Differentiation of Simple ODE Benchmarks
From the paper A Comparison of Automatic Differentiation and Continuous Sensitivity Analysis for Derivatives of Differential Equation Solutions
using ParameterizedFunctions, OrdinaryDiffEq, LinearAlgebra, StaticArrays
using OrdinaryDiffEqRosenbrock
using SciMLSensitivity, ForwardDiff, FiniteDiff, ReverseDiff, BenchmarkTools, Test
using DataFrames, PrettyTables, Markdown
tols = (abstol = 1e-5, reltol = 1e-7)(abstol = 1.0e-5, reltol = 1.0e-7)Define the Test ODEs
function lvdf(du, u, p, t)
a, b, c = p
x, y = u
du[1] = a*x - b*x*y
du[2] = -c*y + x*y
nothing
end
function lvcom_df(du, u, p, t)
a, b, c = p
x, y, s1, s2, s3, s4, s5, s6 = u
du[1] = a*x - b*x*y
du[2] = -c*y + x*y
#####################
# [a-by -bx]
# J = [ ]
# [y x-c]
#####################
J = @SMatrix [a-b*y -b*x
y x-c]
JS = J*@SMatrix[s1 s3 s5
s2 s4 s6]
G = @SMatrix [x -x*y 0
0 0 -y]
du[3:end] .= vec(JS+G)
nothing
end
lvdf_with_jacobian = ODEFunction{true, SciMLBase.FullSpecialize}(lvdf, jac = (
J, u, p, t)->begin
a, b, c = p
x, y = u
J[1] = a-b*y
J[2] = y
J[3] = -b*x
J[4] = x-c
nothing
end)
u0 = [1.0, 1.0];
tspan = (0.0, 10.0);
p = [1.5, 1.0, 3.0];
lvcom_u0 = [u0...; zeros(6)]
lvprob = ODEProblem{true, SciMLBase.FullSpecialize}(lvcom_df, lvcom_u0, tspan, p)ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
Non-trivial mass matrix: false
timespan: (0.0, 10.0)
u0: 8-element Vector{Float64}:
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0pkpdf = @ode_def begin
dEv = -Ka1*Ev
dCent = Ka1*Ev - (CL+Vmax/(Km+(Cent/Vc))+Q)*(Cent/Vc) + Q*(Periph/Vp) - Q2*(Cent/Vc) +
Q2*(Periph2/Vp2)
dPeriph = Q*(Cent/Vc) - Q*(Periph/Vp)
dPeriph2 = Q2*(Cent/Vc) - Q2*(Periph2/Vp2)
dResp = Kin*(1-(IMAX*(Cent/Vc)^γ/(IC50^γ+(Cent/Vc)^γ))) - Kout*Resp
end Ka1 CL Vc Q Vp Kin Kout IC50 IMAX γ Vmax Km Q2 Vp2
pkpdp = [
1, # Ka1 Absorption rate constant 1 (1/time)
1, # CL Clearance (volume/time)
20, # Vc Central volume (volume)
2, # Q Inter-compartmental clearance (volume/time)
10, # Vp Peripheral volume of distribution (volume)
10, # Kin Response in rate constant (1/time)
2, # Kout Response out rate constant (1/time)
2, # IC50 Concentration for 50% of max inhibition (mass/volume)
1, # IMAX Maximum inhibition
1, # γ Emax model sigmoidicity
0, # Vmax Maximum reaction velocity (mass/time)
2, # Km Michaelis constant (mass/volume)
0.5, # Q2 Inter-compartmental clearance2 (volume/time)
100 # Vp2 Peripheral2 volume of distribution (volume)
];
pkpdu0 = [100, eps(), eps(), eps(), 5.0] # exact zero in the initial condition triggers NaN in Jacobian
#pkpdu0 = ones(5)
pkpdcondition = function (u, t, integrator)
t in 0:24:240
end
pkpdaffect! = function (integrator)
integrator.u[1] += 100
end
pkpdcb = DiscreteCallback(pkpdcondition, pkpdaffect!, save_positions = (false, true))
pkpdtspan = (0.0, 240.0)
pkpdprob = ODEProblem{true, SciMLBase.FullSpecialize}(pkpdf.f, pkpdu0, pkpdtspan, pkpdp)
pkpdfcomp = let pkpdf=pkpdf, J=zeros(5, 5), JP=zeros(5, 14), tmpdu=zeros(5, 14)
function (du, u, p, t)
pkpdf.f(@view(du[:, 1]), u, p, t)
pkpdf.jac(J, u, p, t)
pkpdf.paramjac(JP, u, p, t)
mul!(tmpdu, J, @view(u[:, 2:end]))
du[:, 2:end] .= tmpdu .+ JP
nothing
end
end
pkpdcompprob = ODEProblem{true, SciMLBase.FullSpecialize}(
pkpdfcomp, hcat(pkpdprob.u0, zeros(5, 14)), pkpdprob.tspan, pkpdprob.p)ODEProblem with uType Matrix{Float64} and tType Float64. In-place: true
Non-trivial mass matrix: false
timespan: (0.0, 240.0)
u0: 5×15 Matrix{Float64}:
100.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0
.0
2.22045e-16 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0
.0
2.22045e-16 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0
.0
2.22045e-16 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0
.0
5.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0
.0pollution = @ode_def begin
dy1 = -k1 * y1-k10*y11*y1-k14*y1*y6-k23*y1*y4-k24*y19*y1+
k2 * y2 * y4+k3 * y5 * y2+k9 * y11 * y2+k11*y13+k12*y10*y2+k22*y19+k25*y20
dy2 = -k2 * y2 * y4-k3 * y5 * y2-k9 * y11 * y2-k12*y10*y2+k1 * y1+k21*y19
dy3 = -k15*y3+k1 * y1+k17*y4+k19*y16+k22*y19
dy4 = -k2 * y2 * y4-k16*y4-k17*y4-k23*y1*y4+k15*y3
dy5 = -k3 * y5 * y2+k4 * y7+k4 * y7+k6 * y7 * y6+k7 * y9+k13*y14+k20*y17*y6
dy6 = -k6 * y7 * y6-k8 * y9 * y6-k14*y1*y6-k20*y17*y6+k3 * y5 * y2+k18*y16+k18*y16
dy7 = -k4 * y7-k5 * y7-k6 * y7 * y6+k13*y14
dy8 = k4 * y7+k5 * y7+k6 * y7 * y6+k7 * y9
dy9 = -k7 * y9-k8 * y9 * y6
dy10 = -k12*y10*y2+k7 * y9+k9 * y11 * y2
dy11 = -k9 * y11 * y2-k10*y11*y1+k8 * y9 * y6+k11*y13
dy12 = k9 * y11 * y2
dy13 = -k11*y13+k10*y11*y1
dy14 = -k13*y14+k12*y10*y2
dy15 = k14*y1*y6
dy16 = -k18*y16-k19*y16+k16*y4
dy17 = -k20*y17*y6
dy18 = k20*y17*y6
dy19 = -k21*y19-k22*y19-k24*y19*y1+k23*y1*y4+k25*y20
dy20 = -k25*y20+k24*y19*y1
end k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12 k13 k14 k15 k16 k17 k18 k19 k20 k21 k22 k23 k24 k25
function make_pollution()
comp = let pollution = pollution, J = zeros(20, 20), JP = zeros(20, 25),
tmpdu = zeros(20, 25), tmpu = zeros(20, 25)
function comp(du, u, p, t)
tmpu .= @view(u[:, 2:26])
pollution(@view(du[:, 1]), u, p, t)
pollution.jac(J, u, p, t)
pollution.paramjac(JP, u, p, t)
mul!(tmpdu, J, tmpu)
du[:, 2:26] .= tmpdu .+ JP
nothing
end
end
u0 = zeros(20)
p = [.35e0, .266e2, .123e5, .86e-3, .82e-3, .15e5, .13e-3, .24e5, .165e5,
.9e4, .22e-1, .12e5, .188e1, .163e5, .48e7, .35e-3, .175e-1,
.1e9, .444e12, .124e4, .21e1, .578e1, .474e-1, .178e4, .312e1]
u0[2] = 0.2
u0[4] = 0.04
u0[7] = 0.1
u0[8] = 0.3
u0[9] = 0.01
u0[17] = 0.007
compu0 = zeros(20, 26)
compu0[1:20] .= u0
comp, u0, p, compu0
endmake_pollution (generic function with 1 method)function makebrusselator(N = 8)
xyd_brusselator = range(0, stop = 1, length = N)
function limit(a, N)
if a == N+1
return 1
elseif a == 0
return N
else
return a
end
end
brusselator_f(x, y, t) = ifelse(
(((x-0.3)^2 + (y-0.6)^2) <= 0.1^2) &&
(t >= 1.1), 5.0, 0.0)
brusselator_2d_loop = let N=N, xyd=xyd_brusselator, dx=step(xyd_brusselator)
function brusselator_2d_loop(du, u, p, t)
@inbounds begin
ii1 = N^2
ii2 = ii1+N^2
ii3 = ii2+2(N^2)
A = @view p[1:ii1]
B = @view p[(ii1 + 1):ii2]
α = @view p[(ii2 + 1):ii3]
II = LinearIndices((N, N, 2))
for I in CartesianIndices((N, N))
x = xyd[I[1]]
y = xyd[I[2]]
i = I[1]
j = I[2]
ip1 = limit(i+1, N);
im1 = limit(i-1, N)
jp1 = limit(j+1, N);
jm1 = limit(j-1, N)
du[II[i, j, 1]] = α[II[
i, j, 1]]*(u[II[im1, j, 1]] + u[II[ip1, j, 1]] +
u[II[i, jp1, 1]] + u[II[i, jm1, 1]] -
4u[II[i, j, 1]])/dx^2 +
B[II[i, j, 1]] + u[II[i, j, 1]]^2*u[II[i, j, 2]] -
(A[II[i, j, 1]] + 1)*u[II[i, j, 1]] +
brusselator_f(x, y, t)
end
for I in CartesianIndices((N, N))
i = I[1]
j = I[2]
ip1 = limit(i+1, N)
im1 = limit(i-1, N)
jp1 = limit(j+1, N)
jm1 = limit(j-1, N)
du[II[i, j, 2]] = α[II[
i, j, 2]]*(u[II[im1, j, 2]] + u[II[ip1, j, 2]] + u[II[i, jp1, 2]] +
u[II[i, jm1, 2]] - 4u[II[i, j, 2]])/dx^2 +
A[II[i, j, 1]]*u[II[i, j, 1]] -
u[II[i, j, 1]]^2*u[II[i, j, 2]]
end
return nothing
end
end
end
function init_brusselator_2d(xyd)
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
vec(u)
end
dx = step(xyd_brusselator)
e1 = ones(N-1)
off = N-1
e4 = ones(N-off)
T = diagm(0=>-2ones(N), -1=>e1, 1=>e1, off=>e4, -off=>e4) ./ dx^2
Ie = Matrix{Float64}(I, N, N)
# A + df/du
Op = kron(Ie, T) + kron(T, Ie)
brusselator_jac = let N=N
(J, a, p, t) -> begin
ii1 = N^2
ii2 = ii1+N^2
ii3 = ii2+2(N^2)
A = @view p[1:ii1]
B = @view p[(ii1 + 1):ii2]
α = @view p[(ii2 + 1):ii3]
u = @view a[1:(end ÷ 2)]
v = @view a[(end ÷ 2 + 1):end]
N2 = length(a)÷2
α1 = @view α[1:(end ÷ 2)]
α2 = @view α[(end ÷ 2 + 1):end]
fill!(J, 0)
J[1:N2, 1:N2] .= α1 .* Op
J[(N2 + 1):end, (N2 + 1):end] .= α2 .* Op
J1 = @view J[1:N2, 1:N2]
J2 = @view J[(N2 + 1):end, 1:N2]
J3 = @view J[1:N2, (N2 + 1):end]
J4 = @view J[(N2 + 1):end, (N2 + 1):end]
J1[diagind(J1)] .+= @. 2u*v-(A+1)
J2[diagind(J2)] .= @. A-2u*v
J3[diagind(J3)] .= @. u^2
J4[diagind(J4)] .+= @. -u^2
nothing
end
end
Jmat = zeros(2N*N, 2N*N)
dp = zeros(2N*N, 4N*N)
brusselator_comp = let N=N, xyd=xyd_brusselator, dx=step(xyd_brusselator), Jmat=Jmat,
dp=dp, brusselator_jac=brusselator_jac
function brusselator_comp(dus, us, p, t)
@inbounds begin
ii1 = N^2
ii2 = ii1+N^2
ii3 = ii2+2(N^2)
@views u, s = us[1:ii2], us[(ii2 + 1):end]
du = @view dus[1:ii2]
ds = @view dus[(ii2 + 1):end]
fill!(dp, 0)
A = @view p[1:ii1]
B = @view p[(ii1 + 1):ii2]
α = @view p[(ii2 + 1):ii3]
dfdα = @view dp[:, (ii2 + 1):ii3]
diagind(dfdα)
for i in 1:ii1
dp[i, ii1 + i] = 1
end
II = LinearIndices((N, N, 2))
uu = @view u[1:(end ÷ 2)]
for i in eachindex(uu)
dp[i, i] = -uu[i]
dp[i + ii1, i] = uu[i]
end
for I in CartesianIndices((N, N))
x = xyd[I[1]]
y = xyd[I[2]]
i = I[1]
j = I[2]
ip1 = limit(i+1, N);
im1 = limit(i-1, N)
jp1 = limit(j+1, N);
jm1 = limit(j-1, N)
au = dfdα[II[i, j, 1], II[i, j, 1]] = (u[II[im1, j, 1]] +
u[II[ip1, j, 1]] +
u[II[i, jp1, 1]] +
u[II[i, jm1, 1]] -
4u[II[i, j, 1]])/dx^2
du[II[i, j, 1]] = α[II[i, j, 1]]*(au) + B[II[i, j, 1]] +
u[II[i, j, 1]]^2*u[II[i, j, 2]] -
(A[II[i, j, 1]] + 1)*u[II[i, j, 1]] +
brusselator_f(x, y, t)
end
for I in CartesianIndices((N, N))
i = I[1]
j = I[2]
ip1 = limit(i+1, N)
im1 = limit(i-1, N)
jp1 = limit(j+1, N)
jm1 = limit(j-1, N)
av = dfdα[II[i, j, 2], II[i, j, 2]] = (u[II[im1, j, 2]] +
u[II[ip1, j, 2]] +
u[II[i, jp1, 2]] +
u[II[i, jm1, 2]] -
4u[II[i, j, 2]])/dx^2
du[II[i, j, 2]] = α[II[i, j, 2]]*(av) + A[II[i, j, 1]]*u[II[i, j, 1]] -
u[II[i, j, 1]]^2*u[II[i, j, 2]]
end
brusselator_jac(Jmat, u, p, t)
BLAS.gemm!('N', 'N', 1.0, Jmat, reshape(s, 2N*N, 4N*N), 1.0, dp)
copyto!(ds, vec(dp))
return nothing
end
end
end
u0 = init_brusselator_2d(xyd_brusselator)
p = [fill(3.4, N^2); fill(1.0, N^2); fill(10.0, 2*N^2)]
brusselator_2d_loop, u0,
p,
brusselator_jac,
ODEProblem{true, SciMLBase.FullSpecialize}(
brusselator_comp, copy([u0; zeros((N^2*2)*(N^2*4))]), (0.0, 10.0), p)
endmakebrusselator (generic function with 2 methods)Differentiation Setups
function diffeq_sen(prob, args...; kwargs...)
diffeq_sen(prob.f, prob.u0, prob.tspan, prob.p, args...; kwargs...)
end
function auto_sen(prob, args...; kwargs...)
auto_sen(prob.f, prob.u0, prob.tspan, prob.p, args...; kwargs...)
end
function diffeq_sen(
f, u0, tspan, p, alg = Tsit5(); sensalg = ForwardSensitivity(), kwargs...)
prob = ODEForwardSensitivityProblem(f, u0, tspan, p, sensalg)
sol = solve(prob, alg; save_everystep = false, kwargs...)
extract_local_sensitivities(sol, length(sol.t))[2]
end
function auto_sen(f, u0, tspan, p, alg = Tsit5(); kwargs...)
test_f(p) = begin
prob = ODEProblem{true, SciMLBase.FullSpecialize}(f, eltype(p).(u0), tspan, p)
solve(prob, alg; save_everystep = false, kwargs...).u[end]
end
ForwardDiff.jacobian(test_f, p)
end
function numerical_sen(f, u0, tspan, p, alg = Tsit5(); kwargs...)
test_f(out, p) = begin
prob = ODEProblem{true, SciMLBase.FullSpecialize}(f, eltype(p).(u0), tspan, p)
copyto!(out, solve(prob, alg; kwargs...).u[end])
end
J = Matrix{Float64}(undef, length(u0), length(p))
FiniteDiff.finite_difference_jacobian!(
J, test_f, p, FiniteDiff.JacobianCache(p, Array{Float64}(undef, length(u0))))
return J
end
function diffeq_sen_l2(df, u0, tspan, p, t, alg = Tsit5();
abstol = 1e-5, reltol = 1e-7,
sensalg = InterpolatingAdjoint(), kwargs...)
prob = ODEProblem(df, u0, tspan, p)
sol = solve(prob, alg, sensealg = SciMLSensitivity.SensitivityADPassThrough(),
abstol = abstol, reltol = reltol; kwargs...)
dg(out, u, p, t, i) = (out.=u .- 1.0)
adjoint_sensitivities(sol, alg; t, abstol = abstol, dgdu_discrete = dg,
reltol = reltol, sensealg = sensalg)[2]
end
function auto_sen_l2(
f, u0, tspan, p, t, alg = Tsit5(); diffalg = ReverseDiff.gradient, kwargs...)
test_f(p) = begin
prob = ODEProblem{true, SciMLBase.FullSpecialize}(f, eltype(p).(u0), tspan, p)
sol = solve(prob, alg; sensealg = SciMLSensitivity.SensitivityADPassThrough(), kwargs...)(t)
sum(sol.u) do x
sum(z->(1-z)^2/2, x)
end
end
diffalg(test_f, p)
end
function numerical_sen_l2(f, u0, tspan, p, t, alg = Tsit5(); kwargs...)
test_f(p) = begin
prob = ODEProblem(f, eltype(p).(u0), tspan, p)
sol = solve(prob, alg; kwargs...)(t)
sum(sol.u) do x
sum(z->(1-z)^2/2, x)
end
end
FiniteDiff.finite_difference_gradient(test_f, p, Val{:central})
endnumerical_sen_l2 (generic function with 2 methods)_adjoint_methods = ntuple(3) do ii
Alg = (InterpolatingAdjoint, QuadratureAdjoint, BacksolveAdjoint)[ii]
(
user = Alg(autodiff = false, autojacvec = false), # user Jacobian
adjc = Alg(autodiff = true, autojacvec = false), # AD Jacobian
advj = Alg(autodiff = true, autojacvec = EnzymeVJP()) # AD vJ
)
end |> NamedTuple{(:interp, :quad, :backsol)}
@isdefined(ADJOINT_METHODS) ||
(const ADJOINT_METHODS = mapreduce(collect, vcat, _adjoint_methods))9-element Vector{SciMLBase.AbstractAdjointSensitivityAlgorithm{0, AD, Val{:
central}} where AD}:
SciMLSensitivity.InterpolatingAdjoint{0, false, Val{:central}, Bool}(false
, false, false)
SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, Bool}(false,
false, false)
SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensiti
vity.EnzymeVJP{EnzymeCore.ReverseMode{false, false, false, EnzymeCore.FFIAB
I, false, false}}}(SciMLSensitivity.EnzymeVJP{EnzymeCore.ReverseMode{false,
false, false, EnzymeCore.FFIABI, false, false}}(0, EnzymeCore.ReverseMode{
false, false, false, EnzymeCore.FFIABI, false, false}()), false, false)
SciMLSensitivity.QuadratureAdjoint{0, false, Val{:central}, Bool, Val{true
}}(false, 1.0e-6, 0.001, Val{true}())
SciMLSensitivity.QuadratureAdjoint{0, true, Val{:central}, Bool, Val{true}
}(false, 1.0e-6, 0.001, Val{true}())
SciMLSensitivity.QuadratureAdjoint{0, true, Val{:central}, SciMLSensitivit
y.EnzymeVJP{EnzymeCore.ReverseMode{false, false, false, EnzymeCore.FFIABI,
false, false}}, Val{true}}(SciMLSensitivity.EnzymeVJP{EnzymeCore.ReverseMod
e{false, false, false, EnzymeCore.FFIABI, false, false}}(0, EnzymeCore.Reve
rseMode{false, false, false, EnzymeCore.FFIABI, false, false}()), 1.0e-6, 0
.001, Val{true}())
SciMLSensitivity.BacksolveAdjoint{0, false, Val{:central}, Bool}(false, tr
ue, false)
SciMLSensitivity.BacksolveAdjoint{0, true, Val{:central}, Bool}(false, tru
e, false)
SciMLSensitivity.BacksolveAdjoint{0, true, Val{:central}, SciMLSensitivity
.EnzymeVJP{EnzymeCore.ReverseMode{false, false, false, EnzymeCore.FFIABI, f
alse, false}}}(SciMLSensitivity.EnzymeVJP{EnzymeCore.ReverseMode{false, fal
se, false, EnzymeCore.FFIABI, false, false}}(0, EnzymeCore.ReverseMode{fals
e, false, false, EnzymeCore.FFIABI, false, false}()), true, false)Run Forward Mode Benchmarks
These are testing for the construction of the full Jacobian.
forward_lv = let
@info "Running the Lotka-Volterra model:"
@info " Running compile-time CSA"
t1 = @belapsed solve($lvprob, $(Tsit5()); $tols...)
@info " Running DSA"
t2 = @belapsed auto_sen($lvdf, $u0, $tspan, $p, $(Tsit5()); $tols...)
@info " Running CSA user-Jacobian"
t3 = @belapsed diffeq_sen($lvdf_with_jacobian, $u0, $tspan, $p, $(Tsit5());
sensalg = ForwardSensitivity(autodiff = false, autojacvec = false), $tols...)
@info " Running AD-Jacobian"
t4 = @belapsed diffeq_sen($lvdf, $u0, $tspan, $p, $(Tsit5());
sensalg = ForwardSensitivity(autojacvec = false), $tols...)
@info " Running AD-Jv seeding"
t5 = @belapsed diffeq_sen($lvdf, $u0, $tspan, $p, $(Tsit5());
sensalg = ForwardSensitivity(autojacvec = true), $tols...)
@info " Running numerical differentiation"
t6 = @belapsed numerical_sen($lvdf, $u0, $tspan, $p, $(Tsit5()); $tols...)
print('\n')
[t1, t2, t3, t4, t5, t6]
end6-element Vector{Float64}:
7.1329e-5
3.686e-5
0.000249157
0.000302037
0.000194648
0.000173358forward_bruss = let
@info "Running the Brusselator model:"
n = 5
# Run low tolerance to test correctness
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator(n)
sol1 = @time numerical_sen(
bfun, b_u0, (0.0, 10.0), b_p, Rodas5(), abstol = 1.0e-6, reltol = 1.0e-8
)
sol2 = @time auto_sen(
bfun, b_u0, (0.0, 10.0), b_p, Rodas5(), abstol = 1e-5, reltol = 1e-7);
@test sol1 ≈ sol2 atol=1e-2
sol3 = @time diffeq_sen(bfun, b_u0, (0.0, 10.0), b_p, Rodas5(autodiff = AutoFiniteDiff()),
abstol = 1e-5, reltol = 1e-7);
@test sol1 ≈ hcat(sol3...) atol=1e-3
sol4 = @time diffeq_sen(
ODEFunction{true, SciMLBase.FullSpecialize}(bfun, jac = brusselator_jac), b_u0,
(0.0, 10.0), b_p, Rodas5(autodiff = AutoFiniteDiff()), abstol = 1e-5, reltol = 1e-7,
sensalg = ForwardSensitivity(autodiff = false, autojacvec = false));
@test sol1 ≈ hcat(sol4...) atol=1e-2
sol5 = @time solve(brusselator_comp, Rodas5(autodiff = AutoFiniteDiff()), abstol = 1e-5, reltol = 1e-7);
@test sol1 ≈ reshape(sol5.u[end][(2n * n + 1):end], 2n*n, 4n*n) atol=1e-3
# High tolerance to benchmark
@info " Running compile-time CSA"
t1 = @belapsed solve($brusselator_comp, $(Rodas5(autodiff = AutoFiniteDiff())); $tols...);
@info " Running DSA"
t2 = @belapsed auto_sen($bfun, $b_u0, $((0.0, 10.0)), $b_p, $(Rodas5()); $tols...);
@info " Running CSA user-Jacobian"
t3 = @belapsed diffeq_sen(
$(ODEFunction{true, SciMLBase.FullSpecialize}(bfun, jac = brusselator_jac)),
$b_u0, $((0.0, 10.0)), $b_p, $(Rodas5(autodiff = AutoFiniteDiff()));
sensalg = ForwardSensitivity(autodiff = false, autojacvec = false), $tols...);
@info " Running AD-Jacobian"
t4 = @belapsed diffeq_sen(
$bfun, $b_u0, $((0.0, 10.0)), $b_p, $(Rodas5(autodiff = AutoFiniteDiff()));
sensalg = ForwardSensitivity(autojacvec = false), $tols...);
@info " Running AD-Jv seeding"
t5 = @belapsed diffeq_sen(
$bfun, $b_u0, $((0.0, 10.0)), $b_p, $(Rodas5(autodiff = AutoFiniteDiff()));
sensalg = ForwardSensitivity(autojacvec = true), $tols...);
@info " Running numerical differentiation"
t6 = @belapsed numerical_sen($bfun, $b_u0, $((0.0, 10.0)), $b_p, $(Rodas5()); $tols...);
print('\n')
[t1, t2, t3, t4, t5, t6]
end6.185136 seconds (11.95 M allocations: 589.283 MiB, 1.66% gc time, 91.58%
compilation time)
12.175880 seconds (23.76 M allocations: 1.088 GiB, 1.91% gc time, 97.08% c
ompilation time)
130.806484 seconds (13.10 M allocations: 1.182 GiB, 0.26% gc time, 5.14% co
mpilation time)
139.181700 seconds (19.02 M allocations: 1.633 GiB, 0.76% gc time, 5.60% co
mpilation time)
90.175737 seconds (7.85 M allocations: 1.087 GiB, 1.21% gc time, 5.26% com
pilation time)
6-element Vector{Float64}:
106.308739981
0.355186951
141.222430263
124.461914531
121.055864473
0.373140153forward_pollution = let
@info "Running the pollution model:"
pcomp, pu0, pp, pcompu0 = make_pollution()
ptspan = (0.0, 60.0)
@info " Running compile-time CSA"
t1 = 0#@belapsed solve($(ODEProblem(pcomp, pcompu0, ptspan, pp)), $(Rodas5(autodiff=AutoFiniteDiff())),);
@info " Running DSA"
t2 = @belapsed auto_sen($(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)),
$pu0, $ptspan, $pp, $(Rodas5()); $tols...);
@info " Running CSA user-Jacobian"
t3 = @belapsed diffeq_sen(
$(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f, jac = pollution.jac)),
$pu0, $ptspan, $pp, $(Rodas5(autodiff = AutoFiniteDiff()));
sensalg = ForwardSensitivity(autodiff = false, autojacvec = false), $tols...);
@info " Running AD-Jacobian"
t4 = @belapsed diffeq_sen($(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)),
$pu0, $ptspan, $pp, $(Rodas5(autodiff = AutoFiniteDiff()));
sensalg = ForwardSensitivity(autojacvec = false), $tols...);
@info " Running AD-Jv seeding"
t5 = @belapsed diffeq_sen($(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)),
$pu0, $ptspan, $pp, $(Rodas5(autodiff = AutoFiniteDiff()));
sensalg = ForwardSensitivity(autojacvec = true), $tols...);
@info " Running numerical differentiation"
t6 = @belapsed numerical_sen(
$(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)),
$pu0, $ptspan, $pp, $(Rodas5()); $tols...);
print('\n')
[t1, t2, t3, t4, t5, t6]
end6-element Vector{Float64}:
0.0
0.009334642
0.349390763
0.399817024
0.410473982
0.006437182forward_pkpd = let
@info "Running the PKPD model:"
#sol1 = solve(pkpdcompprob, Tsit5(),abstol=1e-5,reltol=1e-7,callback=pkpdcb,tstops=0:24:240,).u[end][6:end]
sol2 = vec(auto_sen(pkpdprob, Tsit5(), abstol = 1e-5, reltol = 1e-7,
callback = pkpdcb, tstops = 0:24:240))
sol3 = vec(hcat(diffeq_sen(pkpdprob, Tsit5(), abstol = 1e-5, reltol = 1e-7,
callback = pkpdcb, tstops = 0:24:240)...))
#@test sol1 ≈ sol2 atol=1e-3
@test sol2 ≈ sol3 atol=1e-3
@info " Running compile-time CSA"
#t1 = @belapsed solve($pkpdcompprob, $(Tsit5()),callback=$pkpdcb,tstops=0:24:240,);
@info " Running DSA"
t2 = @belapsed auto_sen($(pkpdf.f), $pkpdu0, $pkpdtspan, $pkpdp, $(Tsit5());
callback = $pkpdcb, tstops = 0:24:240, $tols...);
@info " Running CSA user-Jacobian"
t3 = @belapsed diffeq_sen(
$(ODEFunction{true, SciMLBase.FullSpecialize}(pkpdf.f, jac = pkpdf.jac)),
$pkpdu0, $pkpdtspan, $pkpdp, $(Tsit5()); callback = $pkpdcb, tstops = 0:24:240,
sensalg = ForwardSensitivity(autodiff = false, autojacvec = false), $tols...);
@info " Running AD-Jacobian"
t4 = @belapsed diffeq_sen($(pkpdf.f), $pkpdu0, $pkpdtspan, $pkpdp,
$(Tsit5()); callback = $pkpdcb, tstops = 0:24:240,
sensalg = ForwardSensitivity(autojacvec = false), $tols...);
@info " Running AD-Jv seeding"
t5 = @belapsed diffeq_sen($(pkpdf.f), $pkpdu0, $pkpdtspan, $pkpdp,
$(Tsit5()); callback = $pkpdcb, tstops = 0:24:240,
sensalg = ForwardSensitivity(autojacvec = true), $tols...);
@info " Running numerical differentiation"
t6 = @belapsed numerical_sen($(pkpdf.f), $pkpdu0, $pkpdtspan, $pkpdp, $(Tsit5());
callback = $pkpdcb, tstops = 0:24:240, $tols...);
print('\n')
[0, t2, t3, t4, t5, t6]
end6-element Vector{Float64}:
0.0
0.001704252
0.006008756
0.004646772
0.00570911
0.00665719forward_methods = ["Compile-time CSA", "DSA", "CSA user-Jacobian",
"AD-Jacobian", "AD-Jv seeding", "Numerical Differentiation"]
forward_timings = DataFrame(
methods = forward_methods, LV = forward_lv, Bruss = forward_bruss,
Pollution = forward_pollution, PKPD = forward_pkpd)
display(forward_timings)6×5 DataFrame
Row │ methods LV Bruss Pollution PKPD
⋯
│ String Float64 Float64 Float64 Floa
t64 ⋯
─────┼─────────────────────────────────────────────────────────────────────
─────
1 │ Compile-time CSA 7.1329e-5 106.309 0.0 0.0
⋯
2 │ DSA 3.686e-5 0.355187 0.00933464 0.00
170
3 │ CSA user-Jacobian 0.000249157 141.222 0.349391 0.00
600
4 │ AD-Jacobian 0.000302037 124.462 0.399817 0.00
464
5 │ AD-Jv seeding 0.000194648 121.056 0.410474 0.00
570 ⋯
6 │ Numerical Differentiation 0.000173358 0.37314 0.00643718 0.00
665
1 column om
ittedRun Adjoint Benchmarks
Adjoint requires a slightly different setup even with forward mode ADs since it requires a loss function choice. For that we simply take the L2 norm of the solution.
adjoint_lv = let
@info "Running the Lotka-Volerra model:"
lvu0 = [1.0, 1.0];
lvtspan = (0.0, 10.0);
lvp = [1.5, 1.0, 3.0];
lvt = 0:0.5:10
@time lsol1 = auto_sen_l2(
lvdf, lvu0, lvtspan, lvp, lvt, (Tsit5()); diffalg = (ForwardDiff.gradient), tols...);
@time lsol2 = auto_sen_l2(
lvdf, lvu0, lvtspan, lvp, lvt, (Tsit5()); diffalg = (ReverseDiff.gradient), tols...);
@time lsol3 = map(ADJOINT_METHODS) do alg
f = SciMLSensitivity.alg_autodiff(alg) ? lvdf : lvdf_with_jacobian
diffeq_sen_l2(f, lvu0, lvtspan, lvp, lvt, (Tsit5()); sensalg = alg, tols...)
end
@time lsol4 = numerical_sen_l2(lvdf, lvu0, lvtspan, lvp, lvt, Tsit5(); tols...);
@test maximum(abs, lsol1 .- lsol2)/maximum(abs, lsol1) < 0.2
@test all(i -> maximum(abs, lsol1 .- lsol3[i]')/maximum(abs, lsol1) < 0.2, eachindex(ADJOINT_METHODS))
@test maximum(abs, lsol1 .- lsol4)/maximum(abs, lsol1) < 0.2
t1 = @belapsed auto_sen_l2($lvdf, $lvu0, $lvtspan, $lvp, $lvt, $(Tsit5());
diffalg = $(ForwardDiff.gradient), $tols...);
t2 = @belapsed auto_sen_l2($lvdf, $lvu0, $lvtspan, $lvp, $lvt, $(Tsit5());
diffalg = $(ReverseDiff.gradient), $tols...);
t3 = map(ADJOINT_METHODS) do alg
f = SciMLSensitivity.alg_autodiff(alg) ? lvdf : lvdf_with_jacobian
@belapsed diffeq_sen_l2(
$f, $lvu0, $lvtspan, $lvp, $lvt, $(Tsit5()); sensalg = $alg, $tols...);
end
t4 = @belapsed numerical_sen_l2(
$lvdf, $lvu0, $lvtspan, $lvp, $lvt, $(Tsit5()); $tols...);
[t1; t2; t3; t4]
end4.594284 seconds (11.81 M allocations: 598.924 MiB, 1.32% gc time, 99.98%
compilation time)
7.282119 seconds (14.90 M allocations: 738.138 MiB, 1.62% gc time, 99.78%
compilation time)
119.408145 seconds (236.04 M allocations: 11.819 GiB, 1.98% gc time, 99.96%
compilation time)
0.789846 seconds (2.01 M allocations: 103.823 MiB, 7.12% gc time, 99.75%
compilation time)
12-element Vector{Float64}:
8.2219e-5
0.010057415
0.000434305
0.000541435
0.000369166
0.000673583
0.001309106
0.002425585
0.000683702
0.00098385
0.000600673
0.000496205adjoint_bruss = let
@info "Running the Brusselator model:"
bt = 0:0.1:10
tspan = (0.0, 10.0)
n = 5
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator(n)
@time bsol1 = auto_sen_l2(
bfun, b_u0, tspan, b_p, bt, (Rodas5()); diffalg = (ForwardDiff.gradient), tols...);
#@time bsol2 = auto_sen_l2(bfun, b_u0, tspan, b_p, bt, (Rodas5(autodiff=AutoFiniteDiff())); diffalg=(ReverseDiff.gradient), tols...);
#@test maximum(abs, bsol1 .- bsol2)/maximum(abs, bsol1) < 1e-2
@time bsol3 = map(ADJOINT_METHODS) do alg
@info "Running $alg"
f = SciMLSensitivity.alg_autodiff(alg) ? bfun :
ODEFunction{true, SciMLBase.FullSpecialize}(bfun, jac = brusselator_jac)
solver = Rodas5(autodiff = AutoFiniteDiff())
diffeq_sen_l2(
f, b_u0, tspan, b_p, bt, solver, reltol = 1e-7; sensalg = alg, tols...)
end
@time bsol4 = numerical_sen_l2(bfun, b_u0, tspan, b_p, bt, (Rodas5()); tols...);
# NOTE: backsolve gives unstable results!!!
@test all(i->maximum(abs, bsol1 .- bsol3[i]')/maximum(abs, bsol1) < 4e-2,
eachindex(ADJOINT_METHODS)[1:(2end ÷ 3)])
@test all(i->maximum(abs, bsol1 .- bsol3[i]')/maximum(abs, bsol1) >= 4e-2,
eachindex(ADJOINT_METHODS)[(2end ÷ 3 + 1):end])
@test maximum(abs, bsol1 .- bsol4)/maximum(abs, bsol1) < 2e-2
t1 = @belapsed auto_sen_l2($bfun, $b_u0, $tspan, $b_p, $bt, $(Rodas5());
diffalg = $(ForwardDiff.gradient), $tols...);
#t2 = @belapsed auto_sen_l2($bfun, $b_u0, $tspan, $b_p, $bt, $(Rodas5(autodiff=AutoFiniteDiff())); diffalg=$(ReverseDiff.gradient), $tols...);
t2 = NaN
t3 = map(ADJOINT_METHODS[1:(2end ÷ 3)]) do alg
@info "Running $alg"
f = SciMLSensitivity.alg_autodiff(alg) ? bfun :
ODEFunction{true, SciMLBase.FullSpecialize}(bfun, jac = brusselator_jac)
solver = Rodas5(autodiff = AutoFiniteDiff())
@elapsed diffeq_sen_l2(f, b_u0, tspan, b_p, bt, solver; sensalg = alg, tols...);
end
t3 = [t3; fill(NaN, length(ADJOINT_METHODS)÷3)]
t4 = @belapsed numerical_sen_l2($bfun, $b_u0, $tspan, $b_p, $bt, $(Rodas5()); $tols...);
[t1; t2; t3; t4]
end11.350090 seconds (20.46 M allocations: 968.789 MiB, 1.10% gc time, 96.80%
compilation time)
100.195275 seconds (110.06 M allocations: 5.430 GiB, 1.19% gc time, 91.10%
compilation time: 1% of which was recompilation)
5.860041 seconds (6.16 M allocations: 343.872 MiB, 2.15% gc time, 80.52%
compilation time: <1% of which was recompilation)
12-element Vector{Float64}:
0.358773007
NaN
2.840470629
1.210936904
0.378493741
0.162833517
0.2371262
0.200567742
NaN
NaN
NaN
1.065763773adjoint_pollution = let
@info "Running the Pollution model:"
pcomp, pu0, pp, pcompu0 = make_pollution();
ptspan = (0.0, 60.0)
pts = 0:0.5:60
@time psol1 = auto_sen_l2(
(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)), pu0, ptspan, pp,
pts, (Rodas5(autodiff = AutoFiniteDiff())); diffalg = (ForwardDiff.gradient), tols...);
#@time psol2 = auto_sen_l2((ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)), pu0, ptspan, pp, pts, (Rodas5(autodiff=AutoFiniteDiff())); diffalg=(ReverseDiff.gradient), tols...);
#@test maximum(abs, psol1 .- psol2)/maximum(abs, psol1) < 1e-2
@time psol3 = map(ADJOINT_METHODS) do alg
@info "Running $alg"
f = SciMLSensitivity.alg_autodiff(alg) ? pollution.f :
ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f, jac = pollution.jac)
solver = Rodas5(autodiff = AutoFiniteDiff())
diffeq_sen_l2(f, pu0, ptspan, pp, pts, solver; sensalg = alg, tols...);
end
@time psol4 = numerical_sen_l2(
(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)),
pu0, ptspan, pp, pts, (Rodas5(autodiff = AutoFiniteDiff())); tols...);
# NOTE: backsolve gives unstable results!!!
@test all(i->maximum(abs, psol1 .- psol3[i]')/maximum(abs, psol1) < 1e-2,
eachindex(ADJOINT_METHODS)[1:(2end ÷ 3)])
@test all(i->maximum(abs, psol1 .- psol3[i]')/maximum(abs, psol1) >= 1e-2,
eachindex(ADJOINT_METHODS)[(2end ÷ 3 + 1):end])
@test maximum(abs, psol1 .- psol4)/maximum(abs, psol1) < 1e-2
t1 = @belapsed auto_sen_l2(
$(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)), $pu0, $ptspan, $pp,
$pts, $(Rodas5(autodiff = AutoFiniteDiff())); diffalg = $(ForwardDiff.gradient), $tols...);
#t2 = @belapsed auto_sen_l2($(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)), $pu0, $ptspan, $pp, $pts, $(Rodas5(autodiff=AutoFiniteDiff())); diffalg=$(ReverseDiff.gradient), $tols...);
t2 = NaN
t3 = map(ADJOINT_METHODS[1:(2end ÷ 3)]) do alg
@info "Running $alg"
f = SciMLSensitivity.alg_autodiff(alg) ? pollution.f :
ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f, jac = pollution.jac)
solver = Rodas5(autodiff = AutoFiniteDiff())
@elapsed diffeq_sen_l2(f, pu0, ptspan, pp, pts, solver; sensalg = alg, tols...);
end
t3 = [t3; fill(NaN, length(ADJOINT_METHODS)÷3)]
t4 = @belapsed numerical_sen_l2(
$(ODEFunction{true, SciMLBase.FullSpecialize}(pollution.f)),
$pu0, $ptspan, $pp, $pts, $(Rodas5(autodiff = AutoFiniteDiff())); $tols...);
[t1; t2; t3; t4]
end10.051955 seconds (18.04 M allocations: 833.678 MiB, 1.03% gc time, 99.92%
compilation time)
80.252121 seconds (85.93 M allocations: 4.057 GiB, 1.06% gc time, 94.27% c
ompilation time)
9.672638 seconds (37.53 M allocations: 1.764 GiB, 5.18% gc time, 99.86% c
ompilation time)
12-element Vector{Float64}:
0.007404692
NaN
0.618838138
1.186713499
0.240339296
0.149494257
0.49532423
0.428487439
NaN
NaN
NaN
0.012336601adjoint_pkpd = let
@info "Running the PKPD model:"
pts = 0:0.5:50
# need to use lower tolerances to avoid running into the complex domain because of exponentiation
pkpdsol1 = @time auto_sen_l2((pkpdf.f), pkpdu0, pkpdtspan, pkpdp, pts,
(Tsit5()); callback = pkpdcb, tstops = 0:24:240,
diffalg = (ForwardDiff.gradient), tols...);
pkpdsol2 = @time auto_sen_l2((pkpdf.f), pkpdu0, pkpdtspan, pkpdp, pts,
(Tsit5()); callback = pkpdcb, tstops = 0:24:240,
diffalg = (ReverseDiff.gradient), tols...);
pkpdsol3 = @time map(ADJOINT_METHODS[1:(2end ÷ 3)]) do alg
f = SciMLSensitivity.alg_autodiff(alg) ? pkpdf.f :
ODEFunction{true, SciMLBase.FullSpecialize}(pkpdf.f, jac = pkpdf.jac)
diffeq_sen_l2(f, pkpdu0, pkpdtspan, pkpdp, pts, (Tsit5()); sensalg = alg,
callback = pkpdcb, tstops = 0:24:240, tols...);
end
pkpdsol4 = @time numerical_sen_l2(
(ODEFunction{true, SciMLBase.FullSpecialize}(pkpdf.f)),
pkpdu0, pkpdtspan, pkpdp, pts, (Tsit5());
callback = pkpdcb, tstops = 0:24:240, tols...);
@test maximum(abs, pkpdsol1 .- pkpdsol2)/maximum(abs, pkpdsol1) < 0.2
@test all(i->maximum(abs, pkpdsol1 .- pkpdsol3[i]')/maximum(abs, pkpdsol1) < 0.2,
eachindex(ADJOINT_METHODS)[1:(2end ÷ 3)])
@test maximum(abs, pkpdsol1 .- pkpdsol4)/maximum(abs, pkpdsol1) < 0.2
t1 = @belapsed auto_sen_l2($(pkpdf.f), $pkpdu0, $pkpdtspan, $pkpdp, $pts,
$(Tsit5()); callback = pkpdcb, tstops = 0:24:240,
diffalg = $(ForwardDiff.gradient), $tols...);
t2 = @belapsed auto_sen_l2($(pkpdf.f), $pkpdu0, $pkpdtspan, $pkpdp, $pts,
$(Tsit5()); callback = pkpdcb, tstops = 0:24:240,
diffalg = $(ReverseDiff.gradient), $tols...);
t3 = map(ADJOINT_METHODS[1:(2end ÷ 3)]) do alg
f = SciMLSensitivity.alg_autodiff(alg) ? pkpdf.f :
ODEFunction{true, SciMLBase.FullSpecialize}(pkpdf.f, jac = pkpdf.jac)
@belapsed diffeq_sen_l2(
$f, $pkpdu0, $pkpdtspan, $pkpdp, $pts, $(Tsit5()); tstops = 0:24:240,
callback = pkpdcb, sensalg = $alg, tols...);
end
t3 = [t3; fill(NaN, length(ADJOINT_METHODS)÷3)]
t4 = @belapsed numerical_sen_l2(
$(ODEFunction{true, SciMLBase.FullSpecialize}(pkpdf.f)), $pkpdu0,
$pkpdtspan, $pkpdp, $pts, $(Tsit5()); tstops = 0:24:240,
callback = $pkpdcb, $tols...);
[t1; t2; t3; t4]
end4.460480 seconds (11.64 M allocations: 582.540 MiB, 1.39% gc time, 99.91%
compilation time)
4.754505 seconds (10.18 M allocations: 494.269 MiB, 1.35% gc time, 92.93%
compilation time)
35.851012 seconds (69.62 M allocations: 3.479 GiB, 1.40% gc time, 99.82% c
ompilation time)
1.201130 seconds (7.45 M allocations: 386.080 MiB, 5.49% gc time, 97.27%
compilation time)
12-element Vector{Float64}:
0.002215357
0.241353314
0.006080418
0.004376025
0.002023179
0.004203717
0.005294625
0.007964608
NaN
NaN
NaN
0.014842027adjoint_methods = ["ForwardDiff", "ReverseDiff",
"InterpolatingAdjoint User Jac", "InterpolatingAdjoint AD Jac", "InterpolatingAdjoint v'J",
"QuadratureAdjoint User Jac", "QuadratureAdjoint AD Jac", "QuadratureAdjoint v'J",
"BacksolveAdjoint User Jac", "BacksolveAdjoint AD Jac", "BacksolveAdjoint v'J",
"Numerical Differentiation"]
adjoint_timings = DataFrame(
methods = adjoint_methods, LV = adjoint_lv, Bruss = adjoint_bruss,
Pollution = adjoint_pollution, PKPD = adjoint_pkpd)
Markdown.parse(PrettyTables.pretty_table(
String, adjoint_timings; backend = :markdown, column_labels = names(adjoint_timings)))| methods | LV | Bruss | Pollution | PKPD |
|---|---|---|---|---|
| ForwardDiff | 8.2219e-5 | 0.358773 | 0.00740469 | 0.00221536 |
| ReverseDiff | 0.0100574 | NaN | NaN | 0.241353 |
| InterpolatingAdjoint User Jac | 0.000434305 | 2.84047 | 0.618838 | 0.00608042 |
| InterpolatingAdjoint AD Jac | 0.000541435 | 1.21094 | 1.18671 | 0.00437603 |
| InterpolatingAdjoint v'J | 0.000369166 | 0.378494 | 0.240339 | 0.00202318 |
| QuadratureAdjoint User Jac | 0.000673583 | 0.162834 | 0.149494 | 0.00420372 |
| QuadratureAdjoint AD Jac | 0.00130911 | 0.237126 | 0.495324 | 0.00529462 |
| QuadratureAdjoint v'J | 0.00242559 | 0.200568 | 0.428487 | 0.00796461 |
| BacksolveAdjoint User Jac | 0.000683702 | NaN | NaN | NaN |
| BacksolveAdjoint AD Jac | 0.00098385 | NaN | NaN | NaN |
| BacksolveAdjoint v'J | 0.000600673 | NaN | NaN | NaN |
| Numerical Differentiation | 0.000496205 | 1.06576 | 0.0123366 | 0.014842 |
Appendix
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/AutomaticDifferentiation","SimpleODEAD.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_DEPOT_PATH = /home/crackauc/github-runners/amdci8-1/.julia
JULIA_NUM_THREADS = auto
Package Information:
Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/AutomaticDifferentiation/Project.toml`
[6e4b80f9] BenchmarkTools v1.8.0
[0ca39b1e] Chairmarks v1.3.1
[a93c6f00] DataFrames v1.8.2
[1313f7d8] DataFramesMeta v0.15.6
[a0c0ee7d] DifferentiationInterface v0.7.21
[a82114a7] DifferentiationInterfaceTest v0.11.0
[7da242da] Enzyme v0.13.199
[6a86dc24] FiniteDiff v2.33.0
[f6369f11] ForwardDiff v1.4.5
[7ed4a6bd] LinearSolve v5.14.1
[da2b9cff] Mooncake v0.5.48
[1dea7af3] OrdinaryDiffEq v7.8.1
[43230ef6] OrdinaryDiffEqRosenbrock v2.7.1
[65888b18] ParameterizedFunctions v5.27.0
[91a5bcdd] Plots v1.41.7
[08abe8d2] PrettyTables v3.4.8
[37e2e3b7] ReverseDiff v1.17.0
⌃ [31c91b34] SciMLBenchmarks v0.1.3 [loaded: v0.2.0]
[1ed8b502] SciMLSensitivity v7.119.1
[90137ffa] StaticArrays v1.9.19
[c3572dad] Sundials v6.6.0
[9f7883ad] Tracker v0.2.38
[e88e6eb3] Zygote v0.7.12
[37e2e46d] LinearAlgebra v1.12.0
[d6f4376e] Markdown v1.11.0
[de0858da] Printf v1.11.0
[8dfed614] Test v1.11.0
Info Packages marked with ⌃ have new versions available and may be upgradable.And the full manifest:
Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/AutomaticDifferentiation/Manifest.toml`
[47edcb42] ADTypes v1.24.0
[14f7f29c] AMD v0.5.3
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[79e6a3ab] Adapt v4.7.0
[66dad0bd] AliasTables v1.1.3
[9b6a8646] AllocCheck v0.2.6
[ec485272] ArnoldiMethod v0.4.0
[4fba245c] ArrayInterface v7.30.0
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[a9b6321e] Atomix v1.1.3
[ab4f0b2a] BFloat16s v0.6.1
[aae01518] BandedMatrices v1.12.0
[6e4b80f9] BenchmarkTools v1.8.0
[e2ed5e7c] Bijections v0.2.2
[b2a6c25c] BinaryHeaps v1.1.0
[caf10ac8] BipartiteGraphs v0.1.12
[8e7c35d0] BlockArrays v1.10.0
[70df07ce] BracketingNonlinearSolve v1.12.6
[fa961155] CEnum v0.5.0
[8be319e6] Chain v1.0.0
[082447d4] ChainRules v1.73.0
[d360d2e6] ChainRulesCore v1.26.1
[0ca39b1e] Chairmarks v1.3.1
[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 v0.2.14
[bbf7d656] CommonSubexpressions v0.3.1
[f70d9fcc] CommonWorldInvalidations v1.2.0
[34da2185] Compat v4.18.1
[b152e2b5] CompositeTypes v0.1.4
[a33af91c] CompositionsBase v0.1.2
[2569d6c7] ConcreteStructs v0.2.8
[8f4d0f93] Conda v1.10.3
[187b0558] ConstructionBase v1.6.0
[d38c429a] Contour v0.6.3
[a8cc5b0e] Crayons v4.2.0
[9a962f9c] DataAPI v1.16.0
[a93c6f00] DataFrames v1.8.2
[1313f7d8] DataFramesMeta v0.15.6
[864edb3b] DataStructures v0.19.6
[e2d170a0] DataValueInterfaces v1.0.0
[8bb1440f] DelimitedFiles v1.9.1
[2b5f629d] DiffEqBase v7.19.0
[459566f4] DiffEqCallbacks v4.19.3
[77a26b50] DiffEqNoiseProcess v5.36.1
[163ba53b] DiffResults v1.1.0
[b552c78f] DiffRules v1.16.0
[a0c0ee7d] DifferentiationInterface v0.7.21
[a82114a7] DifferentiationInterfaceTest v0.11.0
[8d63f2c5] DispatchDoctor v0.4.28
[31c24e10] Distributions v0.25.131
[ffbed154] DocStringExtensions v0.9.5
[5b8099bc] DomainSets v0.8.1
[7c1d4256] DynamicPolynomials v0.6.7
[4e289a0a] EnumX v1.0.7
[7da242da] Enzyme v0.13.199
[f151be2c] EnzymeCore v0.8.21
[e2ba6199] ExprTools v0.1.11
[55351af7] ExproniconLite v0.10.14
[c87230d0] FFMPEG v0.4.5
[7034ab61] FastBroadcast v1.4.0
[9aa1b823] FastClosures v0.3.2
[a4df4552] FastPower v1.5.0
[1a297f60] FillArrays v1.17.0
[64ca27bc] FindFirstFunctions v3.2.1
[6a86dc24] FiniteDiff v2.33.0
⌅ [53c48c17] FixedPointNumbers v0.8.6
[1fa38f19] Format v1.3.7
[f6369f11] ForwardDiff v1.4.5
[a85aefff] FunctionMaps v0.1.2
[f62d2435] FunctionProperties v1.2.0
[069b7b12] FunctionWrappers v1.1.3
[77dc65aa] FunctionWrappersWrappers v1.13.0
[d9f16b24] Functors v0.5.3
[46192b85] GPUArraysCore v0.2.0
⌅ [61eb1bfa] GPUCompiler v1.23.0
[28b8d3ca] GR v0.73.27
⌃ [a0844989] Gamma v1.1.0
[d7ba0133] Git v1.5.0
[86223c79] Graphs v1.14.0
[42e2da0e] Grisu v1.0.2
[076d061b] HashArrayMappedTries v0.2.0
⌅ [eafb193a] Highlights v0.5.3
[34004b35] HypergeometricFunctions v0.3.30
[7073ff75] IJulia v1.34.4
[7869d1d1] IRTools v0.4.20
[3263718b] ImplicitDiscreteSolve v2.2.0
[d25df0c9] Inflate v0.1.5
[842dd82b] InlineStrings v1.4.5
[18e54dd8] IntegerMathUtils v0.1.4
[8197267c] IntervalSets v0.7.14
[3587e190] InverseFunctions v0.1.17
[41ab1584] InvertedIndices v1.3.1
[92d709cd] IrrationalConstants v0.2.6
[82899510] IteratorInterfaceExtensions v1.0.0
[1019f520] JLFzf v0.1.11
[692b3bcd] JLLWrappers v1.8.0
⌅ [682c06a0] JSON v0.21.4
[ae98c720] Jieko v0.2.1
[ccbc3e58] JumpProcesses v9.30.1
[63c18a36] KernelAbstractions v0.9.42
[ba0b0d4f] Krylov v0.10.9
[2faa5264] LHLFactorization v2.2.1
[929cbde3] LLVM v9.13.1
[b964fa9f] LaTeXStrings v1.4.1
[23fbe1c1] Latexify v0.16.12
[87fe0de2] LineSearch v0.1.16
[7ed4a6bd] LinearSolve v5.14.1
⌅ [2ab3a3ac] LogExpFunctions v0.3.29
[e6f89c97] LoggingExtras v1.2.0
[1914dd2f] MacroTools v0.5.16
[bb5d69b7] MaybeInplace v0.1.8
[442fdcdd] Measures v0.3.3
[e1d29d7a] Missings v1.2.0
[dbe65cb8] MistyClosures v2.1.0
[961ee093] ModelingToolkit v11.40.0
⌃ [7771a370] ModelingToolkitBase v1.68.0
[6bb917b9] ModelingToolkitTearing v1.20.6
[da2b9cff] Mooncake v0.5.48
[2e0e35c7] Moshi v0.3.12
[46d2c3a1] MuladdMacro v0.2.7
[102ac46a] MultivariatePolynomials v0.5.19
[ffc61752] Mustache v1.0.21
[d8a4904e] MutableArithmetics v1.8.0
[872c559c] NNlib v0.9.45
[77ba4419] NaNMath v1.1.4
[8913a72c] NonlinearSolve v4.28.1
[be0214bd] NonlinearSolveBase v2.48.2
[5959db7a] NonlinearSolveFirstOrder v2.4.1
[9a2c21bd] NonlinearSolveQuasiNewton v1.15.2
[26075421] NonlinearSolveSpectralMethods v1.8.1
[d8793406] ObjectFile v0.5.1
[6fe1bfb0] OffsetArrays v1.17.0
[3bd65402] Optimisers v0.4.9
⌅ [bac558e1] OrderedCollections v1.8.2 [loaded: v2.0.1]
[1dea7af3] OrdinaryDiffEq v7.8.1
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.5
[bbf590c4] OrdinaryDiffEqCore v4.15.2
[50262376] OrdinaryDiffEqDefault v2.6.0
[4302a76b] OrdinaryDiffEqDifferentiation v3.11.0
⌃ [127b3ac7] OrdinaryDiffEqNonlinearSolve v2.9.2
[43230ef6] OrdinaryDiffEqRosenbrock v2.7.1
[b4bd8bb3] OrdinaryDiffEqRosenbrockTableaus v2.4.2
[2d112036] OrdinaryDiffEqSDIRK v2.9.1
[b1df2697] OrdinaryDiffEqTsit5 v2.1.4
[79d7bb75] OrdinaryDiffEqVerner v2.4.1
[90014a1f] PDMats v0.11.41
[65888b18] ParameterizedFunctions v5.27.0
⌅ [69de0a69] Parsers v2.8.7
[ccf2f8ad] PlotThemes v3.3.0
[995b91a9] PlotUtils v1.4.4
[91a5bcdd] Plots v1.41.7
[e409e4f3] PoissonRandom v0.4.13
[2dfb63ee] PooledArrays v1.4.3
[d236fae5] PreallocationTools v1.7.1
[aea7be01] PrecompileTools v1.3.4
[21216c6a] Preferences v1.5.2
[08abe8d2] PrettyTables v3.4.8
[27ebfcd6] Primes v0.5.7
[92933f4c] ProgressMeter v1.11.0
[43287f4e] PtrArrays v1.4.0
[0c0d3e7f] PureKLU v1.4.1
[1fd47b50] QuadGK v2.11.3
[e6cf234a] RandomNumbers v1.6.0
[988b38a3] ReadOnlyArrays v0.2.0
[795d4caa] ReadOnlyDicts v1.0.1
[c1ae055f] RealDot v0.1.0
[3cdcf5f2] RecipesBase v1.3.4
[01d81517] RecipesPipeline v0.6.12
[731186ca] RecursiveArrayTools v4.5.1
[189a3867] Reexport v1.2.2
[05181044] RelocatableFolders v1.0.1
[ae029012] Requires v1.3.1
[ae5879a3] ResettableStacks v1.4.0
[9fe22ead] RespecializeParams v1.3.0
[37e2e3b7] ReverseDiff v1.17.0
[79098fc4] Rmath v0.9.0
[f2b01f46] Roots v3.0.7
[7e49a35a] RuntimeGeneratedFunctions v0.5.25
[9dfe8606] SCCNonlinearSolve v1.15.1
[0bca4576] SciMLBase v3.50.0
⌃ [31c91b34] SciMLBenchmarks v0.1.3 [loaded: v0.2.0]
[19f34311] SciMLJacobianOperators v0.1.18
[a6db7da4] SciMLLogging v2.1.0
[c0aeaf25] SciMLOperators v1.30.0
[431bcebd] SciMLPublic v1.3.0
[1ed8b502] SciMLSensitivity v7.119.1
[53ae85a6] SciMLStructures v1.10.5
[7e506255] ScopedValues v1.6.2
[6c6a2e73] Scratch v1.3.0
[91c51154] SentinelArrays v1.4.10
[efcf1570] Setfield v1.1.2
[992d4aef] Showoff v1.0.3
[727e6d20] SimpleNonlinearSolve v2.14.1
[699a6c99] SimpleTraits v0.9.6
[a2af1166] SortingAlgorithms v1.2.3
[a57abbd0] SparseColumnPivotedQR v2.1.7
[dc90abb0] SparseInverseSubset v0.1.3
[0a514795] SparseMatrixColorings v0.4.27
[276daf66] SpecialFunctions v2.9.0
[860ef19b] StableRNGs v1.0.4
[0c0c59c1] StarAlgebras v0.3.0
[64909d44] StateSelection v1.11.1
[90137ffa] StaticArrays v1.9.19
[1e83bf80] StaticArraysCore v1.4.4
[10745b16] Statistics v1.11.4
[82ae8749] StatsAPI v1.8.0
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[2efcf032] SymbolicIndexingInterface v0.3.55
[19f23fe9] SymbolicLimits v1.2.0
⌅ [d1185830] SymbolicUtils v4.45.0
[0c5d862f] Symbolics v7.39.0
[9ce81f87] TableMetadataTools v0.1.0
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.14.0
[ed4db957] TaskLocalValues v0.1.3
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v2.0.0
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[9f7883ad] Tracker v0.2.38
[e689c965] Tracy v0.1.6
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[1cfade01] UnicodeFun v0.4.1
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[44d3d7a6] Weave v0.10.12
[ddb6d928] YAML v0.4.16
[c2297ded] ZMQ v1.5.1
[e88e6eb3] Zygote v0.7.12
[700de1a5] ZygoteRules v0.2.8
[6e34b625] Bzip2_jll v1.0.9+0
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[ee1fde0b] Dbus_jll v1.16.2+0
⌅ [7cc45869] Enzyme_jll v0.0.290+0
[2702e6a9] EpollShim_jll v0.0.20230411+1
[2e619515] Expat_jll v2.8.3+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
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[020c3dae] Git_LFS_jll v3.7.1+0
[f8c6e375] Git_jll v2.55.0+0
[7746bdde] Glib_jll v2.88.3+0
[3b182d85] Graphite2_jll v1.3.16+0
[2e76f6c2] HarfBuzz_jll v100.14003.0+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.2.0+1
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.1.0+0
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[1d63c593] LLVMOpenMP_jll v22.1.7+0
[ad6e5548] LibTracyClient_jll v0.13.1+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
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[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[856f044c] MKL_jll v2025.2.0+0
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[9bd350c2] OpenSSH_jll v10.5.1+0
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[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
[ca45d3f4] SuiteSparse32_jll v7.12.1+0
[fb77eaff] Sundials_jll v7.5.0+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.3+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
[8f1865be] ZeroMQ_jll v4.3.6+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
[a9144af2] libsodium_jll v1.0.21+0
[9a156e7d] libva_jll v2.23.0+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.3.0+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`