ 10.0 .^ (6:13)
reltols = 1.0 ./ 10.0 .^ (3:10)

wp = with_logger(filtered_logger) do
    WorkPrecisionSet(
        [prob, staticprob], abstols, reltols, setups;
        names = labels,
        appxsol = [test_sol, test_sol],
        dense = DENSE,
        save_everystep = SAVE_EVERYSTEP,
        numruns = 10,
        maxiters = Int(1e7),
        timeseries_errors = false,
        verbose = false,
    )
end

plot(wp, title = "Non-stiff 1: Lotka-Volterra", color = colors)

Non-Stiff Problem 2: Rigid Body

Code:
f = @ode_def RigidBodyBench begin
  dy1  = -2*y2*y3
  dy2  = 1.25*y1*y3
  dy3  = -0.5*y1*y2 + 0.25*sin(t)^2
end
u0 = [1.0;0.0;0.9]
tspan = (0.0, 10.0)
prob = ODEProblem{true,SciMLBase.FullSpecialize()}(f,u0,tspan)
staticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(f,SVector{3}(u0),tspan)
sol = solve(prob,Vern7(),abstol=1/10^14,reltol=1/10^14,dense=false)
test_sol = sol
plot(sol, title="Rigid Body Solution", legend=false, xticks=:auto, yticks=:auto)

Code:
_setups = [
  "Julia: DP5" => Dict(:alg=>DP5())
  "Julia: Tsit5" => Dict(:alg=>Tsit5())
  "Julia: Vern7" => Dict(:alg=>Vern7())
  "Hairer: dopri5" => Dict(:alg=>dopri5())
  "SciPy: RK45" => Dict(:alg=>SciPyDiffEq.RK45())
  "SciPy: LSODA" => Dict(:alg=>SciPyDiffEq.LSODA())
  "SciPy: odeint" => Dict(:alg=>SciPyDiffEq.odeint())
  "deSolve: lsoda" => Dict(:alg=>deSolveDiffEq.lsoda())
  "deSolve: ode45" => Dict(:alg=>deSolveDiffEq.ode45())
  "Sundials: Adams" => Dict(:alg=>CVODE_Adams())
  "ProbNumDiffEq: EK0(3)" => Dict(:alg=>EK0(order=3, smooth=DENSE))
  "ProbNumDiffEq: EK0(5)" => Dict(:alg=>EK0(order=5, smooth=DENSE))
  "ProbNumDiffEq: EK1(3)" => Dict(:alg=>EK1(order=3, smooth=DENSE))
  "ProbNumDiffEq: EK1(5)" => Dict(:alg=>EK1(order=5, smooth=DENSE))
  "ProbNumDiffEq: EK1(8)" => Dict(:alg=>EK1(order=8, smooth=DENSE))
]

labels = first.(_setups)
setups = last.(_setups)
colors = tocolor.(labels) |> permutedims

abstols = 1.0 ./ 10.0 .^ (6:13)
reltols = 1.0 ./ 10.0 .^ (3:10)

wp = with_logger(filtered_logger) do
    WorkPrecisionSet(
        [prob,staticprob], abstols, reltols, setups;
        names = labels,
        appxsol = [test_sol, test_sol],
        dense = DENSE,
        save_everystep = SAVE_EVERYSTEP,
        numruns = 10,
        maxiters = Int(1e7),
        timeseries_errors = false,
        verbose = false
    )
end

plot(wp, title = "Non-stiff 2: Rigid-Body", color = colors)

Stiff Problem 1: ROBER

Code:
rober = @ode_def begin
  dy₁ = -k₁*y₁+k₃*y₂*y₃
  dy₂ =  k₁*y₁-k₂*y₂^2-k₃*y₂*y₃
  dy₃ =  k₂*y₂^2
end k₁ k₂ k₃
u0 = [1.0,0.0,0.0]
p = [0.04,3e7,1e4]
prob = ODEProblem{true,SciMLBase.FullSpecialize()}(rober,u0,(0.0,1e5),p)
staticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(rober,SVector{3}(u0),(0.0,1e5),SVector{3}(p))
sol = solve(prob,CVODE_BDF(),abstol=1/10^14,reltol=1/10^14,dense=false)
test_sol = sol
plot(sol, title="ROBER Solution", legend=false, xlims=(1e0, 1e5), xticks=:auto, yticks=:auto)

Code:
_setups = [
  "Julia: Rosenbrock23" => Dict(:alg=>Rosenbrock23())
  "Julia: Rodas4" => Dict(:alg=>Rodas4())
  "Julia: Rodas5" => Dict(:alg=>Rodas5())
  "Hairer: rodas" => Dict(:alg=>rodas())
  "Hairer: radau" => Dict(:alg=>radau())
  "SciPy: LSODA" => Dict(:alg=>SciPyDiffEq.LSODA())
  "SciPy: BDF" => Dict(:alg=>SciPyDiffEq.BDF())
  "SciPy: odeint" => Dict(:alg=>SciPyDiffEq.odeint())
  "deSolve: lsoda" => Dict(:alg=>deSolveDiffEq.lsoda())
  "Sundials: CVODE" => Dict(:alg=>CVODE_BDF())
  "ProbNumDiffEq: EK1(3)" => Dict(:alg=>EK1(order=3, smooth=DENSE))
  "ProbNumDiffEq: EK1(5)" => Dict(:alg=>EK1(order=5, smooth=DENSE))
]

labels = first.(_setups)
setups = last.(_setups)
colors = tocolor.(labels) |> permutedims

abstols = 1.0 ./ 10.0 .^ (5:12)
reltols = 1.0 ./ 10.0 .^ (2:9)

wp = with_logger(filtered_logger) do
    WorkPrecisionSet(
        [prob, staticprob], abstols, reltols, setups;
        names = labels,
        dense = DENSE,
        verbose = false,
        save_everystep = SAVE_EVERYSTEP,
        appxsol = [test_sol, test_sol],
        maxiters=Int(1e5)
    )
end

plot(wp, title = "Stiff 1: ROBER", color = colors)

Stiff Problem 2: HIRES

Code:
f = @ode_def Hires begin
  dy1 = -1.71*y1 + 0.43*y2 + 8.32*y3 + 0.0007
  dy2 = 1.71*y1 - 8.75*y2
  dy3 = -10.03*y3 + 0.43*y4 + 0.035*y5
  dy4 = 8.32*y2 + 1.71*y3 - 1.12*y4
  dy5 = -1.745*y5 + 0.43*y6 + 0.43*y7
  dy6 = -280.0*y6*y8 + 0.69*y4 + 1.71*y5 -
           0.43*y6 + 0.69*y7
  dy7 = 280.0*y6*y8 - 1.81*y7
  dy8 = -280.0*y6*y8 + 1.81*y7
end

u0 = zeros(8)
u0[1] = 1
u0[8] = 0.0057
prob = ODEProblem{true,SciMLBase.FullSpecialize()}(f,u0,(0.0,321.8122))
staticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(f,SVector{8}(u0),(0.0,321.8122))

sol = solve(prob,Rodas5(),abstol=1/10^14,reltol=1/10^14, dense=false)
test_sol = sol
plot(sol, title="HIRES Solution", legend=false, xticks=:auto, yticks=:auto)

Code:
_setups = [
  "Julia: Rosenbrock23" => Dict(:alg=>Rosenbrock23())
  "Julia: Rodas4" => Dict(:alg=>Rodas4())
  "Julia: radau" => Dict(:alg=>RadauIIA5())
  "Hairer: rodas" => Dict(:alg=>rodas())
  "Hairer: radau" => Dict(:alg=>radau())
  "SciPy: LSODA" => Dict(:alg=>SciPyDiffEq.LSODA())
  "SciPy: BDF" => Dict(:alg=>SciPyDiffEq.BDF())
  "SciPy: odeint" => Dict(:alg=>SciPyDiffEq.odeint())
  "deSolve: lsoda" => Dict(:alg=>deSolveDiffEq.lsoda())
  "Sundials: CVODE" => Dict(:alg=>CVODE_BDF())
  "ProbNumDiffEq: EK1(2)" => Dict(:alg=>EK1(order=2, smooth=DENSE))
  "ProbNumDiffEq: EK1(3)" => Dict(:alg=>EK1(order=3, smooth=DENSE))
  "ProbNumDiffEq: EK1(5)" => Dict(:alg=>EK1(order=5, smooth=DENSE))
]

labels = first.(_setups)
setups = last.(_setups)
colors = tocolor.(labels) |> permutedims

abstols = 1.0 ./ 10.0 .^ (5:10)
reltols = 1.0 ./ 10.0 .^ (1:6)

wp = with_logger(filtered_logger) do
    WorkPrecisionSet(
        [prob, staticprob], abstols, reltols, setups;
        names = labels,
        dense = false,
        verbose = false,
        save_everystep = false,
        appxsol = [test_sol, test_sol],
        maxiters = Int(1e5),
        numruns=100
    )
end

plot(wp, title = "Stiff 2: Hires", color=colors)

Appendix

Computer information:
using InteractiveUtils
InteractiveUtils.versioninfo()
Julia Version 1.12.4
Commit 01a2eadb047 (2026-01-06 16:56 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD Ryzen Threadripper PRO 7985WX 64-Cores
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 128 virtual cores)
Environment:
  LD_LIBRARY_PATH = /.singularity.d/libs
Package information:
using Pkg
Pkg.status()
Status `/home/nrbosch/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`
  [f3b72e0c] DiffEqDevTools v2.49.0
  [31c24e10] Distributions v0.25.123
  [7073ff75] IJulia v1.34.3
  [7f56f5a3] LSODA v0.7.5
  [e6f89c97] LoggingExtras v1.2.0
  [e2752cbe] MATLABDiffEq v1.4.0
⌃ [961ee093] ModelingToolkit v11.10.0
  [54ca160b] ODEInterface v0.5.0
  [09606e27] ODEInterfaceDiffEq v3.15.0
  [1dea7af3] OrdinaryDiffEq v6.108.0
  [65888b18] ParameterizedFunctions v5.22.0
⌃ [91a5bcdd] Plots v1.41.5
  [bf3e78b0] ProbNumDiffEq v0.16.4 `/home/nrbosch/.julia/dev/ProbNumDiffEq`
⌃ [0bca4576] SciMLBase v2.138.1
  [505e40e9] SciPyDiffEq v0.2.2
  [ce78b400] SimpleUnPack v1.1.0
  [90137ffa] StaticArrays v1.9.16
  [c3572dad] Sundials v5.1.0
  [44d3d7a6] Weave v0.10.12
  [0518478a] deSolveDiffEq v1.1.0
Info Packages marked with ⌃ have new versions available and may be upgradable.
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.
Full manifest:
Pkg.status(mode=Pkg.PKGMODE_MANIFEST)
Status `/home/nrbosch/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`
  [47edcb42] ADTypes v1.21.0
  [621f4979] AbstractFFTs v1.5.0
  [6e696c72] AbstractPlutoDingetjes v1.3.2
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.43
  [79e6a3ab] Adapt v4.4.0
  [66dad0bd] AliasTables v1.1.3
  [ec485272] ArnoldiMethod v0.4.0
  [c9d4266f] ArrayAllocators v0.3.0
  [4fba245c] ArrayInterface v7.22.0
  [4c555306] ArrayLayouts v1.12.2
  [0e736298] Bessels v0.2.8
  [e2ed5e7c] Bijections v0.2.2
⌃ [caf10ac8] BipartiteGraphs v0.1.6
  [d1d4a3ce] BitFlags v0.1.9
  [62783981] BitTwiddlingConvenienceFunctions v0.1.6
  [8e7c35d0] BlockArrays v1.9.3
⌃ [70df07ce] BracketingNonlinearSolve v1.7.1
  [fa961155] CEnum v0.5.0
  [2a0fbf3d] CPUSummary v0.2.7
  [324d7699] CategoricalArrays v1.0.2
  [d360d2e6] ChainRulesCore v1.26.0
  [fb6a15b2] CloseOpenIntervals v0.1.13
  [944b1d66] CodecZlib v0.7.8
  [35d6a980] ColorSchemes v3.31.0
  [3da002f7] ColorTypes v0.12.1
  [c3611d14] ColorVectorSpace v0.11.0
  [5ae59095] Colors v0.13.1
⌅ [861a8166] Combinatorics v1.0.2
  [38540f10] CommonSolve v0.2.6
  [bbf7d656] CommonSubexpressions v0.3.1
  [f70d9fcc] CommonWorldInvalidations v1.0.0
  [34da2185] Compat v4.18.1
  [b152e2b5] CompositeTypes v0.1.4
  [a33af91c] CompositionsBase v0.1.2
  [2569d6c7] ConcreteStructs v0.2.3
⌃ [f0e56b4a] ConcurrentUtilities v2.5.0
  [8f4d0f93] Conda v1.10.3
  [187b0558] ConstructionBase v1.6.0
  [d38c429a] Contour v0.6.3
  [adafc99b] CpuId v0.3.1
  [a8cc5b0e] Crayons v4.1.1
  [717857b8] DSP v0.8.4
  [9a962f9c] DataAPI v1.16.0
  [a93c6f00] DataFrames v1.8.1
  [864edb3b] DataStructures v0.19.3
  [e2d170a0] DataValueInterfaces v1.0.0
  [8bb1440f] DelimitedFiles v1.9.1
⌃ [2b5f629d] DiffEqBase v6.203.0
  [459566f4] DiffEqCallbacks v4.12.0
  [f3b72e0c] DiffEqDevTools v2.49.0
  [77a26b50] DiffEqNoiseProcess v5.27.0
  [163ba53b] DiffResults v1.1.0
  [b552c78f] DiffRules v1.15.1
  [a0c0ee7d] DifferentiationInterface v0.7.16
  [b4f34e82] Distances v0.10.12
  [31c24e10] Distributions v0.25.123
  [ffbed154] DocStringExtensions v0.9.5
  [5b8099bc] DomainSets v0.7.16
  [7c1d4256] DynamicPolynomials v0.6.4
  [4e289a0a] EnumX v1.0.6
  [f151be2c] EnzymeCore v0.8.18
  [6912e4f1] Espresso v0.6.4
  [460bff9d] ExceptionUnwrapping v0.1.11
  [d4d017d3] ExponentialUtilities v1.30.0
  [e2ba6199] ExprTools v0.1.10
  [55351af7] ExproniconLite v0.10.14
  [c87230d0] FFMPEG v0.4.5
  [7a1cc6ca] FFTW v1.10.0
  [7034ab61] FastBroadcast v0.3.5
  [9aa1b823] FastClosures v0.3.2
  [442a2c76] FastGaussQuadrature v1.1.0
  [a4df4552] FastPower v1.3.1
  [1a297f60] FillArrays v1.16.0
  [64ca27bc] FindFirstFunctions v1.8.0
  [6a86dc24] FiniteDiff v2.29.0
  [b59a298d] FiniteHorizonGramians v0.2.1
  [53c48c17] FixedPointNumbers v0.8.5
  [1fa38f19] Format v1.3.7
  [f6369f11] ForwardDiff v1.3.2
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v0.1.3
  [46192b85] GPUArraysCore v0.2.0
  [28b8d3ca] GR v0.73.22
  [c145ed77] GenericSchur v0.5.6
  [86223c79] Graphs v1.13.4
  [42e2da0e] Grisu v1.0.2
  [cd3eb016] HTTP v1.10.19
⌅ [eafb193a] Highlights v0.5.3
  [3e5b6fbb] HostCPUFeatures v0.1.18
  [34004b35] HypergeometricFunctions v0.3.28
  [7073ff75] IJulia v1.34.3
  [615f187c] IfElse v0.1.1
  [3263718b] ImplicitDiscreteSolve v1.7.0
  [d25df0c9] Inflate v0.1.5
  [842dd82b] InlineStrings v1.4.5
  [18e54dd8] IntegerMathUtils v0.1.3
  [8197267c] IntervalSets v0.7.13
  [3587e190] InverseFunctions v0.1.17
  [41ab1584] InvertedIndices v1.3.1
  [92d709cd] IrrationalConstants v0.2.6
  [c8e1da08] IterTools v1.10.0
  [82899510] IteratorInterfaceExtensions v1.0.0
  [1019f520] JLFzf v0.1.11
  [692b3bcd] JLLWrappers v1.7.1
⌅ [682c06a0] JSON v0.21.4
  [ae98c720] Jieko v0.2.1
  [ccbc3e58] JumpProcesses v9.22.0
  [2c470bb0] Kronecker v0.5.5
  [ba0b0d4f] Krylov v0.10.5
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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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