Bruss Scaling PDE Differentaition Benchmarks
From the paper A Comparison of Automatic Differentiation and Continuous Sensitivity Analysis for Derivatives of Differential Equation Solutions
using OrdinaryDiffEq, ReverseDiff, ForwardDiff, FiniteDiff, SciMLSensitivity
using LinearAlgebra, Tracker, Plotsfunction 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.)
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., Jmat, reshape(s, 2N*N, 4N*N), 1., dp)
copyto!(ds, vec(dp))
return nothing
end
end
end
u0 = init_brusselator_2d(xyd_brusselator)
p = [fill(3.4,N^2); fill(1.,N^2); fill(10.,2*N^2)]
brusselator_2d_loop, u0, p, brusselator_jac, ODEProblem(brusselator_comp, copy([u0;zeros((N^2*2)*(N^2*4))]), (0.,10.), p)
end
Base.eps(::Type{Tracker.TrackedReal{T}}) where T = eps(T)
Base.vec(v::Adjoint{<:Real, <:AbstractVector}) = vec(v') # bad bad hackSetup AutoDiff
bt = 0:0.1:1
tspan = (0.0, 1.0)
forwarddiffn = vcat(2:10,12,15)
reversediffn = 2:10
numdiffn = vcat(2:10,12)
csan = vcat(2:10,12,15,17)
#csaseedn = 2:10
tols = (abstol=1e-5, reltol=1e-7)
@isdefined(PROBS) || (const PROBS = Dict{Int,Any}())
makebrusselator!(dict, n) = get!(()->makebrusselator(n), dict, n)
_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))
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,convert.(eltype(p),u0),tspan,p)
sol = solve(prob,alg,saveat=t; kwargs...)
sum(sol.u) do x
sum(z->(1-z)^2/2, x)
end
end
diffalg(test_f, p)
end
@inline function diffeq_sen_l2(df, u0, tspan, p, t, alg=Tsit5();
abstol=1e-5, reltol=1e-7, iabstol=abstol, ireltol=reltol,
sensalg=SensitivityAlg(), kwargs...)
prob = ODEProblem{true, SciMLBase.FullSpecialize}(df,u0,tspan,p)
saveat = tspan[1] != t[1] && tspan[end] != t[end] ? vcat(tspan[1],t,tspan[end]) : t
sol = solve(prob, alg, abstol=abstol, reltol=reltol, saveat=saveat; 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)
enddiffeq_sen_l2 (generic function with 2 methods)AD Choice Benchmarks
forwarddiff = map(forwarddiffn) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
@elapsed auto_sen_l2(bfun, b_u0, tspan, b_p, bt, (Rodas5()); diffalg=(ForwardDiff.gradient), tols...)
t = @elapsed auto_sen_l2(bfun, b_u0, tspan, b_p, bt, (Rodas5()); diffalg=(ForwardDiff.gradient), tols...)
@show n,t
t
end(n, t) = (2, 0.000638025)
(n, t) = (3, 0.014818023)
(n, t) = (4, 0.037541015)
(n, t) = (5, 0.520653741)
(n, t) = (6, 1.731218393)
(n, t) = (7, 5.512084862)
(n, t) = (8, 15.127203746)
(n, t) = (9, 36.06489506)
(n, t) = (10, 82.559742306)
(n, t) = (12, 343.941457935)
(n, t) = (15, 2076.784127286)
11-element Vector{Float64}:
0.000638025
0.014818023
0.037541015
0.520653741
1.731218393
5.512084862
15.127203746
36.06489506
82.559742306
343.941457935
2076.784127286#=
reversediff = map(reversediffn) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
@elapsed auto_sen_l2(bfun, b_u0, tspan, b_p, bt, (Rodas5(autodiff=false)); diffalg=(ReverseDiff.gradient), tols...)
t = @elapsed auto_sen_l2(bfun, b_u0, tspan, b_p, bt, (Rodas5(autodiff=false)); diffalg=(ReverseDiff.gradient), tols...)
@show n,t
t
end
=#numdiff = map(numdiffn) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
@elapsed auto_sen_l2(bfun, b_u0, tspan, b_p, bt, (Rodas5()); diffalg=(FiniteDiff.finite_difference_gradient), tols...)
t = @elapsed auto_sen_l2(bfun, b_u0, tspan, b_p, bt, (Rodas5()); diffalg=(FiniteDiff.finite_difference_gradient), tols...)
@show n,t
t
end(n, t) = (2, 0.002366211)
(n, t) = (3, 0.021170203)
(n, t) = (4, 0.07246591)
(n, t) = (5, 0.240821835)
(n, t) = (6, 0.59219555)
(n, t) = (7, 1.317970678)
(n, t) = (8, 3.074216759)
(n, t) = (9, 6.692397508)
(n, t) = (10, 12.334107618)
(n, t) = (12, 44.152922056)
10-element Vector{Float64}:
0.002366211
0.021170203
0.07246591
0.240821835
0.59219555
1.317970678
3.074216759
6.692397508
12.334107618
44.152922056csa = map(csan) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
@time ts = map(ADJOINT_METHODS[1:2end÷3]) do alg
@info "Running $alg"
f = SciMLSensitivity.alg_autodiff(alg) ? bfun : ODEFunction(bfun, jac=brusselator_jac)
solver = Rodas5(autodiff=false)
@time diffeq_sen_l2(bfun, b_u0, tspan, b_p, bt, solver; sensalg=alg, tols...)
t = @elapsed diffeq_sen_l2(bfun, b_u0, tspan, b_p, bt, solver; sensalg=alg, tols...)
return t
end
@show n,ts
ts
end11.673866 seconds (22.77 M allocations: 1.431 GiB, 4.17% gc time, 99.91% c
ompilation time)
7.373040 seconds (9.37 M allocations: 609.410 MiB, 2.26% gc time, 99.92%
compilation time)
16.021497 seconds (17.51 M allocations: 1.153 GiB, 1.63% gc time, 99.97% c
ompilation time)
10.329365 seconds (16.39 M allocations: 1.045 GiB, 2.60% gc time, 99.95% c
ompilation time)
5.536433 seconds (6.29 M allocations: 407.921 MiB, 2.12% gc time, 99.92%
compilation time)
5.969226 seconds (9.63 M allocations: 633.380 MiB, 3.11% gc time, 99.92%
compilation time)
57.554926 seconds (82.61 M allocations: 5.286 GiB, 2.58% gc time, 99.89% c
ompilation time: <1% of which was recompilation)
(n, ts) = (2, [0.005766315, 0.002615139, 0.001530478, 0.002469601, 0.001214
4, 0.001077771])
0.077516 seconds (62.26 k allocations: 5.428 MiB)
10.177735 seconds (4.93 M allocations: 315.954 MiB, 0.73% gc time, 99.87%
compilation time)
0.003136 seconds (1.33 k allocations: 335.484 KiB)
0.015205 seconds (10.37 k allocations: 526.766 KiB)
5.345636 seconds (3.19 M allocations: 200.407 MiB, 1.32% gc time, 99.88%
compilation time)
0.001653 seconds (2.00 k allocations: 324.484 KiB)
15.738828 seconds (8.29 M allocations: 531.136 MiB, 0.92% gc time, 98.50%
compilation time)
(n, ts) = (3, [0.079083065, 0.010416808, 0.002884667, 0.016020024, 0.003188
155, 0.001517188])
0.607980 seconds (171.07 k allocations: 14.870 MiB)
10.232305 seconds (4.77 M allocations: 304.974 MiB, 0.71% gc time, 99.58%
compilation time)
0.006891 seconds (1.34 k allocations: 644.125 KiB)
0.105528 seconds (25.35 k allocations: 1.129 MiB)
6.096430 seconds (3.19 M allocations: 200.691 MiB, 0.99% gc time, 99.80%
compilation time)
0.002837 seconds (2.01 k allocations: 476.203 KiB)
17.824796 seconds (8.39 M allocations: 542.482 MiB, 0.75% gc time, 91.30%
compilation time)
(n, ts) = (4, [0.605376479, 0.039999894, 0.006600048, 0.105607206, 0.008933
949, 0.002645529])
2.146124 seconds (258.31 k allocations: 22.838 MiB)
10.337101 seconds (3.94 M allocations: 250.977 MiB, 0.58% gc time, 98.60%
compilation time)
0.012939 seconds (1.37 k allocations: 1.187 MiB)
0.395402 seconds (39.99 k allocations: 1.873 MiB)
5.752005 seconds (3.19 M allocations: 200.889 MiB, 1.42% gc time, 99.43%
compilation time)
0.004713 seconds (2.02 k allocations: 713.219 KiB)
21.376260 seconds (7.77 M allocations: 509.239 MiB, 0.66% gc time, 74.44%
compilation time)
(n, ts) = (5, [2.144412444, 0.140900077, 0.01260177, 0.392291652, 0.0294178
78, 0.004479414])
9.252132 seconds (550.87 k allocations: 48.491 MiB)
12.602450 seconds (3.88 M allocations: 248.069 MiB, 0.66% gc time, 96.91%
compilation time)
0.023654 seconds (1.38 k allocations: 2.126 MiB)
1.258592 seconds (63.10 k allocations: 3.091 MiB)
6.679730 seconds (3.19 M allocations: 201.619 MiB, 98.78% compilation tim
e)
0.007698 seconds (2.02 k allocations: 1005.625 KiB)
40.832238 seconds (8.36 M allocations: 565.820 MiB, 0.20% gc time, 46.07%
compilation time)
(n, ts) = (6, [9.247024686, 0.386754816, 0.023369405, 1.260113049, 0.077969
535, 0.00759652])
20.180575 seconds (645.69 k allocations: 58.354 MiB)
11.203397 seconds (2.80 M allocations: 178.722 MiB, 91.75% compilation tim
e)
0.048933 seconds (1.38 k allocations: 3.629 MiB)
3.763778 seconds (101.24 k allocations: 4.883 MiB)
1.440449 seconds (486.53 k allocations: 33.353 MiB, 4.18% gc time, 90.11%
compilation time)
0.012181 seconds (2.02 k allocations: 1.348 MiB)
61.882581 seconds (4.87 M allocations: 358.811 MiB, 0.36% gc time, 18.71%
compilation time)
(n, ts) = (7, [20.336134035, 0.924284653, 0.049776787, 3.763606015, 0.14175
1231, 0.011959805])
47.518351 seconds (900.95 k allocations: 82.711 MiB)
2.236290 seconds (87.09 k allocations: 11.261 MiB)
0.085626 seconds (1.38 k allocations: 5.840 MiB)
7.832869 seconds (124.44 k allocations: 6.799 MiB)
0.406060 seconds (15.28 k allocations: 4.193 MiB)
0.020294 seconds (2.02 k allocations: 1.853 MiB)
116.146916 seconds (2.26 M allocations: 225.530 MiB)
(n, ts) = (8, [47.458886539, 2.23977471, 0.084984069, 7.831247874, 0.406233
653, 0.019794413])
111.669495 seconds (1.32 M allocations: 122.362 MiB, 0.10% gc time)
5.333110 seconds (129.01 k allocations: 17.277 MiB)
0.162652 seconds (1.38 k allocations: 9.038 MiB)
18.439632 seconds (181.83 k allocations: 10.563 MiB)
0.805259 seconds (18.54 k allocations: 6.093 MiB)
0.031802 seconds (2.02 k allocations: 2.484 MiB)
272.894875 seconds (3.32 M allocations: 335.848 MiB, 0.06% gc time)
(n, ts) = (9, [111.640496255, 5.363320235, 0.162026892, 18.444480809, 0.803
039225, 0.031353593])
185.273775 seconds (1.43 M allocations: 137.566 MiB)
8.882475 seconds (138.58 k allocations: 23.825 MiB)
0.355799 seconds (1.39 k allocations: 13.525 MiB)
36.501456 seconds (234.78 k allocations: 14.327 MiB)
1.477429 seconds (22.19 k allocations: 8.639 MiB)
0.046857 seconds (2.02 k allocations: 3.280 MiB)
464.913812 seconds (3.67 M allocations: 402.536 MiB, 0.01% gc time)
(n, ts) = (10, [185.071636528, 8.943698357, 0.432105674, 36.391540075, 1.46
793774, 0.045954298])
742.390015 seconds (2.81 M allocations: 271.145 MiB, 0.01% gc time)
28.242135 seconds (204.72 k allocations: 45.189 MiB, 0.26% gc time)
0.520924 seconds (1.39 k allocations: 27.085 MiB)
100.797741 seconds (315.80 k allocations: 23.681 MiB)
4.427439 seconds (30.64 k allocations: 16.156 MiB)
0.105995 seconds (2.02 k allocations: 5.388 MiB)
1751.522064 seconds (6.73 M allocations: 777.501 MiB, 0.01% gc time)
(n, ts) = (12, [741.096580009, 28.073572982, 0.653599631, 100.676165224, 4.
415155522, 0.105222724])
3271.910010 seconds (5.14 M allocations: 516.645 MiB, 0.01% gc time)
111.513792 seconds (328.37 k allocations: 102.016 MiB)
1.362499 seconds (1.40 k allocations: 64.528 MiB)
453.781448 seconds (586.28 k allocations: 50.502 MiB)
16.777105 seconds (46.19 k allocations: 36.239 MiB)
0.299543 seconds (2.02 k allocations: 10.569 MiB)
7712.407907 seconds (12.20 M allocations: 1.525 GiB, 0.02% gc time)
(n, ts) = (15, [3273.139537912, 111.489407172, 1.403090087, 453.610881798,
16.813436676, 0.295176203])
7828.649315 seconds (7.45 M allocations: 766.079 MiB, 0.00% gc time)
219.838040 seconds (392.62 k allocations: 161.462 MiB)
2.819736 seconds (1.41 k allocations: 105.404 MiB, 1.04% gc time)
1114.664890 seconds (872.20 k allocations: 80.006 MiB, 0.00% gc time)
35.293195 seconds (58.48 k allocations: 57.750 MiB)
1.180990 seconds (2.03 k allocations: 15.774 MiB)
18388.113853 seconds (17.55 M allocations: 2.318 GiB, 0.01% gc time)
(n, ts) = (17, [7810.521708444, 219.537046931, 2.991678379, 1116.379010265,
35.393436637, 0.832696167])
12-element Vector{Vector{Float64}}:
[0.005766315, 0.002615139, 0.001530478, 0.002469601, 0.0012144, 0.00107777
1]
[0.079083065, 0.010416808, 0.002884667, 0.016020024, 0.003188155, 0.001517
188]
[0.605376479, 0.039999894, 0.006600048, 0.105607206, 0.008933949, 0.002645
529]
[2.144412444, 0.140900077, 0.01260177, 0.392291652, 0.029417878, 0.0044794
14]
[9.247024686, 0.386754816, 0.023369405, 1.260113049, 0.077969535, 0.007596
52]
[20.336134035, 0.924284653, 0.049776787, 3.763606015, 0.141751231, 0.01195
9805]
[47.458886539, 2.23977471, 0.084984069, 7.831247874, 0.406233653, 0.019794
413]
[111.640496255, 5.363320235, 0.162026892, 18.444480809, 0.803039225, 0.031
353593]
[185.071636528, 8.943698357, 0.432105674, 36.391540075, 1.46793774, 0.0459
54298]
[741.096580009, 28.073572982, 0.653599631, 100.676165224, 4.415155522, 0.1
05222724]
[3273.139537912, 111.489407172, 1.403090087, 453.610881798, 16.813436676,
0.295176203]
[7810.521708444, 219.537046931, 2.991678379, 1116.379010265, 35.393436637,
0.832696167]n_to_param(n) = 4n^2
lw = 2
ms = 0.5
plt1 = plot(title="Sensitivity Scaling on Brusselator");
plot!(plt1, n_to_param.(forwarddiffn), forwarddiff, lab="Forward-Mode DSAAD", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
#plot!(plt1, n_to_param.(reversediffn), reversediff, lab="Reverse-Mode DSAAD", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
csadata = [[csa[j][i] for j in eachindex(csa)] for i in eachindex(csa[1])]
plot!(plt1, n_to_param.(csan), csadata[1], lab="Interpolating CASA user-Jacobian", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
plot!(plt1, n_to_param.(csan), csadata[2], lab="Interpolating CASA AD-Jacobian", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
plot!(plt1, n_to_param.(csan), csadata[3], lab=raw"Interpolating CASA AD-$v^{T}J$ seeding", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
plot!(plt1, n_to_param.(csan), csadata[1+3], lab="Quadrature CASA user-Jacobian", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
plot!(plt1, n_to_param.(csan), csadata[2+3], lab="Quadrature CASA AD-Jacobian", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
plot!(plt1, n_to_param.(csan), csadata[3+3], lab=raw"Quadrature CASA AD-$v^{T}J$ seeding", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
plot!(plt1, n_to_param.(numdiffn), numdiff, lab="Numerical Differentiation", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
xaxis!(plt1, "Number of Parameters", :log10);
yaxis!(plt1, "Runtime (s)", :log10);
plot!(plt1, legend=:outertopleft, size=(1200, 600))
VJP Choice Benchmarks
bt = 0:0.1:1
tspan = (0.0, 1.0)
csan = vcat(2:10,12,15,17)
tols = (abstol=1e-5, reltol=1e-7)
_adjoint_methods = ntuple(2) do ii
Alg = (InterpolatingAdjoint, QuadratureAdjoint)[ii]
(
advj1 = Alg(autodiff=true,autojacvec=EnzymeVJP()), # AD vJ
advj2 = Alg(autodiff=true,autojacvec=ReverseDiffVJP(false)), # AD vJ
advj3 = Alg(autodiff=true,autojacvec=ReverseDiffVJP(true)), # AD vJ
)
end |> NamedTuple{(:interp, :quad)}
adjoint_methods = mapreduce(collect, vcat, _adjoint_methods)
csavjp = map(csan) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
@time ts = map(adjoint_methods) do alg
@info "Running $alg"
f = SciMLSensitivity.alg_autodiff(alg) ? bfun : ODEFunction(bfun, jac=brusselator_jac)
solver = Rodas5(autodiff=false)
@time diffeq_sen_l2(bfun, b_u0, tspan, b_p, bt, solver; sensalg=alg, tols...)
t = @elapsed diffeq_sen_l2(bfun, b_u0, tspan, b_p, bt, solver; sensalg=alg, tols...)
return t
end
@show n,ts
ts
end0.001690 seconds (1.27 k allocations: 190.828 KiB)
5.570353 seconds (9.30 M allocations: 588.848 MiB, 1.77% gc time, 99.14%
compilation time)
4.800770 seconds (6.89 M allocations: 448.154 MiB, 2.65% gc time, 99.79%
compilation time)
0.001104 seconds (2.02 k allocations: 216.688 KiB)
4.698080 seconds (7.03 M allocations: 448.099 MiB, 1.78% gc time, 99.36%
compilation time)
4.897851 seconds (6.68 M allocations: 432.411 MiB, 2.70% gc time, 99.86%
compilation time)
20.460323 seconds (31.16 M allocations: 1.933 GiB, 2.33% gc time, 99.07% c
ompilation time)
(n, ts) = (2, [0.00134782, 0.045370017, 0.005664228, 0.000861954, 0.0289168
51, 0.003877491])
0.003174 seconds (1.33 k allocations: 335.484 KiB)
0.143369 seconds (2.09 M allocations: 95.600 MiB)
0.016478 seconds (5.41 k allocations: 549.000 KiB)
0.001609 seconds (2.00 k allocations: 324.484 KiB)
0.080354 seconds (1.10 M allocations: 50.154 MiB)
0.009897 seconds (7.24 k allocations: 600.453 KiB)
0.544487 seconds (6.41 M allocations: 295.261 MiB, 6.62% gc time)
(n, ts) = (3, [0.002918817, 0.175614832, 0.016079348, 0.001447749, 0.079940
026, 0.009716347])
0.006638 seconds (1.34 k allocations: 644.125 KiB)
0.457758 seconds (6.23 M allocations: 268.498 MiB, 9.07% gc time)
0.045981 seconds (8.54 k allocations: 1017.906 KiB)
0.002802 seconds (2.01 k allocations: 476.203 KiB)
0.222059 seconds (2.94 M allocations: 126.581 MiB, 7.38% gc time)
0.023726 seconds (11.63 k allocations: 957.094 KiB)
1.474712 seconds (18.39 M allocations: 796.423 MiB, 5.02% gc time)
(n, ts) = (4, [0.006365751, 0.428318034, 0.045904303, 0.002571931, 0.205848
574, 0.023545472])
0.012787 seconds (1.37 k allocations: 1.187 MiB)
1.005298 seconds (14.51 M allocations: 666.822 MiB, 4.83% gc time)
0.107025 seconds (12.50 k allocations: 1.762 MiB)
0.004581 seconds (2.02 k allocations: 713.219 KiB)
0.470657 seconds (6.63 M allocations: 304.667 MiB, 3.44% gc time)
0.051861 seconds (17.30 k allocations: 1.453 MiB)
3.361923 seconds (42.35 M allocations: 1.908 GiB, 5.33% gc time)
(n, ts) = (5, [0.02905132, 1.019605045, 0.109611032, 0.004368717, 0.4911791
68, 0.052094806])
0.023941 seconds (1.38 k allocations: 2.126 MiB)
2.030913 seconds (29.30 M allocations: 1.260 GiB, 5.51% gc time)
0.216941 seconds (17.32 k allocations: 2.941 MiB)
0.007488 seconds (2.02 k allocations: 1005.625 KiB)
0.938821 seconds (12.88 M allocations: 567.154 MiB, 6.37% gc time)
0.102241 seconds (24.17 k allocations: 2.051 MiB)
6.624316 seconds (84.44 M allocations: 3.643 GiB, 5.00% gc time)
(n, ts) = (6, [0.023715081, 2.034729794, 0.218926176, 0.007279226, 0.917513
287, 0.098336767])
0.049382 seconds (1.38 k allocations: 3.629 MiB)
3.860072 seconds (55.53 M allocations: 2.326 GiB, 5.23% gc time)
0.421175 seconds (23.32 k allocations: 4.747 MiB)
0.012493 seconds (2.02 k allocations: 1.348 MiB)
1.614930 seconds (22.90 M allocations: 981.691 MiB, 4.92% gc time)
0.175317 seconds (32.28 k allocations: 2.785 MiB)
12.262925 seconds (156.98 M allocations: 6.593 GiB, 4.62% gc time)
(n, ts) = (7, [0.048996969, 3.840106902, 0.435524889, 0.011982559, 1.614185
443, 0.174397882])
0.084674 seconds (1.38 k allocations: 5.840 MiB)
6.521814 seconds (93.69 M allocations: 4.182 GiB, 5.48% gc time)
0.706281 seconds (29.97 k allocations: 7.322 MiB)
0.019816 seconds (2.02 k allocations: 1.853 MiB)
2.761892 seconds (39.29 M allocations: 1.753 GiB, 5.21% gc time)
0.292543 seconds (41.80 k allocations: 3.789 MiB)
20.764954 seconds (266.11 M allocations: 11.907 GiB, 4.81% gc time)
(n, ts) = (8, [0.084454822, 6.492965447, 0.703652362, 0.019438883, 2.762388
087, 0.309297833])
0.162697 seconds (1.38 k allocations: 9.038 MiB)
12.643907 seconds (181.84 M allocations: 7.891 GiB, 4.97% gc time)
1.422938 seconds (40.48 k allocations: 11.079 MiB)
0.031657 seconds (2.02 k allocations: 2.484 MiB)
4.374755 seconds (61.70 M allocations: 2.677 GiB, 5.17% gc time)
0.474672 seconds (52.44 k allocations: 4.905 MiB)
38.271864 seconds (487.28 M allocations: 21.190 GiB, 4.59% gc time)
(n, ts) = (9, [0.166029446, 12.704010424, 1.397609859, 0.031407873, 4.37985
4015, 0.476869296])
0.322541 seconds (1.39 k allocations: 13.525 MiB)
16.593448 seconds (233.92 M allocations: 9.952 GiB, 5.79% gc time)
1.971747 seconds (46.53 k allocations: 15.842 MiB)
0.047572 seconds (2.02 k allocations: 3.280 MiB)
6.720136 seconds (92.71 M allocations: 3.942 GiB, 6.15% gc time)
0.694388 seconds (64.33 k allocations: 6.243 MiB)
52.981381 seconds (653.49 M allocations: 27.864 GiB, 5.30% gc time)
(n, ts) = (10, [0.366210753, 16.849548053, 1.975394454, 0.046164092, 6.6928
40149, 0.694055986])
0.536341 seconds (1.39 k allocations: 27.085 MiB)
35.845385 seconds (498.79 M allocations: 22.283 GiB, 6.11% gc time)
4.135898 seconds (67.17 k allocations: 30.513 MiB)
0.107482 seconds (2.02 k allocations: 5.388 MiB)
13.812169 seconds (188.66 M allocations: 8.423 GiB, 6.34% gc time)
1.394373 seconds (91.88 k allocations: 9.739 MiB)
112.304933 seconds (1.38 G allocations: 61.554 GiB, 5.67% gc time)
(n, ts) = (12, [0.535702803, 36.576058422, 4.110717811, 0.105450193, 13.731
807029, 1.404409329])
1.346842 seconds (1.40 k allocations: 64.528 MiB, 1.43% gc time)
94.751902 seconds (1.25 G allocations: 53.273 GiB, 7.78% gc time)
10.440717 seconds (105.39 k allocations: 69.894 MiB)
0.297721 seconds (2.02 k allocations: 10.569 MiB)
35.724916 seconds (453.40 M allocations: 19.275 GiB, 10.19% gc time)
3.434800 seconds (142.59 k allocations: 17.231 MiB)
292.931394 seconds (3.41 G allocations: 145.412 GiB, 7.42% gc time)
(n, ts) = (15, [1.404007982, 97.010873436, 10.30252896, 0.295888695, 34.474
083925, 3.436738948])
2.795526 seconds (1.41 k allocations: 105.404 MiB)
134.015879 seconds (1.92 G allocations: 80.231 GiB, 5.86% gc time)
15.912583 seconds (131.41 k allocations: 112.046 MiB, 0.10% gc time)
0.679523 seconds (2.03 k allocations: 15.774 MiB)
53.960011 seconds (743.36 M allocations: 30.973 GiB, 6.73% gc time)
6.061710 seconds (182.65 k allocations: 24.263 MiB)
429.984940 seconds (5.34 G allocations: 222.910 GiB, 5.34% gc time)
(n, ts) = (17, [2.751485266, 136.515721937, 16.370492397, 0.677623282, 54.2
55790305, 5.977512346])
12-element Vector{Vector{Float64}}:
[0.00134782, 0.045370017, 0.005664228, 0.000861954, 0.028916851, 0.0038774
91]
[0.002918817, 0.175614832, 0.016079348, 0.001447749, 0.079940026, 0.009716
347]
[0.006365751, 0.428318034, 0.045904303, 0.002571931, 0.205848574, 0.023545
472]
[0.02905132, 1.019605045, 0.109611032, 0.004368717, 0.491179168, 0.0520948
06]
[0.023715081, 2.034729794, 0.218926176, 0.007279226, 0.917513287, 0.098336
767]
[0.048996969, 3.840106902, 0.435524889, 0.011982559, 1.614185443, 0.174397
882]
[0.084454822, 6.492965447, 0.703652362, 0.019438883, 2.762388087, 0.309297
833]
[0.166029446, 12.704010424, 1.397609859, 0.031407873, 4.379854015, 0.47686
9296]
[0.366210753, 16.849548053, 1.975394454, 0.046164092, 6.692840149, 0.69405
5986]
[0.535702803, 36.576058422, 4.110717811, 0.105450193, 13.731807029, 1.4044
09329]
[1.404007982, 97.010873436, 10.30252896, 0.295888695, 34.474083925, 3.4367
38948]
[2.751485266, 136.515721937, 16.370492397, 0.677623282, 54.255790305, 5.97
7512346]plt2 = plot(title="Brusselator quadrature adjoint scaling");
csacompare = [[csavjp[j][i] for j in eachindex(csavjp)] for i in eachindex(csavjp[1])]
plot!(plt2, n_to_param.(csan), csadata[2+3], lab="AD-Jacobian", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
plot!(plt2, n_to_param.(csan), csacompare[1+3], lab=raw"EnzymeVJP", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
plot!(plt2, n_to_param.(csan), csacompare[2+3], lab=raw"ReverseDiffVJP", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
plot!(plt2, n_to_param.(csan), csacompare[3+3], lab=raw"Compiled ReverseDiffVJP", lw=lw, marksize=ms, linestyle=:auto, marker=:auto);
xaxis!(plt2, "Number of Parameters", :log10);
yaxis!(plt2, "Runtime (s)", :log10);
plot!(plt2, legend=:outertopleft, size=(1200, 600))
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","BrussScaling.jmd")Computer Information:
Julia Version 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 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
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
JULIA_CPU_THREADS = 128
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
Package Information:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/AutomaticDifferentiation/Project.toml`
⌃ [6e4b80f9] BenchmarkTools v1.5.0
⌃ [a93c6f00] DataFrames v1.6.1
⌃ [1313f7d8] DataFramesMeta v0.15.3
⌅ [a0c0ee7d] DifferentiationInterface v0.5.9
⌅ [a82114a7] DifferentiationInterfaceTest v0.5.0
⌅ [7da242da] Enzyme v0.12.25
⌃ [6a86dc24] FiniteDiff v2.23.1
⌅ [f6369f11] ForwardDiff v0.10.36
⌃ [1dea7af3] OrdinaryDiffEq v6.86.0
⌃ [65888b18] ParameterizedFunctions v5.17.0
⌃ [91a5bcdd] Plots v1.40.5
⌃ [08abe8d2] PrettyTables v2.3.2
[37e2e3b7] ReverseDiff v1.15.3
[31c91b34] SciMLBenchmarks v0.1.3
⌃ [1ed8b502] SciMLSensitivity v7.64.0
⌃ [90137ffa] StaticArrays v1.9.7
⌃ [07d77754] Tapir v0.2.26
⌃ [9f7883ad] Tracker v0.2.34
⌅ [e88e6eb3] Zygote v0.6.70
[37e2e46d] LinearAlgebra
[d6f4376e] Markdown
[de0858da] Printf
[8dfed614] Test
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`
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.And the full manifest:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/AutomaticDifferentiation/Manifest.toml`
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[621f4979] AbstractFFTs v1.5.0
[1520ce14] AbstractTrees v0.4.5
⌃ [7d9f7c33] Accessors v0.1.37
⌃ [79e6a3ab] Adapt v4.0.4
[66dad0bd] AliasTables v1.1.3
[ec485272] ArnoldiMethod v0.4.0
⌃ [4fba245c] ArrayInterface v7.12.0
⌃ [4c555306] ArrayLayouts v1.10.2
⌅ [a9b6321e] Atomix v0.1.0
⌃ [6e4b80f9] BenchmarkTools v1.5.0
⌃ [e2ed5e7c] Bijections v0.1.7
[d1d4a3ce] BitFlags v0.1.9
[62783981] BitTwiddlingConvenienceFunctions v0.1.6
[fa961155] CEnum v0.5.0
[2a0fbf3d] CPUSummary v0.2.6
[00ebfdb7] CSTParser v3.4.3
⌃ [49dc2e85] Calculus v0.5.1
⌃ [7057c7e9] Cassette v0.3.13
[8be319e6] Chain v0.6.0
⌃ [082447d4] ChainRules v1.69.0
⌃ [d360d2e6] ChainRulesCore v1.24.0
⌃ [0ca39b1e] Chairmarks v1.2.1
[fb6a15b2] CloseOpenIntervals v0.1.13
⌃ [da1fd8a2] CodeTracking v1.3.5
⌃ [944b1d66] CodecZlib v0.7.5
⌃ [35d6a980] ColorSchemes v3.26.0
⌅ [3da002f7] ColorTypes v0.11.5
⌅ [c3611d14] ColorVectorSpace v0.10.0
⌅ [5ae59095] Colors v0.12.11
[861a8166] Combinatorics v1.0.2
⌅ [a80b9123] CommonMark v0.8.12
[38540f10] CommonSolve v0.2.4
⌃ [bbf7d656] CommonSubexpressions v0.3.0
[f70d9fcc] CommonWorldInvalidations v1.0.0
⌃ [34da2185] Compat v4.15.0
⌃ [b0b7db55] ComponentArrays v0.15.14
[b152e2b5] CompositeTypes v0.1.4
[a33af91c] CompositionsBase v0.1.2
[2569d6c7] ConcreteStructs v0.2.3
⌃ [f0e56b4a] ConcurrentUtilities v2.4.2
[8f4d0f93] Conda v1.10.2
⌅ [187b0558] ConstructionBase v1.5.6
[d38c429a] Contour v0.6.3
[adafc99b] CpuId v0.3.1
[a8cc5b0e] Crayons v4.1.1
[9a962f9c] DataAPI v1.16.0
⌃ [a93c6f00] DataFrames v1.6.1
⌃ [1313f7d8] DataFramesMeta v0.15.3
⌃ [864edb3b] DataStructures v0.18.20
[e2d170a0] DataValueInterfaces v1.0.0
[8bb1440f] DelimitedFiles v1.9.1
⌃ [2b5f629d] DiffEqBase v6.151.5
⌅ [459566f4] DiffEqCallbacks v3.6.2
⌃ [77a26b50] DiffEqNoiseProcess v5.22.0
[163ba53b] DiffResults v1.1.0
[b552c78f] DiffRules v1.15.1
[de460e47] DiffTests v0.1.2
⌅ [a0c0ee7d] DifferentiationInterface v0.5.9
⌅ [a82114a7] DifferentiationInterfaceTest v0.5.0
⌃ [b4f34e82] Distances v0.10.11
⌃ [31c24e10] Distributions v0.25.109
[ffbed154] DocStringExtensions v0.9.3
⌃ [5b8099bc] DomainSets v0.7.14
⌃ [fa6b7ba4] DualNumbers v0.6.8
⌅ [7c1d4256] DynamicPolynomials v0.5.7
⌅ [06fc5a27] DynamicQuantities v0.13.2
[da5c29d0] EllipsisNotation v1.8.0
[4e289a0a] EnumX v1.0.4
⌅ [7da242da] Enzyme v0.12.25
⌅ [f151be2c] EnzymeCore v0.7.7
⌃ [460bff9d] ExceptionUnwrapping v0.1.10
⌃ [d4d017d3] ExponentialUtilities v1.26.1
[e2ba6199] ExprTools v0.1.10
⌃ [c87230d0] FFMPEG v0.4.1
⌃ [7034ab61] FastBroadcast v0.3.4
[9aa1b823] FastClosures v0.3.2
[29a986be] FastLapackInterface v2.0.4
⌃ [1a297f60] FillArrays v1.11.0
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[53c48c17] FixedPointNumbers v0.8.5
[1fa38f19] Format v1.3.7
⌅ [f6369f11] ForwardDiff v0.10.36
[f62d2435] FunctionProperties v0.1.2
[069b7b12] FunctionWrappers v1.1.3
[77dc65aa] FunctionWrappersWrappers v0.1.3
⌅ [d9f16b24] Functors v0.4.11
⌅ [0c68f7d7] GPUArrays v10.3.0
⌅ [46192b85] GPUArraysCore v0.1.6
⌅ [61eb1bfa] GPUCompiler v0.26.7
⌃ [28b8d3ca] GR v0.73.7
[c145ed77] GenericSchur v0.5.4
[d7ba0133] Git v1.3.1
[c27321d9] Glob v1.3.1
⌃ [86223c79] Graphs v1.11.2
[42e2da0e] Grisu v1.0.2
⌃ [cd3eb016] HTTP v1.10.8
[eafb193a] Highlights v0.5.3
[3e5b6fbb] HostCPUFeatures v0.1.17
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[615f187c] IfElse v0.1.1
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⌃ [92d709cd] IrrationalConstants v0.2.2
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⌃ [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
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[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
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[2a0f44e3] Base64
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[b77e0a4c] InteractiveUtils
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[8f399da3] Libdl
[37e2e46d] LinearAlgebra
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[4607b0f0] SuiteSparse
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[e66e0078] CompilerSupportLibraries_jll v1.1.1+0
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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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