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 OrdinaryDiffEqRosenbrock
using LinearAlgebra, Tracker, Mooncake, 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, 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(brusselator_comp, copy([u0; zeros((N^2*2)*(N^2*4))]), (0.0, 10.0), 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_iq = ntuple(2) do ii
Alg = (InterpolatingAdjoint, QuadratureAdjoint)[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)}
# GaussAdjoint/GaussKronrodAdjoint do not support user-provided Jacobians (autodiff=false)
_adjoint_methods_g = ntuple(2) do ii
Alg = (GaussAdjoint, GaussKronrodAdjoint)[ii]
(
adjc = Alg(autodiff = true, autojacvec = false), # AD Jacobian
advj = Alg(autodiff = true, autojacvec = EnzymeVJP()) # AD vJ
)
end |> NamedTuple{(:gauss, :gausskronrod)}
@isdefined(ADJOINT_METHODS_IQ) ||
(const ADJOINT_METHODS_IQ = mapreduce(collect, vcat, _adjoint_methods_iq))
@isdefined(ADJOINT_METHODS_G) ||
(const ADJOINT_METHODS_G = mapreduce(collect, vcat, _adjoint_methods_g))
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, 0.001459562)
(n, t) = (3, 0.009954915)
(n, t) = (4, 0.039400813)
(n, t) = (5, 0.181939321)
(n, t) = (6, 0.400181132)
(n, t) = (7, 1.172793899)
(n, t) = (8, 2.035467911)
(n, t) = (9, 3.969346768)
(n, t) = (10, 9.470611813)
(n, t) = (12, 31.640812253)
(n, t) = (15, 116.313197846)
11-element Vector{Float64}:
0.001459562
0.009954915
0.039400813
0.181939321
0.400181132
1.172793899
2.035467911
3.969346768
9.470611813
31.640812253
116.313197846#=
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=AutoFiniteDiff())); diffalg=(ReverseDiff.gradient), tols...)
t = @elapsed auto_sen_l2(bfun, b_u0, tspan, b_p, bt, (Rodas5(autodiff=AutoFiniteDiff())); 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, 0.003868278)
(n, t) = (3, 0.029838454)
(n, t) = (4, 0.099196068)
(n, t) = (5, 0.319011285)
(n, t) = (6, 0.839714196)
(n, t) = (7, 2.164495402)
(n, t) = (8, 4.333828015)
(n, t) = (9, 10.174363259)
(n, t) = (10, 18.35778455)
(n, t) = (12, 89.58415216)
10-element Vector{Float64}:
0.003868278
0.029838454
0.099196068
0.319011285
0.839714196
2.164495402
4.333828015
10.174363259
18.35778455
89.58415216Warmup: run each adjoint method once at the smallest size to ensure all compilation is complete before we start timing.
let n = first(csan)
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
solver = Rodas5(autodiff = AutoFiniteDiff())
for alg in ADJOINT_METHODS_IQ
f = SciMLSensitivity.alg_autodiff(alg) ? bfun :
ODEFunction(bfun, jac = brusselator_jac)
diffeq_sen_l2(f, b_u0, tspan, b_p, bt, solver; sensalg = alg, tols...)
end
for alg in ADJOINT_METHODS_G
diffeq_sen_l2(bfun, b_u0, tspan, b_p, bt, solver; sensalg = alg, tols...)
end
endcsa_iq = map(csan) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
@time ts = map(ADJOINT_METHODS_IQ) do alg
@info "Running $alg"
f = SciMLSensitivity.alg_autodiff(alg) ? bfun :
ODEFunction(bfun, jac = brusselator_jac)
solver = Rodas5(autodiff = AutoFiniteDiff())
@time diffeq_sen_l2(f, b_u0, tspan, b_p, bt, solver; sensalg = alg, tols...)
t = @elapsed diffeq_sen_l2(f, b_u0, tspan, b_p, bt, solver; sensalg = alg, tols...)
return t
end
@show n, ts
ts
end0.005960 seconds (9.61 k allocations: 1.117 MiB)
0.003477 seconds (6.32 k allocations: 2.050 MiB)
0.003669 seconds (7.34 k allocations: 569.445 KiB)
0.001867 seconds (4.58 k allocations: 304.148 KiB)
0.001644 seconds (2.30 k allocations: 230.867 KiB)
0.002895 seconds (7.90 k allocations: 499.922 KiB)
0.963717 seconds (1.20 M allocations: 69.093 MiB, 94.37% compilation time
: 6% of which was recompilation)
(n, ts) = (2, [0.006171065, 0.002836634, 0.003528211, 0.001600151, 0.001332
193, 0.002764545])
0.025804 seconds (14.74 k allocations: 2.059 MiB)
10.087194 seconds (8.09 M allocations: 388.772 MiB, 2.56% gc time, 99.85%
compilation time)
0.010339 seconds (10.96 k allocations: 994.891 KiB)
0.004046 seconds (5.60 k allocations: 550.633 KiB)
10.231215 seconds (6.85 M allocations: 326.929 MiB, 0.57% gc time, 99.94%
compilation time)
0.006048 seconds (9.10 k allocations: 695.078 KiB)
20.437824 seconds (15.03 M allocations: 726.195 MiB, 1.55% gc time, 99.31%
compilation time)
(n, ts) = (3, [0.02525832, 0.011758195, 0.010520651, 0.003880729, 0.0038506
39, 0.006290635])
0.111492 seconds (22.87 k allocations: 4.004 MiB)
9.872295 seconds (8.10 M allocations: 389.665 MiB, 0.95% gc time, 99.54%
compilation time)
0.022005 seconds (16.77 k allocations: 1.758 MiB)
0.008869 seconds (7.44 k allocations: 1.057 MiB)
10.330958 seconds (6.85 M allocations: 327.161 MiB, 0.89% gc time, 99.88%
compilation time)
0.010017 seconds (11.39 k allocations: 1005.797 KiB)
20.569703 seconds (15.08 M allocations: 735.333 MiB, 0.90% gc time, 97.94%
compilation time)
(n, ts) = (4, [0.107809398, 0.043582287, 0.022071127, 0.008899321, 0.010518
741, 0.010023664])
0.360105 seconds (32.72 k allocations: 6.980 MiB)
9.512170 seconds (7.12 M allocations: 343.415 MiB, 0.85% gc time, 98.16%
compilation time)
0.046810 seconds (23.80 k allocations: 3.015 MiB)
0.020013 seconds (9.86 k allocations: 1.938 MiB)
10.364029 seconds (6.84 M allocations: 327.658 MiB, 1.54% gc time, 99.65%
compilation time)
0.019518 seconds (14.40 k allocations: 1.447 MiB)
20.988198 seconds (14.14 M allocations: 702.280 MiB, 1.15% gc time, 93.70%
compilation time)
(n, ts) = (5, [0.361169763, 0.172788135, 0.047564305, 0.020227567, 0.033864
811, 0.01961637])
0.994153 seconds (44.78 k allocations: 12.106 MiB)
9.929286 seconds (7.13 M allocations: 345.993 MiB, 1.87% gc time, 95.05%
compilation time)
0.086210 seconds (32.42 k allocations: 5.000 MiB)
0.039311 seconds (12.60 k allocations: 3.539 MiB)
10.207443 seconds (6.84 M allocations: 327.941 MiB, 0.51% gc time, 99.31%
compilation time)
0.033539 seconds (17.82 k allocations: 2.003 MiB)
23.050965 seconds (14.22 M allocations: 726.433 MiB, 1.19% gc time, 84.92%
compilation time)
(n, ts) = (6, [0.996815545, 0.488680971, 0.086696966, 0.039261421, 0.104752
605, 0.033367834])
2.529728 seconds (61.29 k allocations: 19.862 MiB)
9.842722 seconds (6.05 M allocations: 297.285 MiB, 0.92% gc time, 88.88%
compilation time)
0.172945 seconds (44.24 k allocations: 7.965 MiB)
0.075276 seconds (15.82 k allocations: 5.740 MiB)
2.235917 seconds (789.87 k allocations: 41.042 MiB, 92.94% compilation ti
me)
0.054366 seconds (21.85 k allocations: 2.693 MiB)
19.077269 seconds (7.16 M allocations: 421.874 MiB, 0.83% gc time, 56.75%
compilation time)
(n, ts) = (7, [2.530754208, 1.090251481, 0.24535913, 0.074840353, 0.1576308
69, 0.054149718])
5.498522 seconds (78.33 k allocations: 30.015 MiB, 0.97% gc time)
2.764278 seconds (24.18 k allocations: 11.810 MiB)
0.287365 seconds (56.40 k allocations: 12.095 MiB)
0.179019 seconds (20.61 k allocations: 9.053 MiB, 26.99% gc time)
0.344099 seconds (7.27 k allocations: 4.423 MiB)
0.086916 seconds (27.71 k allocations: 3.636 MiB)
18.292580 seconds (431.11 k allocations: 142.930 MiB, 0.88% gc time)
(n, ts) = (8, [5.508283047, 2.76269396, 0.286064982, 0.127681997, 0.3461244
37, 0.084475698])
12.549997 seconds (111.52 k allocations: 48.409 MiB, 0.58% gc time)
6.535358 seconds (34.65 k allocations: 18.375 MiB, 0.44% gc time)
0.845336 seconds (107.37 k allocations: 20.228 MiB, 18.03% gc time)
0.385244 seconds (24.97 k allocations: 13.632 MiB, 43.23% gc time)
0.683794 seconds (8.36 k allocations: 6.499 MiB)
0.136613 seconds (33.16 k allocations: 4.793 MiB)
41.875855 seconds (642.17 k allocations: 224.738 MiB, 1.08% gc time)
(n, ts) = (9, [12.491102848, 6.508444383, 0.690010995, 0.219140206, 0.68078
4867, 0.136480417])
21.028911 seconds (123.39 k allocations: 67.505 MiB, 0.13% gc time)
10.829744 seconds (37.06 k allocations: 26.776 MiB, 0.30% gc time)
0.813356 seconds (88.59 k allocations: 25.620 MiB, 4.44% gc time)
0.383398 seconds (29.83 k allocations: 20.837 MiB, 8.69% gc time)
1.258185 seconds (9.58 k allocations: 9.225 MiB)
0.192865 seconds (39.23 k allocations: 6.158 MiB)
69.205073 seconds (657.47 k allocations: 313.107 MiB, 0.38% gc time)
(n, ts) = (10, [21.255836426, 10.797951878, 0.806370854, 0.377695749, 1.249
375577, 0.19322269])
65.442142 seconds (181.92 k allocations: 130.713 MiB, 0.70% gc time)
32.586795 seconds (54.28 k allocations: 53.241 MiB, 0.07% gc time)
2.255074 seconds (130.62 k allocations: 48.696 MiB, 13.47% gc time)
0.959829 seconds (41.39 k allocations: 40.368 MiB, 6.45% gc time)
3.936722 seconds (12.68 k allocations: 17.459 MiB)
0.424866 seconds (53.61 k allocations: 9.886 MiB)
209.515725 seconds (951.12 k allocations: 601.591 MiB, 0.59% gc time)
(n, ts) = (12, [63.872513055, 32.67609872, 2.030823259, 0.898381484, 3.9445
00053, 0.459260821])
229.391276 seconds (271.09 k allocations: 298.637 MiB, 0.19% gc time)
122.963619 seconds (82.50 k allocations: 125.349 MiB, 0.06% gc time)
4.352613 seconds (207.92 k allocations: 109.802 MiB, 4.96% gc time)
3.002017 seconds (62.13 k allocations: 96.879 MiB, 8.75% gc time)
14.816399 seconds (17.87 k allocations: 39.433 MiB, 0.23% gc time)
1.012913 seconds (79.53 k allocations: 18.622 MiB)
751.734557 seconds (1.44 M allocations: 1.346 GiB, 0.30% gc time)
(n, ts) = (15, [229.61943815, 123.700041422, 4.184588132, 2.751999032, 14.7
98665264, 1.119510035])
590.008927 seconds (475.80 k allocations: 479.804 MiB, 0.17% gc time)
646.649764 seconds (101.29 k allocations: 203.644 MiB, 0.08% gc time)
7.032174 seconds (248.09 k allocations: 173.691 MiB, 3.92% gc time)
5.703297 seconds (116.40 k allocations: 149.955 MiB, 4.53% gc time)
32.259683 seconds (21.97 k allocations: 63.005 MiB, 0.04% gc time)
1.954583 seconds (100.02 k allocations: 27.201 MiB, 5.40% gc time)
2563.290144 seconds (2.13 M allocations: 2.144 GiB, 0.16% gc time)
(n, ts) = (17, [587.996828594, 645.393233337, 6.780198853, 5.500113209, 32.
211955373, 1.77538201])
12-element Vector{Vector{Float64}}:
[0.006171065, 0.002836634, 0.003528211, 0.001600151, 0.001332193, 0.002764
545]
[0.02525832, 0.011758195, 0.010520651, 0.003880729, 0.003850639, 0.0062906
35]
[0.107809398, 0.043582287, 0.022071127, 0.008899321, 0.010518741, 0.010023
664]
[0.361169763, 0.172788135, 0.047564305, 0.020227567, 0.033864811, 0.019616
37]
[0.996815545, 0.488680971, 0.086696966, 0.039261421, 0.104752605, 0.033367
834]
[2.530754208, 1.090251481, 0.24535913, 0.074840353, 0.157630869, 0.0541497
18]
[5.508283047, 2.76269396, 0.286064982, 0.127681997, 0.346124437, 0.0844756
98]
[12.491102848, 6.508444383, 0.690010995, 0.219140206, 0.680784867, 0.13648
0417]
[21.255836426, 10.797951878, 0.806370854, 0.377695749, 1.249375577, 0.1932
2269]
[63.872513055, 32.67609872, 2.030823259, 0.898381484, 3.944500053, 0.45926
0821]
[229.61943815, 123.700041422, 4.184588132, 2.751999032, 14.798665264, 1.11
9510035]
[587.996828594, 645.393233337, 6.780198853, 5.500113209, 32.211955373, 1.7
7538201]csa_g = map(csan) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
@time ts = map(ADJOINT_METHODS_G) do alg
@info "Running $alg"
solver = Rodas5(autodiff = AutoFiniteDiff())
@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
end0.001806 seconds (2.54 k allocations: 289.445 KiB)
0.002849 seconds (6.17 k allocations: 488.875 KiB)
0.002105 seconds (4.75 k allocations: 338.898 KiB)
0.003188 seconds (8.92 k allocations: 575.453 KiB)
0.470901 seconds (441.66 k allocations: 24.257 MiB, 94.39% compilation ti
me)
(n, ts) = (2, [0.001299582, 0.002450377, 0.001583211, 0.002916453])
9.165433 seconds (7.52 M allocations: 372.279 MiB, 2.56% gc time, 99.93%
compilation time)
0.005951 seconds (7.19 k allocations: 682.125 KiB)
9.270521 seconds (7.73 M allocations: 382.894 MiB, 1.06% gc time, 99.91%
compilation time)
0.007348 seconds (13.23 k allocations: 910.031 KiB)
18.479189 seconds (15.31 M allocations: 759.973 MiB, 1.80% gc time, 99.68%
compilation time)
(n, ts) = (3, [0.003959128, 0.005819448, 0.00540886, 0.007030011])
9.093432 seconds (7.52 M allocations: 372.629 MiB, 1.02% gc time, 99.86%
compilation time)
0.010141 seconds (9.26 k allocations: 1.067 MiB)
9.190527 seconds (7.74 M allocations: 383.313 MiB, 0.97% gc time, 99.80%
compilation time)
0.013337 seconds (18.10 k allocations: 1.426 MiB)
18.362876 seconds (15.33 M allocations: 763.509 MiB, 0.99% gc time, 99.40%
compilation time)
(n, ts) = (4, [0.009938495, 0.009941075, 0.015590993, 0.013444855])
8.423085 seconds (5.83 M allocations: 279.143 MiB, 1.01% gc time, 99.60%
compilation time)
0.020250 seconds (11.74 k allocations: 1.742 MiB)
8.737583 seconds (6.20 M allocations: 297.419 MiB, 1.57% gc time, 99.16%
compilation time)
0.023570 seconds (20.58 k allocations: 2.155 MiB)
17.357630 seconds (12.10 M allocations: 588.358 MiB, 1.28% gc time, 98.25%
compilation time)
(n, ts) = (5, [0.031967453, 0.020054849, 0.071021427, 0.02348135])
8.676998 seconds (5.83 M allocations: 280.514 MiB, 2.29% gc time, 99.21%
compilation time)
0.035212 seconds (14.75 k allocations: 2.842 MiB)
8.754428 seconds (6.20 M allocations: 298.816 MiB, 1.08% gc time, 98.84%
compilation time)
0.041764 seconds (27.09 k allocations: 3.487 MiB)
17.757278 seconds (12.13 M allocations: 598.843 MiB, 1.65% gc time, 97.21%
compilation time)
(n, ts) = (6, [0.066605202, 0.034677438, 0.09929653, 0.040888984])
8.475536 seconds (5.64 M allocations: 273.301 MiB, 1.10% gc time, 98.29%
compilation time)
0.057168 seconds (18.22 k allocations: 4.432 MiB)
8.768203 seconds (6.01 M allocations: 291.727 MiB, 1.18% gc time, 97.44%
compilation time)
0.064860 seconds (32.50 k allocations: 5.265 MiB)
17.858263 seconds (11.78 M allocations: 595.182 MiB, 1.10% gc time, 94.49%
compilation time)
(n, ts) = (7, [0.143510306, 0.056491407, 0.220610989, 0.063846087])
0.310545 seconds (6.60 k allocations: 7.829 MiB)
0.087917 seconds (22.29 k allocations: 6.766 MiB)
0.454546 seconds (19.32 k allocations: 8.627 MiB)
0.098933 seconds (40.35 k allocations: 7.927 MiB)
2.184856 seconds (178.47 k allocations: 62.874 MiB, 12.50% gc time)
(n, ts) = (8, [0.409239615, 0.087388036, 0.629126559, 0.098297667])
0.608577 seconds (7.51 k allocations: 12.114 MiB)
0.176661 seconds (26.88 k allocations: 10.085 MiB, 17.82% gc time)
0.886739 seconds (22.78 k allocations: 13.252 MiB)
0.160880 seconds (47.70 k allocations: 11.604 MiB)
3.708159 seconds (211.10 k allocations: 94.687 MiB, 2.46% gc time)
(n, ts) = (9, [0.611990584, 0.144533841, 0.916550719, 0.190512426])
1.154813 seconds (8.77 k allocations: 17.895 MiB)
0.217186 seconds (34.16 k allocations: 14.581 MiB)
1.568501 seconds (23.96 k allocations: 19.216 MiB)
0.234477 seconds (55.54 k allocations: 16.334 MiB)
6.446957 seconds (246.20 k allocations: 136.628 MiB, 1.58% gc time)
(n, ts) = (10, [1.161188145, 0.249260181, 1.584077804, 0.265996047])
4.156729 seconds (13.50 k allocations: 35.815 MiB, 0.45% gc time)
0.498971 seconds (46.18 k allocations: 27.760 MiB, 2.15% gc time)
5.400423 seconds (31.86 k allocations: 37.959 MiB, 0.29% gc time)
0.693351 seconds (72.60 k allocations: 30.481 MiB, 24.09% gc time)
21.493533 seconds (329.63 k allocations: 264.610 MiB, 1.93% gc time)
(n, ts) = (12, [4.17988877, 0.538733149, 5.462228645, 0.552399873])
13.421478 seconds (16.55 k allocations: 84.682 MiB, 0.22% gc time)
1.462577 seconds (73.53 k allocations: 63.578 MiB, 11.98% gc time)
17.382767 seconds (40.63 k allocations: 88.781 MiB, 0.28% gc time)
1.344119 seconds (104.84 k allocations: 68.044 MiB, 2.30% gc time)
67.533788 seconds (472.47 k allocations: 610.750 MiB, 1.24% gc time)
(n, ts) = (15, [13.58970723, 1.328753451, 17.560043902, 1.432725345])
27.700902 seconds (19.56 k allocations: 138.003 MiB, 0.48% gc time)
2.515945 seconds (89.78 k allocations: 101.685 MiB, 6.42% gc time)
51.880001 seconds (42.30 k allocations: 142.853 MiB, 0.24% gc time)
2.581125 seconds (126.33 k allocations: 108.001 MiB, 9.77% gc time)
169.304522 seconds (557.30 k allocations: 981.664 MiB, 0.83% gc time)
(n, ts) = (17, [27.892838065, 2.495646676, 51.890351576, 2.332406881])
12-element Vector{Vector{Float64}}:
[0.001299582, 0.002450377, 0.001583211, 0.002916453]
[0.003959128, 0.005819448, 0.00540886, 0.007030011]
[0.009938495, 0.009941075, 0.015590993, 0.013444855]
[0.031967453, 0.020054849, 0.071021427, 0.02348135]
[0.066605202, 0.034677438, 0.09929653, 0.040888984]
[0.143510306, 0.056491407, 0.220610989, 0.063846087]
[0.409239615, 0.087388036, 0.629126559, 0.098297667]
[0.611990584, 0.144533841, 0.916550719, 0.190512426]
[1.161188145, 0.249260181, 1.584077804, 0.265996047]
[4.17988877, 0.538733149, 5.462228645, 0.552399873]
[13.58970723, 1.328753451, 17.560043902, 1.432725345]
[27.892838065, 2.495646676, 51.890351576, 2.332406881]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_iq = [[csa_iq[j][i] for j in eachindex(csa_iq)] for i in eachindex(csa_iq[1])]
csadata_g = [[csa_g[j][i] for j in eachindex(csa_g)] for i in eachindex(csa_g[1])]
plot!(plt1, n_to_param.(csan), csadata_iq[1], lab = "Interpolating CASA user-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt1, n_to_param.(csan), csadata_iq[2], lab = "Interpolating CASA AD-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(
plt1, n_to_param.(csan), csadata_iq[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_iq[1 + 3], lab = "Quadrature CASA user-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt1, n_to_param.(csan), csadata_iq[2 + 3], lab = "Quadrature CASA AD-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(
plt1, n_to_param.(csan), csadata_iq[3 + 3], lab = raw"Quadrature CASA AD-$v^{T}J$ seeding",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt1, n_to_param.(csan), csadata_g[1], lab = "Gauss CASA AD-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(
plt1, n_to_param.(csan), csadata_g[2], lab = raw"Gauss CASA AD-$v^{T}J$ seeding",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt1, n_to_param.(csan), csadata_g[1 + 2], lab = "GaussKronrod CASA AD-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(
plt1, n_to_param.(csan), csadata_g[2 + 2], lab = raw"GaussKronrod 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(4) do ii
Alg = (InterpolatingAdjoint, QuadratureAdjoint, GaussAdjoint, GaussKronrodAdjoint)[ii]
(
advj1 = Alg(autodiff = true, autojacvec = EnzymeVJP()), # AD vJ (Enzyme)
advj2 = Alg(autodiff = true, autojacvec = ReverseDiffVJP(false)), # AD vJ (ReverseDiff)
advj3 = Alg(autodiff = true, autojacvec = ReverseDiffVJP(true)), # AD vJ (Compiled ReverseDiff)
advj4 = Alg(autodiff = true, autojacvec = SciMLSensitivity.MooncakeVJP()) # AD vJ (Mooncake)
)
end |> NamedTuple{(:interp, :quad, :gauss, :gausskronrod)}
adjoint_methods = mapreduce(collect, vcat, _adjoint_methods)16-element Vector{SciMLBase.AbstractAdjointSensitivityAlgorithm{0, true, Va
l{:central}}}:
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.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensiti
vity.ReverseDiffVJP{false}}(SciMLSensitivity.ReverseDiffVJP{false}(), false
, false)
SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensiti
vity.ReverseDiffVJP{true}}(SciMLSensitivity.ReverseDiffVJP{true}(), false,
false)
SciMLSensitivity.InterpolatingAdjoint{0, true, Val{:central}, SciMLSensiti
vity.MooncakeVJP}(SciMLSensitivity.MooncakeVJP(), false, false)
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.QuadratureAdjoint{0, true, Val{:central}, SciMLSensitivit
y.ReverseDiffVJP{false}, Val{true}}(SciMLSensitivity.ReverseDiffVJP{false}(
), 1.0e-6, 0.001, Val{true}())
SciMLSensitivity.QuadratureAdjoint{0, true, Val{:central}, SciMLSensitivit
y.ReverseDiffVJP{true}, Val{true}}(SciMLSensitivity.ReverseDiffVJP{true}(),
1.0e-6, 0.001, Val{true}())
SciMLSensitivity.QuadratureAdjoint{0, true, Val{:central}, SciMLSensitivit
y.MooncakeVJP, Val{true}}(SciMLSensitivity.MooncakeVJP(), 1.0e-6, 0.001, Va
l{true}())
SciMLSensitivity.GaussAdjoint{0, true, Val{:central}, SciMLSensitivity.Enz
ymeVJP{EnzymeCore.ReverseMode{false, false, false, EnzymeCore.FFIABI, false
, false}}, Val{true}}(SciMLSensitivity.EnzymeVJP{EnzymeCore.ReverseMode{fal
se, false, false, EnzymeCore.FFIABI, false, false}}(0, EnzymeCore.ReverseMo
de{false, false, false, EnzymeCore.FFIABI, false, false}()), false, Val{tru
e}())
SciMLSensitivity.GaussAdjoint{0, true, Val{:central}, SciMLSensitivity.Rev
erseDiffVJP{false}, Val{true}}(SciMLSensitivity.ReverseDiffVJP{false}(), fa
lse, Val{true}())
SciMLSensitivity.GaussAdjoint{0, true, Val{:central}, SciMLSensitivity.Rev
erseDiffVJP{true}, Val{true}}(SciMLSensitivity.ReverseDiffVJP{true}(), fals
e, Val{true}())
SciMLSensitivity.GaussAdjoint{0, true, Val{:central}, SciMLSensitivity.Moo
ncakeVJP, Val{true}}(SciMLSensitivity.MooncakeVJP(), false, Val{true}())
SciMLSensitivity.GaussKronrodAdjoint{0, true, Val{:central}, SciMLSensitiv
ity.EnzymeVJP{EnzymeCore.ReverseMode{false, false, false, EnzymeCore.FFIABI
, false, false}}}(SciMLSensitivity.EnzymeVJP{EnzymeCore.ReverseMode{false,
false, false, EnzymeCore.FFIABI, false, false}}(0, EnzymeCore.ReverseMode{f
alse, false, false, EnzymeCore.FFIABI, false, false}()), false)
SciMLSensitivity.GaussKronrodAdjoint{0, true, Val{:central}, SciMLSensitiv
ity.ReverseDiffVJP{false}}(SciMLSensitivity.ReverseDiffVJP{false}(), false)
SciMLSensitivity.GaussKronrodAdjoint{0, true, Val{:central}, SciMLSensitiv
ity.ReverseDiffVJP{true}}(SciMLSensitivity.ReverseDiffVJP{true}(), false)
SciMLSensitivity.GaussKronrodAdjoint{0, true, Val{:central}, SciMLSensitiv
ity.MooncakeVJP}(SciMLSensitivity.MooncakeVJP(), false)Warmup: compile all VJP backends before benchmarking.
let n = first(csan)
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
solver = Rodas5(autodiff = AutoFiniteDiff())
for alg in adjoint_methods
f = SciMLSensitivity.alg_autodiff(alg) ? bfun :
ODEFunction(bfun, jac = brusselator_jac)
diffeq_sen_l2(f, b_u0, tspan, b_p, bt, solver; sensalg = alg, tols...)
end
endcsavjp = 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 = AutoFiniteDiff())
@time diffeq_sen_l2(f, b_u0, tspan, b_p, bt, solver; sensalg = alg, tols...)
t = @elapsed diffeq_sen_l2(f, b_u0, tspan, b_p, bt, solver; sensalg = alg, tols...)
return t
end
@show n, ts
ts
end0.003472 seconds (7.34 k allocations: 569.445 KiB)
0.123174 seconds (567.59 k allocations: 23.797 MiB)
0.008846 seconds (3.63 k allocations: 295.617 KiB)
1.211385 seconds (2.71 M allocations: 124.079 MiB, 65.28% compilation tim
e: <1% of which was recompilation)
0.002648 seconds (7.90 k allocations: 499.922 KiB)
0.092986 seconds (450.86 k allocations: 18.901 MiB)
0.062070 seconds (4.40 k allocations: 307.938 KiB, 88.58% gc time)
0.002642 seconds (3.98 k allocations: 474.500 KiB)
0.002573 seconds (6.17 k allocations: 488.875 KiB)
0.081982 seconds (422.27 k allocations: 17.729 MiB)
0.007201 seconds (4.74 k allocations: 343.312 KiB)
0.002821 seconds (4.32 k allocations: 520.328 KiB)
0.003300 seconds (8.92 k allocations: 575.453 KiB)
0.087951 seconds (500.39 k allocations: 20.936 MiB)
0.008077 seconds (6.82 k allocations: 386.578 KiB)
0.003437 seconds (6.67 k allocations: 604.844 KiB)
3.678478 seconds (7.97 M allocations: 363.135 MiB, 2.66% gc time, 62.33%
compilation time: <1% of which was recompilation)
(n, ts) = (2, [0.00361374, 0.13941255, 0.008303054, 0.003268732, 0.00258191
5, 0.073138846, 0.006432454, 0.002191308, 0.002372366, 0.090886087, 0.00654
6363, 0.002448847, 0.003303522, 0.078120968, 0.007549068, 0.002921654])
0.010715 seconds (10.96 k allocations: 994.922 KiB)
0.456049 seconds (2.11 M allocations: 92.709 MiB, 13.44% gc time)
0.029405 seconds (5.10 k allocations: 556.945 KiB)
0.009134 seconds (6.10 k allocations: 993.820 KiB)
0.005902 seconds (9.10 k allocations: 695.078 KiB)
0.247037 seconds (1.29 M allocations: 56.838 MiB, 12.97% gc time)
0.018209 seconds (7.19 k allocations: 556.734 KiB)
0.005341 seconds (4.45 k allocations: 671.109 KiB)
0.005861 seconds (7.19 k allocations: 682.125 KiB)
0.238043 seconds (1.13 M allocations: 49.919 MiB, 14.32% gc time)
0.017019 seconds (7.72 k allocations: 615.203 KiB)
0.005471 seconds (4.83 k allocations: 716.219 KiB)
0.007426 seconds (13.23 k allocations: 910.031 KiB)
0.355085 seconds (1.62 M allocations: 71.162 MiB, 9.51% gc time)
0.024421 seconds (11.85 k allocations: 717.047 KiB)
0.007426 seconds (9.72 k allocations: 938.125 KiB)
2.882323 seconds (12.51 M allocations: 561.240 MiB, 8.76% gc time)
(n, ts) = (3, [0.010507832, 0.456075969, 0.029429637, 0.009015331, 0.005827
037, 0.26410017, 0.017874291, 0.005028922, 0.005796197, 0.220319642, 0.0167
92477, 0.005299771, 0.007610037, 0.330758412, 0.024394034, 0.00721734])
0.021730 seconds (16.77 k allocations: 1.758 MiB)
1.252436 seconds (6.26 M allocations: 263.560 MiB, 7.86% gc time)
0.084339 seconds (7.08 k allocations: 1.045 MiB)
0.023253 seconds (8.47 k allocations: 1.764 MiB)
0.009896 seconds (11.39 k allocations: 1005.797 KiB)
0.669561 seconds (3.28 M allocations: 138.065 MiB, 4.99% gc time)
0.045197 seconds (11.13 k allocations: 893.750 KiB)
0.011199 seconds (5.44 k allocations: 1.003 MiB)
0.010413 seconds (9.26 k allocations: 1.067 MiB)
0.586237 seconds (2.90 M allocations: 122.115 MiB, 5.78% gc time)
0.040587 seconds (11.68 k allocations: 1.042 MiB)
0.011392 seconds (5.75 k allocations: 1.145 MiB)
0.013009 seconds (18.10 k allocations: 1.426 MiB)
0.845008 seconds (4.19 M allocations: 176.445 MiB, 8.09% gc time)
0.057304 seconds (17.60 k allocations: 1.214 MiB)
0.015790 seconds (12.84 k allocations: 1.495 MiB)
7.360607 seconds (33.54 M allocations: 1.399 GiB, 6.77% gc time)
(n, ts) = (4, [0.022025398, 1.252281567, 0.083749416, 0.022984393, 0.010180
723, 0.646644736, 0.046507693, 0.011080109, 0.010152164, 0.559158369, 0.040
791524, 0.011312927, 0.013460266, 0.838309926, 0.056885436, 0.015582534])
0.047044 seconds (23.80 k allocations: 3.015 MiB)
2.883171 seconds (14.56 M allocations: 645.065 MiB, 7.99% gc time)
0.199217 seconds (9.63 k allocations: 1.973 MiB)
0.052936 seconds (11.29 k allocations: 3.000 MiB)
0.019205 seconds (14.40 k allocations: 1.447 MiB)
1.388445 seconds (7.16 M allocations: 317.339 MiB, 7.10% gc time)
0.095613 seconds (16.21 k allocations: 1.387 MiB)
0.022606 seconds (6.66 k allocations: 1.460 MiB)
0.020205 seconds (11.74 k allocations: 1.742 MiB)
1.242414 seconds (6.20 M allocations: 275.098 MiB, 8.34% gc time)
0.094449 seconds (16.75 k allocations: 1.802 MiB)
0.022852 seconds (6.74 k allocations: 1.815 MiB)
0.023849 seconds (20.58 k allocations: 2.155 MiB)
1.576231 seconds (8.21 M allocations: 364.008 MiB, 6.07% gc time)
0.140551 seconds (22.67 k allocations: 2.027 MiB)
0.028335 seconds (13.83 k allocations: 2.218 MiB)
15.740935 seconds (72.62 M allocations: 3.177 GiB, 7.02% gc time)
(n, ts) = (5, [0.045819467, 2.909324098, 0.199773585, 0.052205821, 0.019497
272, 1.39468882, 0.095082294, 0.022423617, 0.019641702, 1.192356089, 0.0954
57303, 0.022573345, 0.023824158, 1.602353592, 0.136287997, 0.028805291])
0.086944 seconds (32.42 k allocations: 5.000 MiB)
5.580199 seconds (29.36 M allocations: 1.230 GiB, 6.56% gc time)
0.400064 seconds (12.76 k allocations: 3.543 MiB)
0.102726 seconds (14.84 k allocations: 5.060 MiB)
0.033159 seconds (17.82 k allocations: 2.003 MiB)
2.585226 seconds (13.65 M allocations: 585.186 MiB, 6.51% gc time)
0.182559 seconds (22.39 k allocations: 1.996 MiB)
0.041794 seconds (8.16 k allocations: 2.207 MiB)
0.034340 seconds (14.75 k allocations: 2.842 MiB)
2.267800 seconds (11.83 M allocations: 508.290 MiB, 7.23% gc time)
0.168149 seconds (22.97 k allocations: 2.982 MiB)
0.041322 seconds (8.11 k allocations: 3.107 MiB)
0.040205 seconds (27.09 k allocations: 3.487 MiB)
3.012420 seconds (15.96 M allocations: 685.153 MiB, 6.55% gc time)
0.286655 seconds (31.13 k allocations: 3.359 MiB)
0.052760 seconds (17.94 k allocations: 3.739 MiB)
29.828166 seconds (142.04 M allocations: 6.014 GiB, 5.88% gc time)
(n, ts) = (6, [0.087048488, 5.581079745, 0.397664201, 0.103423198, 0.032759
969, 2.600944951, 0.181199178, 0.04161923, 0.03431266, 2.2231238, 0.1637631
05, 0.041262762, 0.03983651, 3.016810291, 0.288201676, 0.052403801])
0.179635 seconds (44.24 k allocations: 7.964 MiB)
10.625923 seconds (55.63 M allocations: 2.474 GiB, 6.83% gc time)
0.784731 seconds (16.48 k allocations: 5.938 MiB)
0.214418 seconds (19.60 k allocations: 7.989 MiB)
0.052983 seconds (21.85 k allocations: 2.693 MiB)
4.606611 seconds (23.94 M allocations: 1.064 GiB, 7.22% gc time)
0.419871 seconds (29.70 k allocations: 2.794 MiB)
0.071463 seconds (9.77 k allocations: 2.887 MiB)
0.056625 seconds (18.22 k allocations: 4.432 MiB)
3.933408 seconds (20.72 M allocations: 945.126 MiB, 6.78% gc time)
0.332488 seconds (30.28 k allocations: 4.716 MiB)
0.072627 seconds (9.50 k allocations: 4.688 MiB)
0.064053 seconds (32.50 k allocations: 5.265 MiB)
5.171406 seconds (27.37 M allocations: 1.218 GiB, 6.32% gc time)
0.431902 seconds (39.80 k allocations: 5.243 MiB)
0.089674 seconds (20.70 k allocations: 5.495 MiB)
54.532379 seconds (255.93 M allocations: 11.479 GiB, 6.81% gc time)
(n, ts) = (7, [0.170317159, 10.585433809, 0.777226412, 0.216174304, 0.05269
0309, 4.59330884, 0.418179807, 0.071603244, 0.05608409, 4.272567993, 0.3225
71636, 0.07235445, 0.06335653, 5.196515183, 0.435314463, 0.089440925])
0.279071 seconds (56.40 k allocations: 12.095 MiB)
17.778492 seconds (93.83 M allocations: 4.048 GiB, 6.67% gc time)
1.335491 seconds (20.71 k allocations: 9.473 MiB)
0.359945 seconds (24.48 k allocations: 12.083 MiB)
0.084253 seconds (27.71 k allocations: 3.636 MiB)
7.735270 seconds (40.66 M allocations: 1.752 GiB, 6.40% gc time)
0.609803 seconds (38.62 k allocations: 3.761 MiB)
0.120301 seconds (12.41 k allocations: 3.817 MiB)
0.087511 seconds (22.29 k allocations: 6.766 MiB)
6.431936 seconds (34.12 M allocations: 1.474 GiB, 6.77% gc time)
0.478763 seconds (38.73 k allocations: 7.160 MiB)
0.117642 seconds (11.14 k allocations: 7.013 MiB)
0.098242 seconds (40.35 k allocations: 7.927 MiB)
8.501917 seconds (45.24 M allocations: 1.953 GiB, 6.73% gc time)
0.619764 seconds (51.03 k allocations: 7.950 MiB)
0.145261 seconds (24.90 k allocations: 8.106 MiB)
89.780298 seconds (428.43 M allocations: 18.631 GiB, 6.04% gc time)
(n, ts) = (8, [0.278147102, 17.840925802, 1.333375775, 0.411650163, 0.08407
3865, 7.809434981, 0.592889691, 0.11934694, 0.087670476, 6.412567376, 0.480
473932, 0.116774134, 0.099939517, 8.532365066, 0.618991086, 0.146104162])
0.668823 seconds (107.37 k allocations: 20.228 MiB)
30.717385 seconds (160.08 M allocations: 6.759 GiB, 8.34% gc time)
2.309240 seconds (25.49 k allocations: 14.548 MiB)
0.656716 seconds (31.97 k allocations: 18.106 MiB, 5.68% gc time)
0.136033 seconds (33.16 k allocations: 4.793 MiB)
12.083068 seconds (63.44 M allocations: 2.676 GiB, 6.49% gc time)
0.904192 seconds (48.17 k allocations: 4.988 MiB, 4.00% gc time)
0.189321 seconds (14.59 k allocations: 4.984 MiB)
0.143452 seconds (26.88 k allocations: 10.085 MiB)
10.004703 seconds (53.29 M allocations: 2.254 GiB, 6.56% gc time)
0.756772 seconds (48.28 k allocations: 10.603 MiB)
0.189670 seconds (12.98 k allocations: 10.345 MiB)
0.159825 seconds (47.70 k allocations: 11.604 MiB)
13.293057 seconds (68.74 M allocations: 2.905 GiB, 9.85% gc time)
1.201789 seconds (61.73 k allocations: 11.646 MiB)
0.229088 seconds (29.22 k allocations: 11.814 MiB)
146.195951 seconds (692.06 M allocations: 29.450 GiB, 6.57% gc time)
(n, ts) = (9, [0.706431551, 30.16606051, 2.294413262, 0.618776356, 0.135549
89, 11.950760286, 0.904923691, 0.18977602, 0.143154308, 9.939428693, 0.7506
95665, 0.223405964, 0.161263537, 12.902374414, 1.203922575, 0.229141061])
0.755396 seconds (88.59 k allocations: 25.620 MiB, 5.17% gc time)
44.508793 seconds (234.14 M allocations: 9.730 GiB, 6.52% gc time)
3.904849 seconds (30.84 k allocations: 21.377 MiB)
0.984076 seconds (37.36 k allocations: 25.523 MiB, 4.10% gc time)
0.193005 seconds (39.23 k allocations: 6.158 MiB)
18.076197 seconds (94.85 M allocations: 3.937 GiB, 6.51% gc time)
1.397158 seconds (58.85 k allocations: 6.430 MiB, 2.35% gc time)
0.280856 seconds (17.02 k allocations: 6.333 MiB)
0.215839 seconds (34.16 k allocations: 14.581 MiB)
16.022582 seconds (85.57 M allocations: 3.561 GiB, 6.55% gc time)
1.243664 seconds (58.97 k allocations: 15.100 MiB, 2.70% gc time)
0.298666 seconds (15.90 k allocations: 14.821 MiB)
0.232980 seconds (55.54 k allocations: 16.333 MiB)
20.007104 seconds (104.93 M allocations: 4.364 GiB, 8.50% gc time)
1.497438 seconds (72.68 k allocations: 16.351 MiB, 2.32% gc time)
0.346367 seconds (32.59 k allocations: 16.521 MiB)
220.029876 seconds (1.04 G allocations: 43.549 GiB, 6.37% gc time)
(n, ts) = (10, [0.718035705, 45.007271432, 4.184487615, 0.986516006, 0.1925
43374, 17.877754083, 1.370441489, 0.280697685, 0.215208848, 15.965065824, 1
.209687548, 0.29770552, 0.266546783, 19.649636674, 1.460936135, 0.345842513
])
1.810605 seconds (130.62 k allocations: 48.696 MiB, 1.89% gc time)
95.713872 seconds (499.22 M allocations: 21.553 GiB, 7.00% gc time)
8.169386 seconds (43.90 k allocations: 42.393 MiB)
2.438446 seconds (54.66 k allocations: 48.844 MiB, 0.55% gc time)
0.522102 seconds (53.61 k allocations: 9.886 MiB, 19.37% gc time)
36.996231 seconds (191.77 M allocations: 8.270 GiB, 7.49% gc time)
2.699760 seconds (83.88 k allocations: 10.438 MiB, 1.26% gc time)
0.604649 seconds (23.09 k allocations: 10.766 MiB)
0.481880 seconds (46.18 k allocations: 27.760 MiB)
31.928776 seconds (167.31 M allocations: 7.234 GiB, 8.16% gc time)
3.324429 seconds (84.47 k allocations: 28.740 MiB)
0.677660 seconds (21.31 k allocations: 28.713 MiB, 5.03% gc time)
0.511944 seconds (72.60 k allocations: 30.481 MiB)
38.895637 seconds (204.15 M allocations: 8.823 GiB, 7.64% gc time)
2.904504 seconds (102.30 k allocations: 30.938 MiB)
0.750218 seconds (43.04 k allocations: 31.504 MiB)
454.100777 seconds (2.13 G allocations: 92.446 GiB, 6.47% gc time)
(n, ts) = (12, [1.783814185, 95.235942276, 8.173629281, 2.388099616, 0.4193
40934, 36.115468862, 2.713958114, 0.603202313, 0.509908561, 31.407736198, 2
.89952596, 0.673656871, 0.54436349, 38.417911535, 2.953413686, 0.79634494])
4.436341 seconds (207.92 k allocations: 109.801 MiB, 6.23% gc time)
238.907946 seconds (1.25 G allocations: 52.110 GiB, 7.15% gc time)
21.535459 seconds (66.69 k allocations: 99.539 MiB, 0.17% gc time)
6.692668 seconds (104.65 k allocations: 112.650 MiB, 0.42% gc time)
1.031336 seconds (79.53 k allocations: 18.622 MiB)
88.015483 seconds (458.25 M allocations: 19.037 GiB, 7.48% gc time)
6.647030 seconds (129.40 k allocations: 19.505 MiB)
1.538462 seconds (33.47 k allocations: 19.487 MiB, 2.87% gc time)
1.322483 seconds (73.53 k allocations: 63.577 MiB, 5.09% gc time)
78.525239 seconds (414.43 M allocations: 17.262 GiB, 7.88% gc time)
6.316001 seconds (130.02 k allocations: 64.799 MiB, 0.63% gc time)
1.756263 seconds (31.34 k allocations: 64.371 MiB, 1.64% gc time)
1.687656 seconds (104.84 k allocations: 68.044 MiB, 17.96% gc time)
92.155068 seconds (484.08 M allocations: 20.156 GiB, 7.20% gc time)
8.663405 seconds (151.47 k allocations: 68.739 MiB, 0.47% gc time)
1.933720 seconds (58.26 k allocations: 69.166 MiB, 0.82% gc time)
1118.787890 seconds (5.22 G allocations: 218.650 GiB, 6.67% gc time)
(n, ts) = (15, [4.24788943, 239.356738771, 19.706477595, 7.0467489, 1.03581
9545, 87.516289708, 6.476677465, 1.49842556, 1.293256141, 77.71851507, 6.66
3641966, 1.763134963, 1.368057221, 91.763055259, 8.202273278, 1.920254193])
8.611386 seconds (248.09 k allocations: 173.692 MiB, 3.18% gc time)
365.176599 seconds (1.92 G allocations: 83.774 GiB, 7.72% gc time)
32.736689 seconds (84.69 k allocations: 161.743 MiB, 0.18% gc time)
8.875047 seconds (101.67 k allocations: 173.513 MiB, 0.36% gc time)
1.818401 seconds (100.02 k allocations: 27.201 MiB)
143.852181 seconds (749.62 M allocations: 32.584 GiB, 7.82% gc time)
10.662632 seconds (165.37 k allocations: 28.552 MiB)
2.560827 seconds (41.66 k allocations: 28.016 MiB)
2.391044 seconds (89.78 k allocations: 101.685 MiB, 4.07% gc time)
129.604663 seconds (678.24 M allocations: 29.556 GiB, 7.51% gc time)
11.501967 seconds (166.01 k allocations: 103.646 MiB, 0.33% gc time)
3.250167 seconds (38.78 k allocations: 102.582 MiB, 5.80% gc time)
2.480234 seconds (126.33 k allocations: 108.000 MiB, 3.28% gc time)
148.445979 seconds (772.63 M allocations: 33.660 GiB, 8.19% gc time)
14.920421 seconds (188.60 k allocations: 108.863 MiB, 0.26% gc time)
3.503452 seconds (66.17 k allocations: 108.531 MiB, 8.51% gc time)
1784.531866 seconds (8.25 G allocations: 361.545 GiB, 6.91% gc time)
(n, ts) = (17, [8.456774942, 366.187780845, 34.460836359, 9.089973576, 1.93
1337333, 145.807917744, 10.708935184, 2.674365185, 2.46675501, 128.26010347
4, 14.133592402, 3.064827466, 2.57251467, 147.914193369, 13.114946171, 3.24
462207])
12-element Vector{Vector{Float64}}:
[0.00361374, 0.13941255, 0.008303054, 0.003268732, 0.002581915, 0.07313884
6, 0.006432454, 0.002191308, 0.002372366, 0.090886087, 0.006546363, 0.00244
8847, 0.003303522, 0.078120968, 0.007549068, 0.002921654]
[0.010507832, 0.456075969, 0.029429637, 0.009015331, 0.005827037, 0.264100
17, 0.017874291, 0.005028922, 0.005796197, 0.220319642, 0.016792477, 0.0052
99771, 0.007610037, 0.330758412, 0.024394034, 0.00721734]
[0.022025398, 1.252281567, 0.083749416, 0.022984393, 0.010180723, 0.646644
736, 0.046507693, 0.011080109, 0.010152164, 0.559158369, 0.040791524, 0.011
312927, 0.013460266, 0.838309926, 0.056885436, 0.015582534]
[0.045819467, 2.909324098, 0.199773585, 0.052205821, 0.019497272, 1.394688
82, 0.095082294, 0.022423617, 0.019641702, 1.192356089, 0.095457303, 0.0225
73345, 0.023824158, 1.602353592, 0.136287997, 0.028805291]
[0.087048488, 5.581079745, 0.397664201, 0.103423198, 0.032759969, 2.600944
951, 0.181199178, 0.04161923, 0.03431266, 2.2231238, 0.163763105, 0.0412627
62, 0.03983651, 3.016810291, 0.288201676, 0.052403801]
[0.170317159, 10.585433809, 0.777226412, 0.216174304, 0.052690309, 4.59330
884, 0.418179807, 0.071603244, 0.05608409, 4.272567993, 0.322571636, 0.0723
5445, 0.06335653, 5.196515183, 0.435314463, 0.089440925]
[0.278147102, 17.840925802, 1.333375775, 0.411650163, 0.084073865, 7.80943
4981, 0.592889691, 0.11934694, 0.087670476, 6.412567376, 0.480473932, 0.116
774134, 0.099939517, 8.532365066, 0.618991086, 0.146104162]
[0.706431551, 30.16606051, 2.294413262, 0.618776356, 0.13554989, 11.950760
286, 0.904923691, 0.18977602, 0.143154308, 9.939428693, 0.750695665, 0.2234
05964, 0.161263537, 12.902374414, 1.203922575, 0.229141061]
[0.718035705, 45.007271432, 4.184487615, 0.986516006, 0.192543374, 17.8777
54083, 1.370441489, 0.280697685, 0.215208848, 15.965065824, 1.209687548, 0.
29770552, 0.266546783, 19.649636674, 1.460936135, 0.345842513]
[1.783814185, 95.235942276, 8.173629281, 2.388099616, 0.419340934, 36.1154
68862, 2.713958114, 0.603202313, 0.509908561, 31.407736198, 2.89952596, 0.6
73656871, 0.54436349, 38.417911535, 2.953413686, 0.79634494]
[4.24788943, 239.356738771, 19.706477595, 7.0467489, 1.035819545, 87.51628
9708, 6.476677465, 1.49842556, 1.293256141, 77.71851507, 6.663641966, 1.763
134963, 1.368057221, 91.763055259, 8.202273278, 1.920254193]
[8.456774942, 366.187780845, 34.460836359, 9.089973576, 1.931337333, 145.8
07917744, 10.708935184, 2.674365185, 2.46675501, 128.260103474, 14.13359240
2, 3.064827466, 2.57251467, 147.914193369, 13.114946171, 3.24462207]csacompare = [[csavjp[j][i] for j in eachindex(csavjp)] for i in eachindex(csavjp[1])]
plt_interp = plot(title = "Brusselator interpolating adjoint VJP scaling");
plot!(plt_interp, n_to_param.(csan), csadata_iq[2], lab = "AD-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_interp, n_to_param.(csan), csacompare[1], lab = raw"EnzymeVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_interp, n_to_param.(csan), csacompare[2], lab = raw"ReverseDiffVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_interp, n_to_param.(csan), csacompare[3], lab = raw"Compiled ReverseDiffVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_interp, n_to_param.(csan), csacompare[4], lab = raw"MooncakeVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
xaxis!(plt_interp, "Number of Parameters", :log10);
yaxis!(plt_interp, "Runtime (s)", :log10);
plot!(plt_interp, legend = :outertopleft, size = (1200, 600))
plt2 = plot(title = "Brusselator quadrature adjoint VJP scaling");
plot!(plt2, n_to_param.(csan), csadata_iq[2 + 3], lab = "AD-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt2, n_to_param.(csan), csacompare[1 + 4], lab = raw"EnzymeVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt2, n_to_param.(csan), csacompare[2 + 4], lab = raw"ReverseDiffVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt2, n_to_param.(csan), csacompare[3 + 4], lab = raw"Compiled ReverseDiffVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt2, n_to_param.(csan), csacompare[4 + 4], lab = raw"MooncakeVJP",
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))
plt_gauss = plot(title = "Brusselator Gauss adjoint VJP scaling");
plot!(plt_gauss, n_to_param.(csan), csadata_g[1], lab = "AD-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_gauss, n_to_param.(csan), csacompare[1 + 8], lab = raw"EnzymeVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_gauss, n_to_param.(csan), csacompare[2 + 8], lab = raw"ReverseDiffVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_gauss, n_to_param.(csan), csacompare[3 + 8], lab = raw"Compiled ReverseDiffVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_gauss, n_to_param.(csan), csacompare[4 + 8], lab = raw"MooncakeVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
xaxis!(plt_gauss, "Number of Parameters", :log10);
yaxis!(plt_gauss, "Runtime (s)", :log10);
plot!(plt_gauss, legend = :outertopleft, size = (1200, 600))
plt_gk = plot(title = "Brusselator GaussKronrod adjoint VJP scaling");
plot!(plt_gk, n_to_param.(csan), csadata_g[1 + 2], lab = "AD-Jacobian",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_gk, n_to_param.(csan), csacompare[1 + 12], lab = raw"EnzymeVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_gk, n_to_param.(csan), csacompare[2 + 12], lab = raw"ReverseDiffVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_gk, n_to_param.(csan), csacompare[3 + 12], lab = raw"Compiled ReverseDiffVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_gk, n_to_param.(csan), csacompare[4 + 12], lab = raw"MooncakeVJP",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
xaxis!(plt_gk, "Number of Parameters", :log10);
yaxis!(plt_gk, "Runtime (s)", :log10);
plot!(plt_gk, legend = :outertopleft, size = (1200, 600))
SUNDIALS CVODES C Adjoint Benchmarks
SundialsAdjoint uses the SUNDIALS CVODES C adjoint interface (CVodeAdjInit/CVodeF/CVodeB): the forward pass is re-integrated by CVODES with checkpointing, and both the backward (adjoint) pass and the parameter-gradient quadrature are integrated by CVODES itself. The vector-Jacobian products inside the backward pass use the same autojacvec machinery as the native Julia adjoints, so comparing against GaussAdjoint with the same vjp choice measures the difference of the surrounding ODE-integration and checkpointing machinery (CVODES in C vs the native Julia adjoints). Note that SundialsAdjoint requires CVODE_BDF/CVODE_Adams as the solver, while the native adjoint rows above use Rodas5, so the forward solver differs as well.
using Sundials
sundials_configs = [
("CVODES Dense EnzymeVJP", CVODE_BDF(),
SundialsAdjoint(autojacvec = EnzymeVJP())),
("CVODES Dense Compiled ReverseDiffVJP", CVODE_BDF(),
SundialsAdjoint(autojacvec = ReverseDiffVJP(true))),
("CVODES GMRES EnzymeVJP", CVODE_BDF(linear_solver = :GMRES),
SundialsAdjoint(autojacvec = EnzymeVJP())),
("CVODES GMRES Compiled ReverseDiffVJP", CVODE_BDF(linear_solver = :GMRES),
SundialsAdjoint(autojacvec = ReverseDiffVJP(true))),
]4-element Vector{Tuple{String, Sundials.CVODE_BDF{:Newton, LinearSolver, No
thing, Nothing} where LinearSolver, SciMLSensitivity.SundialsAdjoint{0, tru
e, Val{:central}}}}:
("CVODES Dense EnzymeVJP", Sundials.CVODE_BDF{:Newton, :Dense, Nothing, No
thing}(0, 0, 0, false, 10, 5, 7, 3, 10, nothing, nothing, 0), SciMLSensitiv
ity.SundialsAdjoint{0, true, Val{:central}, SciMLSensitivity.EnzymeVJP{Enzy
meCore.ReverseMode{false, false, false, EnzymeCore.FFIABI, false, false}}}(
SciMLSensitivity.EnzymeVJP{EnzymeCore.ReverseMode{false, false, false, Enzy
meCore.FFIABI, false, false}}(0, EnzymeCore.ReverseMode{false, false, false
, EnzymeCore.FFIABI, false, false}()), 150, :hermite, true))
("CVODES Dense Compiled ReverseDiffVJP", Sundials.CVODE_BDF{:Newton, :Dens
e, Nothing, Nothing}(0, 0, 0, false, 10, 5, 7, 3, 10, nothing, nothing, 0),
SciMLSensitivity.SundialsAdjoint{0, true, Val{:central}, SciMLSensitivity.
ReverseDiffVJP{true}}(SciMLSensitivity.ReverseDiffVJP{true}(), 150, :hermit
e, true))
("CVODES GMRES EnzymeVJP", Sundials.CVODE_BDF{:Newton, :GMRES, Nothing, No
thing}(0, 0, 0, false, 10, 5, 7, 3, 10, nothing, nothing, 0), SciMLSensitiv
ity.SundialsAdjoint{0, true, Val{:central}, SciMLSensitivity.EnzymeVJP{Enzy
meCore.ReverseMode{false, false, false, EnzymeCore.FFIABI, false, false}}}(
SciMLSensitivity.EnzymeVJP{EnzymeCore.ReverseMode{false, false, false, Enzy
meCore.FFIABI, false, false}}(0, EnzymeCore.ReverseMode{false, false, false
, EnzymeCore.FFIABI, false, false}()), 150, :hermite, true))
("CVODES GMRES Compiled ReverseDiffVJP", Sundials.CVODE_BDF{:Newton, :GMRE
S, Nothing, Nothing}(0, 0, 0, false, 10, 5, 7, 3, 10, nothing, nothing, 0),
SciMLSensitivity.SundialsAdjoint{0, true, Val{:central}, SciMLSensitivity.
ReverseDiffVJP{true}}(SciMLSensitivity.ReverseDiffVJP{true}(), 150, :hermit
e, true))Check that the CVODES C adjoint returns the same gradient as the native GaussAdjoint before timing it:
let n = first(csan)
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
solver = Rodas5(autodiff = AutoFiniteDiff())
gauss = GaussAdjoint(autodiff = true, autojacvec = EnzymeVJP())
du0_ref, dp_ref = diffeq_sen_l2(
bfun, b_u0, tspan, b_p, bt, solver; sensalg = gauss, tols...)
for (name, alg, sensealg) in sundials_configs
du0, dp = diffeq_sen_l2(
bfun, b_u0, tspan, b_p, bt, alg; sensalg = sensealg, tols...)
err_du0 = norm(du0 - du0_ref) / norm(du0_ref)
err_dp = norm(vec(dp) - vec(dp_ref)) / norm(vec(dp_ref))
@show name, err_du0, err_dp
end
end(name, err_du0, err_dp) = ("CVODES Dense EnzymeVJP", 1.576392245414933e-6,
1.1362910965483166e-5)
(name, err_du0, err_dp) = ("CVODES Dense Compiled ReverseDiffVJP", 1.576392
118216258e-6, 1.1362910575670025e-5)
(name, err_du0, err_dp) = ("CVODES GMRES EnzymeVJP", 1.4691746698600681e-6,
1.08063323821113e-5)
(name, err_du0, err_dp) = ("CVODES GMRES Compiled ReverseDiffVJP", 1.469174
542203823e-6, 1.0806332171648812e-5)csa_sundials = map(csan) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
@time ts = map(sundials_configs) do (name, alg, sensealg)
@info "Running $name"
@time diffeq_sen_l2(bfun, b_u0, tspan, b_p, bt, alg; sensalg = sensealg, tols...)
t = @elapsed diffeq_sen_l2(
bfun, b_u0, tspan, b_p, bt, alg; sensalg = sensealg, tols...)
return t
end
@show n, ts
ts
end0.002254 seconds (7.28 k allocations: 319.281 KiB)
0.005884 seconds (6.96 k allocations: 249.562 KiB)
0.002911 seconds (9.43 k allocations: 405.531 KiB)
0.006926 seconds (8.32 k allocations: 286.875 KiB)
0.411032 seconds (510.40 k allocations: 25.015 MiB, 11.78% gc time, 91.19
% compilation time)
(n, ts) = (2, [0.00183678, 0.00532172, 0.002630326, 0.006280965])
0.005005 seconds (11.21 k allocations: 475.281 KiB)
0.015614 seconds (12.46 k allocations: 460.938 KiB)
0.006127 seconds (15.21 k allocations: 618.844 KiB)
0.018740 seconds (15.56 k allocations: 545.438 KiB)
0.091486 seconds (109.43 k allocations: 4.146 MiB)
(n, ts) = (3, [0.005080591, 0.015367953, 0.006208965, 0.018711376])
0.007546 seconds (13.55 k allocations: 577.594 KiB)
0.031428 seconds (17.89 k allocations: 684.312 KiB)
0.007529 seconds (16.26 k allocations: 667.219 KiB)
0.034186 seconds (20.04 k allocations: 741.531 KiB)
0.161573 seconds (136.03 k allocations: 5.259 MiB)
(n, ts) = (4, [0.007802367, 0.030955487, 0.007928925, 0.033798811])
0.012927 seconds (15.82 k allocations: 673.938 KiB)
0.056775 seconds (24.53 k allocations: 983.406 KiB)
0.011046 seconds (18.01 k allocations: 728.906 KiB)
0.057761 seconds (26.79 k allocations: 1.018 MiB)
0.275518 seconds (170.84 k allocations: 6.739 MiB)
(n, ts) = (5, [0.012844889, 0.057778578, 0.011146358, 0.054837914])
0.022197 seconds (19.61 k allocations: 842.422 KiB)
0.103085 seconds (33.08 k allocations: 1.304 MiB)
0.017572 seconds (21.76 k allocations: 869.609 KiB)
0.117228 seconds (35.90 k allocations: 1.373 MiB)
0.537792 seconds (221.26 k allocations: 8.742 MiB)
(n, ts) = (6, [0.022454845, 0.127030141, 0.017593611, 0.110177455])
0.033584 seconds (22.45 k allocations: 964.359 KiB)
0.161578 seconds (42.32 k allocations: 1.732 MiB)
0.021244 seconds (22.60 k allocations: 906.984 KiB)
0.133569 seconds (44.07 k allocations: 1.771 MiB)
0.704647 seconds (263.45 k allocations: 10.704 MiB)
(n, ts) = (7, [0.03397384, 0.15768068, 0.021229632, 0.141042663])
0.056076 seconds (28.34 k allocations: 1.193 MiB)
0.324117 seconds (54.50 k allocations: 2.215 MiB)
0.033555 seconds (35.38 k allocations: 1.257 MiB)
0.232314 seconds (64.20 k allocations: 2.451 MiB)
1.292829 seconds (365.40 k allocations: 14.276 MiB)
(n, ts) = (8, [0.056600525, 0.329599551, 0.033647932, 0.226354778])
0.092781 seconds (34.60 k allocations: 1.442 MiB)
0.490026 seconds (68.48 k allocations: 2.765 MiB)
0.045023 seconds (36.66 k allocations: 1.325 MiB)
0.304748 seconds (74.45 k allocations: 2.900 MiB)
1.846602 seconds (428.93 k allocations: 16.909 MiB)
(n, ts) = (9, [0.091864218, 0.477933803, 0.04471587, 0.298878422])
0.142494 seconds (39.46 k allocations: 1.668 MiB)
0.655841 seconds (81.90 k allocations: 3.322 MiB)
0.053916 seconds (32.61 k allocations: 1.296 MiB)
0.366377 seconds (79.64 k allocations: 3.245 MiB)
2.437129 seconds (467.79 k allocations: 19.104 MiB)
(n, ts) = (10, [0.141302711, 0.665727226, 0.05564256, 0.355051519])
0.329276 seconds (53.48 k allocations: 2.260 MiB)
1.196274 seconds (115.78 k allocations: 4.784 MiB)
0.108298 seconds (50.52 k allocations: 1.957 MiB)
0.769534 seconds (119.00 k allocations: 4.847 MiB)
4.805530 seconds (678.09 k allocations: 27.741 MiB)
(n, ts) = (12, [0.328821866, 1.189259415, 0.107593889, 0.775427843])
1.044998 seconds (78.77 k allocations: 3.339 MiB)
3.041670 seconds (177.69 k allocations: 7.285 MiB)
0.190320 seconds (61.42 k allocations: 2.427 MiB)
1.470264 seconds (172.19 k allocations: 7.103 MiB)
11.659864 seconds (980.72 k allocations: 40.351 MiB, 0.47% gc time)
(n, ts) = (15, [1.035520923, 3.021333253, 0.18992173, 1.664434614])
1.893901 seconds (98.57 k allocations: 4.185 MiB)
5.994477 seconds (226.44 k allocations: 9.461 MiB)
0.247590 seconds (61.78 k allocations: 2.495 MiB)
2.365282 seconds (207.07 k allocations: 8.895 MiB)
21.661627 seconds (1.19 M allocations: 50.114 MiB)
(n, ts) = (17, [1.89093375, 6.437804369, 0.246023017, 2.583206237])
12-element Vector{Vector{Float64}}:
[0.00183678, 0.00532172, 0.002630326, 0.006280965]
[0.005080591, 0.015367953, 0.006208965, 0.018711376]
[0.007802367, 0.030955487, 0.007928925, 0.033798811]
[0.012844889, 0.057778578, 0.011146358, 0.054837914]
[0.022454845, 0.127030141, 0.017593611, 0.110177455]
[0.03397384, 0.15768068, 0.021229632, 0.141042663]
[0.056600525, 0.329599551, 0.033647932, 0.226354778]
[0.091864218, 0.477933803, 0.04471587, 0.298878422]
[0.141302711, 0.665727226, 0.05564256, 0.355051519]
[0.328821866, 1.189259415, 0.107593889, 0.775427843]
[1.035520923, 3.021333253, 0.18992173, 1.664434614]
[1.89093375, 6.437804369, 0.246023017, 2.583206237]csadata_sundials = [[csa_sundials[j][i] for j in eachindex(csa_sundials)]
for i in eachindex(csa_sundials[1])]
plt_sundials = plot(title = "Brusselator CVODES C adjoint vs native adjoints");
plot!(plt_sundials, n_to_param.(csan), csadata_g[2],
lab = raw"GaussAdjoint EnzymeVJP (Rodas5)",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_sundials, n_to_param.(csan), csacompare[3 + 8],
lab = raw"GaussAdjoint Compiled ReverseDiffVJP (Rodas5)",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
plot!(plt_sundials, n_to_param.(csan), csacompare[1],
lab = raw"InterpolatingAdjoint EnzymeVJP (Rodas5)",
lw = lw, marksize = ms, linestyle = :auto, marker = :auto);
for (i, (name, _, _)) in enumerate(sundials_configs)
plot!(plt_sundials, n_to_param.(csan), csadata_sundials[i], lab = name,
lw = lw, marksize = ms, linestyle = :auto, marker = :auto)
end
xaxis!(plt_sundials, "Number of Parameters", :log10);
yaxis!(plt_sundials, "Runtime (s)", :log10);
plot!(plt_sundials, legend = :outertopleft, size = (1200, 600))
Same-solver adjoint comparison
The plot above conflates the forward solver choice with the adjoint method. To isolate the adjoint machinery itself, compare the CVODES C adjoint against the native GaussAdjoint with the same ODE solver: CVODE_BDF with dense and GMRES linear solvers, using EnzymeVJP for both adjoints. FBDF (the native Julia fixed-leading-coefficient BDF analogue of CVODE) with GaussAdjoint is included as the all-Julia BDF counterpart, with dense LU and matrix-free KrylovJL_GMRES linear solvers.
GaussAdjoint with a Sundials solver currently requires a dense forward solution (with saveat the checkpointing path errors on the interpolation type), so the GaussAdjoint rows use dense = true forward solves while the SundialsAdjoint rows use the saveat forward solve plus the internal checkpointed CVodeF re-integration; each method is measured in its natural mode of use.
using LinearSolve
function diffeq_sen_l2_dense(df, u0, tspan, p, t, alg;
abstol = 1e-5, reltol = 1e-7, sensalg, kwargs...)
prob = ODEProblem{true, SciMLBase.FullSpecialize}(df, u0, tspan, p)
sol = solve(prob, alg, abstol = abstol, reltol = reltol, dense = true; 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)
end
gauss_enz = GaussAdjoint(autodiff = true, autojacvec = EnzymeVJP())
sundials_enz = SundialsAdjoint(autojacvec = EnzymeVJP())
same_solver_configs = [
("CVODE_BDF Dense + SundialsAdjoint", CVODE_BDF(), sundials_enz, false),
("CVODE_BDF Dense + GaussAdjoint", CVODE_BDF(), gauss_enz, true),
("CVODE_BDF GMRES + SundialsAdjoint",
CVODE_BDF(linear_solver = :GMRES), sundials_enz, false),
("CVODE_BDF GMRES + GaussAdjoint",
CVODE_BDF(linear_solver = :GMRES), gauss_enz, true),
("FBDF + GaussAdjoint", FBDF(autodiff = AutoFiniteDiff()), gauss_enz, true),
("FBDF GMRES + GaussAdjoint",
FBDF(linsolve = KrylovJL_GMRES(), autodiff = AutoFiniteDiff()), gauss_enz, true),
]
function run_same_solver(bfun, b_u0, b_p, alg, sensealg, dense_forward)
return dense_forward ?
diffeq_sen_l2_dense(bfun, b_u0, tspan, b_p, bt, alg; sensalg = sensealg, tols...) :
diffeq_sen_l2(bfun, b_u0, tspan, b_p, bt, alg; sensalg = sensealg, tols...)
endrun_same_solver (generic function with 1 method)let n = first(csan)
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
solver = Rodas5(autodiff = AutoFiniteDiff())
du0_ref, dp_ref = diffeq_sen_l2(
bfun, b_u0, tspan, b_p, bt, solver; sensalg = gauss_enz, tols...)
for (name, alg, sensealg, dense_forward) in same_solver_configs
du0, dp = run_same_solver(bfun, b_u0, b_p, alg, sensealg, dense_forward)
err_du0 = norm(du0 - du0_ref) / norm(du0_ref)
err_dp = norm(vec(dp) - vec(dp_ref)) / norm(vec(dp_ref))
@show name, err_du0, err_dp
end
end(name, err_du0, err_dp) = ("CVODE_BDF Dense + SundialsAdjoint", 1.576392245
414933e-6, 1.1362910965483166e-5)
(name, err_du0, err_dp) = ("CVODE_BDF Dense + GaussAdjoint", 1.572195778601
5652e-6, 1.8830159827863253e-5)
(name, err_du0, err_dp) = ("CVODE_BDF GMRES + SundialsAdjoint", 1.469174669
8600681e-6, 1.08063323821113e-5)
(name, err_du0, err_dp) = ("CVODE_BDF GMRES + GaussAdjoint", 1.578966610320
9478e-6, 1.8891961107264642e-5)
(name, err_du0, err_dp) = ("FBDF + GaussAdjoint", 4.1711912972772977e-7, 2.
3283254883689787e-5)
(name, err_du0, err_dp) = ("FBDF GMRES + GaussAdjoint", 9.621092548258918e-
7, 2.387569401387661e-5)csa_same_solver = map(csan) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
@time ts = map(same_solver_configs) do (name, alg, sensealg, dense_forward)
@info "Running $name"
@time run_same_solver(bfun, b_u0, b_p, alg, sensealg, dense_forward)
t = @elapsed run_same_solver(bfun, b_u0, b_p, alg, sensealg, dense_forward)
return t
end
@show n, ts
ts
end0.002547 seconds (7.28 k allocations: 319.281 KiB)
0.002890 seconds (7.56 k allocations: 426.625 KiB)
0.002518 seconds (9.43 k allocations: 405.531 KiB)
0.003746 seconds (9.22 k allocations: 495.453 KiB)
0.002477 seconds (5.50 k allocations: 405.812 KiB)
0.010041 seconds (36.68 k allocations: 1.940 MiB)
0.477326 seconds (817.84 k allocations: 40.909 MiB, 10.34% gc time, 89.77
% compilation time)
(n, ts) = (2, [0.002149589, 0.002593146, 0.002285169, 0.003530381, 0.002218
059, 0.009112583])
0.004972 seconds (11.21 k allocations: 475.281 KiB)
0.007023 seconds (11.07 k allocations: 617.133 KiB)
0.005761 seconds (15.21 k allocations: 618.844 KiB)
0.007362 seconds (14.03 k allocations: 732.977 KiB)
0.004679 seconds (7.43 k allocations: 609.484 KiB)
0.025623 seconds (64.76 k allocations: 3.796 MiB)
0.110450 seconds (248.24 k allocations: 13.620 MiB)
(n, ts) = (3, [0.004726295, 0.006178137, 0.005629141, 0.007078093, 0.004368
237, 0.026223994])
0.007819 seconds (13.55 k allocations: 577.594 KiB)
0.009107 seconds (12.57 k allocations: 714.039 KiB)
0.008233 seconds (16.26 k allocations: 667.219 KiB)
0.010012 seconds (14.58 k allocations: 793.398 KiB)
0.006886 seconds (7.54 k allocations: 737.359 KiB)
0.039299 seconds (80.79 k allocations: 5.176 MiB)
0.161450 seconds (291.40 k allocations: 17.231 MiB)
(n, ts) = (4, [0.007866299, 0.009154472, 0.008465566, 0.009840299, 0.006530
396, 0.037552865])
0.013356 seconds (15.82 k allocations: 673.938 KiB)
0.014501 seconds (14.85 k allocations: 873.758 KiB)
0.011877 seconds (18.01 k allocations: 728.906 KiB)
0.013303 seconds (15.90 k allocations: 900.789 KiB)
0.010408 seconds (7.63 k allocations: 942.953 KiB)
0.055676 seconds (94.58 k allocations: 6.476 MiB)
0.237505 seconds (334.40 k allocations: 21.064 MiB)
(n, ts) = (5, [0.012890114, 0.014559434, 0.011828959, 0.013354741, 0.009964
788, 0.055065725])
0.022396 seconds (19.61 k allocations: 842.422 KiB)
0.022855 seconds (17.33 k allocations: 1.009 MiB)
0.017980 seconds (21.76 k allocations: 869.609 KiB)
0.019229 seconds (18.18 k allocations: 1.051 MiB)
0.015591 seconds (7.83 k allocations: 1.236 MiB)
0.082358 seconds (114.08 k allocations: 7.992 MiB)
0.360883 seconds (398.43 k allocations: 25.985 MiB)
(n, ts) = (6, [0.022620963, 0.022869871, 0.018150276, 0.018892301, 0.015290
761, 0.081875115])
0.034017 seconds (22.45 k allocations: 964.359 KiB)
0.034645 seconds (20.42 k allocations: 1.200 MiB)
0.020978 seconds (22.60 k allocations: 906.984 KiB)
0.022664 seconds (19.18 k allocations: 1.179 MiB)
0.026596 seconds (8.25 k allocations: 1.701 MiB)
0.104956 seconds (134.25 k allocations: 9.484 MiB)
0.490595 seconds (455.14 k allocations: 30.847 MiB)
(n, ts) = (7, [0.034233252, 0.034665091, 0.02128273, 0.022735362, 0.0256957
97, 0.107307314])
0.058325 seconds (28.34 k allocations: 1.193 MiB)
0.052916 seconds (24.20 k allocations: 1.461 MiB)
0.033499 seconds (35.38 k allocations: 1.257 MiB)
0.029757 seconds (23.19 k allocations: 1.527 MiB)
0.035642 seconds (8.07 k allocations: 2.166 MiB)
0.201709 seconds (161.03 k allocations: 12.050 MiB, 26.03% gc time)
0.758505 seconds (561.25 k allocations: 39.373 MiB, 6.92% gc time)
(n, ts) = (8, [0.057031195, 0.053012555, 0.033497076, 0.029668946, 0.035435
957, 0.1370588])
0.092250 seconds (34.60 k allocations: 1.442 MiB)
0.083211 seconds (29.31 k allocations: 1.761 MiB)
0.043834 seconds (36.66 k allocations: 1.325 MiB)
0.037919 seconds (24.50 k allocations: 1.751 MiB)
0.062960 seconds (8.32 k allocations: 2.881 MiB)
0.188368 seconds (182.26 k allocations: 14.536 MiB)
1.017706 seconds (632.14 k allocations: 47.457 MiB)
(n, ts) = (9, [0.092080473, 0.083297899, 0.043524255, 0.038264382, 0.063091
293, 0.188000405])
0.142457 seconds (39.46 k allocations: 1.668 MiB)
0.128463 seconds (32.80 k allocations: 1.974 MiB)
0.053363 seconds (32.61 k allocations: 1.296 MiB)
0.047055 seconds (24.70 k allocations: 1.831 MiB)
0.085461 seconds (8.44 k allocations: 3.785 MiB)
0.309035 seconds (214.93 k allocations: 17.122 MiB, 20.14% gc time)
1.463966 seconds (706.70 k allocations: 55.415 MiB, 4.25% gc time)
(n, ts) = (10, [0.141700506, 0.12747261, 0.053191495, 0.047351215, 0.085382
168, 0.242084426])
0.326411 seconds (53.48 k allocations: 2.260 MiB)
0.283427 seconds (43.99 k allocations: 2.706 MiB)
0.108462 seconds (50.52 k allocations: 1.957 MiB)
0.103280 seconds (38.95 k allocations: 2.892 MiB)
0.236948 seconds (9.66 k allocations: 6.391 MiB)
0.421256 seconds (265.69 k allocations: 22.871 MiB, 8.93% gc time)
2.921088 seconds (925.42 k allocations: 78.218 MiB, 1.29% gc time)
(n, ts) = (12, [0.328917835, 0.285966748, 0.108171159, 0.105191944, 0.23379
9438, 0.377995432])
1.054349 seconds (78.77 k allocations: 3.339 MiB)
0.889923 seconds (63.86 k allocations: 4.027 MiB)
0.192349 seconds (61.42 k allocations: 2.427 MiB)
0.185581 seconds (48.43 k allocations: 4.218 MiB)
0.678442 seconds (10.67 k allocations: 13.194 MiB)
0.707350 seconds (385.68 k allocations: 36.270 MiB, 5.30% gc time)
7.428039 seconds (1.30 M allocations: 127.019 MiB, 1.19% gc time)
(n, ts) = (15, [1.036059742, 0.8993121, 0.190723311, 0.186467903, 0.7371385
2, 0.668385507])
1.907196 seconds (98.57 k allocations: 4.185 MiB)
1.642142 seconds (79.31 k allocations: 4.983 MiB)
0.248583 seconds (61.78 k allocations: 2.495 MiB)
0.230912 seconds (48.09 k allocations: 4.928 MiB)
1.238176 seconds (11.41 k allocations: 20.257 MiB)
0.979468 seconds (446.16 k allocations: 46.480 MiB, 3.61% gc time)
12.596462 seconds (1.49 M allocations: 166.723 MiB, 1.39% gc time)
(n, ts) = (17, [1.897472356, 1.718322554, 0.247664066, 0.232154447, 1.28148
4197, 0.970747668])
12-element Vector{Vector{Float64}}:
[0.002149589, 0.002593146, 0.002285169, 0.003530381, 0.002218059, 0.009112
583]
[0.004726295, 0.006178137, 0.005629141, 0.007078093, 0.004368237, 0.026223
994]
[0.007866299, 0.009154472, 0.008465566, 0.009840299, 0.006530396, 0.037552
865]
[0.012890114, 0.014559434, 0.011828959, 0.013354741, 0.009964788, 0.055065
725]
[0.022620963, 0.022869871, 0.018150276, 0.018892301, 0.015290761, 0.081875
115]
[0.034233252, 0.034665091, 0.02128273, 0.022735362, 0.025695797, 0.1073073
14]
[0.057031195, 0.053012555, 0.033497076, 0.029668946, 0.035435957, 0.137058
8]
[0.092080473, 0.083297899, 0.043524255, 0.038264382, 0.063091293, 0.188000
405]
[0.141700506, 0.12747261, 0.053191495, 0.047351215, 0.085382168, 0.2420844
26]
[0.328917835, 0.285966748, 0.108171159, 0.105191944, 0.233799438, 0.377995
432]
[1.036059742, 0.8993121, 0.190723311, 0.186467903, 0.73713852, 0.668385507
]
[1.897472356, 1.718322554, 0.247664066, 0.232154447, 1.281484197, 0.970747
668]csadata_same_solver = [[csa_same_solver[j][i] for j in eachindex(csa_same_solver)]
for i in eachindex(csa_same_solver[1])]
plt_same = plot(title = "Brusselator same-solver adjoint comparison (EnzymeVJP)");
for (i, (name, _, _, _)) in enumerate(same_solver_configs)
plot!(plt_same, n_to_param.(csan), csadata_same_solver[i], lab = name,
lw = lw, marksize = ms, linestyle = :auto, marker = :auto)
end
xaxis!(plt_same, "Number of Parameters", :log10);
yaxis!(plt_same, "Runtime (s)", :log10);
plot!(plt_same, legend = :outertopleft, size = (1200, 600))
Gradient accuracy scaling
The Brusselator's stiffness grows with the grid refinement (the diffusion rate scales as 1/dx^2 ∝ N^2), so sweeping the size also sweeps the stiffness. This measures the gradient accuracy of each adjoint configuration at the benchmark tolerances against a tight (abstol = reltol = 1e-12) reference. The reference is cross-validated at the smallest size by computing it with two completely independent implementations (the CVODES C adjoint and the native GaussAdjoint), which agree to ~1e-10 at every size.
accuracy_configs = vcat(
same_solver_configs,
[("Rodas5 + GaussAdjoint", Rodas5(autodiff = AutoFiniteDiff()), gauss_enz, false)],
)
relerr(a, b) = norm(vec(a) .- vec(b)) / norm(vec(b))
csa_accuracy = map(csan) do n
bfun, b_u0, b_p, brusselator_jac, brusselator_comp = makebrusselator!(PROBS, n)
prob = ODEProblem{true, SciMLBase.FullSpecialize}(bfun, b_u0, tspan, b_p)
dg(out, u, p, t, i) = (out .= u .- 1.0)
ref_sol = solve(prob, CVODE_BDF(linear_solver = :GMRES),
abstol = 1e-12, reltol = 1e-12, dense = true)
du0_ref, dp_ref = adjoint_sensitivities(
ref_sol, CVODE_BDF(linear_solver = :GMRES); t = collect(bt),
dgdu_discrete = dg, abstol = 1e-12, reltol = 1e-12, sensealg = gauss_enz)
ref_sol2 = solve(prob, CVODE_BDF(linear_solver = :GMRES),
abstol = 1e-12, reltol = 1e-12, saveat = bt)
du0_ref2, dp_ref2 = adjoint_sensitivities(
ref_sol2, CVODE_BDF(linear_solver = :GMRES); t = collect(bt),
dgdu_discrete = dg, abstol = 1e-12, reltol = 1e-12,
sensealg = SundialsAdjoint(autojacvec = EnzymeVJP()))
agreement = max(relerr(dp_ref2, dp_ref), relerr(du0_ref2, du0_ref))
errs = map(accuracy_configs) do (name, alg, sensealg, dense_forward)
du0, dp = run_same_solver(bfun, b_u0, b_p, alg, sensealg, dense_forward)
relerr(dp, dp_ref)
end
@show n, agreement, errs
errs
end(n, agreement, errs) = (2, 2.1963974702304678e-11, [1.1360036542730342e-5,
1.8833104332738728e-5, 1.0803476551076878e-5, 1.8894905218340053e-5, 2.3286
287656629697e-5, 2.3878725506819976e-5, 3.0394201713227987e-9])
(n, agreement, errs) = (3, 3.010588434012082e-10, [0.00028737837337994515,
2.80086857116937e-5, 0.00023912428713242524, 3.714568563653246e-5, 0.001505
4829082979052, 0.0014766321747167997, 4.117603080409098e-5])
(n, agreement, errs) = (4, 1.947292936038399e-10, [0.0001687926610456094, 1
.874971527376774e-5, 0.00015883934120616695, 3.691915020869242e-5, 0.001018
0981813033077, 0.0010587913364942991, 1.581329703357047e-5])
(n, agreement, errs) = (5, 1.1458770290019084e-10, [0.00012861447619119944,
2.1358797849169975e-5, 0.000139683208049175, 1.55745766292639e-5, 0.000778
2160218531484, 0.0007979527317029229, 1.132856087354418e-5])
(n, agreement, errs) = (6, 1.1084599668766109e-10, [0.00010956844524867612,
2.0269472136113163e-5, 0.00010943230107359006, 2.3143298643346274e-5, 0.00
06166101072026013, 0.0006496243246276735, 8.017922704767997e-6])
(n, agreement, errs) = (7, 9.654055320116652e-11, [9.417453384377807e-5, 1.
6873080971675968e-5, 9.234561714067285e-5, 1.3570524356883525e-5, 0.0005370
742416252596, 0.0005447209207241839, 5.06464820260714e-6])
(n, agreement, errs) = (8, 9.794632556352656e-11, [8.831634072154492e-5, 1.
6851933957859684e-5, 8.557186511651593e-5, 9.322418419792044e-6, 0.00055794
43464255073, 0.0005556296118236512, 3.5306893395213555e-6])
(n, agreement, errs) = (9, 8.892003424263953e-11, [8.565053270513868e-5, 1.
462096461377351e-5, 8.083127239464406e-5, 4.795229093527008e-6, 0.000464146
1425833273, 0.0004611103756681643, 2.9710570433722095e-6])
(n, agreement, errs) = (10, 9.066173619606881e-11, [7.682476620291043e-5, 1
.8437614241217117e-5, 8.89644356071134e-5, 3.104678797113946e-5, 0.00047807
01676914919, 0.0004783628725488691, 2.794065437343273e-6])
(n, agreement, errs) = (12, 7.67140688387645e-11, [7.411141828954412e-5, 2.
2215058788081397e-5, 6.526502659179764e-5, 1.2430558885657634e-5, 0.0004238
766249800418, 0.00042744908731910143, 2.730558096856138e-6])
(n, agreement, errs) = (15, 6.721550805195372e-11, [7.380143188028393e-5, 9
.487618540479155e-6, 9.757134118760027e-5, 4.297504300599217e-5, 0.00044138
559604753525, 0.0004439295999529228, 2.9139178684502948e-6])
(n, agreement, errs) = (17, 7.560204822537469e-11, [6.830653284894397e-5, 1
.0787115927338272e-5, 0.0001115941045792728, 7.272612520132917e-5, 0.000414
3204376360607, 0.00041593354967809336, 3.89049367453392e-6])
12-element Vector{Vector{Float64}}:
[1.1360036542730342e-5, 1.8833104332738728e-5, 1.0803476551076878e-5, 1.88
94905218340053e-5, 2.3286287656629697e-5, 2.3878725506819976e-5, 3.03942017
13227987e-9]
[0.00028737837337994515, 2.80086857116937e-5, 0.00023912428713242524, 3.71
4568563653246e-5, 0.0015054829082979052, 0.0014766321747167997, 4.117603080
409098e-5]
[0.0001687926610456094, 1.874971527376774e-5, 0.00015883934120616695, 3.69
1915020869242e-5, 0.0010180981813033077, 0.0010587913364942991, 1.581329703
357047e-5]
[0.00012861447619119944, 2.1358797849169975e-5, 0.000139683208049175, 1.55
745766292639e-5, 0.0007782160218531484, 0.0007979527317029229, 1.1328560873
54418e-5]
[0.00010956844524867612, 2.0269472136113163e-5, 0.00010943230107359006, 2.
3143298643346274e-5, 0.0006166101072026013, 0.0006496243246276735, 8.017922
704767997e-6]
[9.417453384377807e-5, 1.6873080971675968e-5, 9.234561714067285e-5, 1.3570
524356883525e-5, 0.0005370742416252596, 0.0005447209207241839, 5.0646482026
0714e-6]
[8.831634072154492e-5, 1.6851933957859684e-5, 8.557186511651593e-5, 9.3224
18419792044e-6, 0.0005579443464255073, 0.0005556296118236512, 3.53068933952
13555e-6]
[8.565053270513868e-5, 1.462096461377351e-5, 8.083127239464406e-5, 4.79522
9093527008e-6, 0.0004641461425833273, 0.0004611103756681643, 2.971057043372
2095e-6]
[7.682476620291043e-5, 1.8437614241217117e-5, 8.89644356071134e-5, 3.10467
8797113946e-5, 0.0004780701676914919, 0.0004783628725488691, 2.794065437343
273e-6]
[7.411141828954412e-5, 2.2215058788081397e-5, 6.526502659179764e-5, 1.2430
558885657634e-5, 0.0004238766249800418, 0.00042744908731910143, 2.730558096
856138e-6]
[7.380143188028393e-5, 9.487618540479155e-6, 9.757134118760027e-5, 4.29750
4300599217e-5, 0.00044138559604753525, 0.0004439295999529228, 2.91391786845
02948e-6]
[6.830653284894397e-5, 1.0787115927338272e-5, 0.0001115941045792728, 7.272
612520132917e-5, 0.0004143204376360607, 0.00041593354967809336, 3.890493674
53392e-6]csadata_accuracy = [[csa_accuracy[j][i] for j in eachindex(csa_accuracy)]
for i in eachindex(csa_accuracy[1])]
plt_acc = plot(title = "Brusselator adjoint gradient accuracy scaling");
for (i, (name, _, _, _)) in enumerate(accuracy_configs)
plot!(plt_acc, n_to_param.(csan), csadata_accuracy[i], lab = name,
lw = lw, marksize = ms, linestyle = :auto, marker = :auto)
end
xaxis!(plt_acc, "Number of Parameters", :log10);
yaxis!(plt_acc, "Relative L2 error of dG/dp", :log10);
plot!(plt_acc, legend = :outertopleft, size = (1200, 600))
Peak Memory Benchmarks
Measures the memory consumed by each sensitivity computation. Each configuration runs in a separate subprocess. We measure current RSS (from /proc/self/statm) before and after the computation to isolate the memory used by the sensitivity solve from the large fixed cost of package loading.
const CHILD_PREAMBLE = raw"""
using OrdinaryDiffEq, OrdinaryDiffEqRosenbrock, ReverseDiff, ForwardDiff, FiniteDiff,
SciMLSensitivity
using LinearAlgebra, Mooncake
function get_rss_mib()
statm = read("/proc/self/statm", String)
resident_pages = parse(Int, split(statm)[2])
return resident_pages * 4096 / (1024^2)
end
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)
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
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
end
Base.vec(v::Adjoint{<:Real, <:AbstractVector}) = vec(v')
bt = 0:0.1:1
tspan = (0.0, 1.0)
tols = (abstol = 1e-5, reltol = 1e-7)
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)
end
"""
const PROJECT_DIR = @__DIR__
function run_memory_benchmark(n::Int, method_setup::String)
child_script = CHILD_PREAMBLE * """
n = $(n)
bfun, b_u0, b_p, brusselator_jac = makebrusselator(n)
GC.gc(); GC.gc()
rss_before = get_rss_mib()
""" * method_setup * """
GC.gc(); GC.gc()
rss_after = get_rss_mib()
println("BRUSSMEM_TIMING:", t)
println("BRUSSMEM_RSS_BEFORE:", rss_before)
println("BRUSSMEM_RSS_AFTER:", rss_after)
"""
try
output = read(
`$(Base.julia_cmd()) --project=$(PROJECT_DIR) -e $(child_script)`, String)
time_m = match(r"BRUSSMEM_TIMING:([\d.eE+-]+)", output)
before_m = match(r"BRUSSMEM_RSS_BEFORE:([\d.eE+-]+)", output)
after_m = match(r"BRUSSMEM_RSS_AFTER:([\d.eE+-]+)", output)
if time_m === nothing || before_m === nothing || after_m === nothing
@warn "Failed to parse subprocess output" n output
return (; rss_before = NaN, rss_after = NaN, delta_mib = NaN, timing = NaN)
end
timing = parse(Float64, time_m.captures[1])
rss_before = parse(Float64, before_m.captures[1])
rss_after = parse(Float64, after_m.captures[1])
delta_mib = rss_after - rss_before
return (; rss_before, rss_after, delta_mib, timing)
catch e
@warn "Subprocess failed" n exception = (e, catch_backtrace())
return (; rss_before = NaN, rss_after = NaN, delta_mib = NaN, timing = NaN)
end
end
mem_sizes = [2, 4, 6, 8, 10, 12]6-element Vector{Int64}:
2
4
6
8
10
12forwarddiff_mem = map(mem_sizes) do n
result = run_memory_benchmark(n, """
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, result
result
end(n, result) = (2, (rss_before = 1053.9375, rss_after = 1204.18359375, delta
_mib = 150.24609375, timing = 0.003111414))
(n, result) = (4, (rss_before = 1052.5859375, rss_after = 1226.33203125, de
lta_mib = 173.74609375, timing = 0.039805451))
(n, result) = (6, (rss_before = 1057.484375, rss_after = 1229.24609375, del
ta_mib = 171.76171875, timing = 0.448656414))
(n, result) = (8, (rss_before = 1073.2421875, rss_after = 1231.90625, delta
_mib = 158.6640625, timing = 2.326156929))
(n, result) = (10, (rss_before = 1056.43359375, rss_after = 1233.49609375,
delta_mib = 177.0625, timing = 11.260277147))
(n, result) = (12, (rss_before = 1046.56640625, rss_after = 1238.30078125,
delta_mib = 191.734375, timing = 31.386858693))
6-element Vector{@NamedTuple{rss_before::Float64, rss_after::Float64, delta
_mib::Float64, timing::Float64}}:
(rss_before = 1053.9375, rss_after = 1204.18359375, delta_mib = 150.246093
75, timing = 0.003111414)
(rss_before = 1052.5859375, rss_after = 1226.33203125, delta_mib = 173.746
09375, timing = 0.039805451)
(rss_before = 1057.484375, rss_after = 1229.24609375, delta_mib = 171.7617
1875, timing = 0.448656414)
(rss_before = 1073.2421875, rss_after = 1231.90625, delta_mib = 158.664062
5, timing = 2.326156929)
(rss_before = 1056.43359375, rss_after = 1233.49609375, delta_mib = 177.06
25, timing = 11.260277147)
(rss_before = 1046.56640625, rss_after = 1238.30078125, delta_mib = 191.73
4375, timing = 31.386858693)numdiff_mem = map(mem_sizes) do n
result = run_memory_benchmark(n, """
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, result
result
end(n, result) = (2, (rss_before = 1042.02734375, rss_after = 1147.80859375, d
elta_mib = 105.78125, timing = 0.004071329))
(n, result) = (4, (rss_before = 1043.4140625, rss_after = 1155.3515625, del
ta_mib = 111.9375, timing = 0.099820814))
(n, result) = (6, (rss_before = 1062.9765625, rss_after = 1165.4765625, del
ta_mib = 102.5, timing = 0.899974812))
(n, result) = (8, (rss_before = 1074.765625, rss_after = 1186.1171875, delt
a_mib = 111.3515625, timing = 4.436777599))
(n, result) = (10, (rss_before = 1057.4609375, rss_after = 1164.49609375, d
elta_mib = 107.03515625, timing = 18.467509641))
(n, result) = (12, (rss_before = 1057.69140625, rss_after = 1178.18359375,
delta_mib = 120.4921875, timing = 88.274741172))
6-element Vector{@NamedTuple{rss_before::Float64, rss_after::Float64, delta
_mib::Float64, timing::Float64}}:
(rss_before = 1042.02734375, rss_after = 1147.80859375, delta_mib = 105.78
125, timing = 0.004071329)
(rss_before = 1043.4140625, rss_after = 1155.3515625, delta_mib = 111.9375
, timing = 0.099820814)
(rss_before = 1062.9765625, rss_after = 1165.4765625, delta_mib = 102.5, t
iming = 0.899974812)
(rss_before = 1074.765625, rss_after = 1186.1171875, delta_mib = 111.35156
25, timing = 4.436777599)
(rss_before = 1057.4609375, rss_after = 1164.49609375, delta_mib = 107.035
15625, timing = 18.467509641)
(rss_before = 1057.69140625, rss_after = 1178.18359375, delta_mib = 120.49
21875, timing = 88.274741172)adjoint_ad_configs = [
("Interp user-Jacobian",
"InterpolatingAdjoint(autodiff = false, autojacvec = false)", true),
("Interp AD-Jacobian",
"InterpolatingAdjoint(autodiff = true, autojacvec = false)", false),
("Quad user-Jacobian",
"QuadratureAdjoint(autodiff = false, autojacvec = false)", true),
("Quad AD-Jacobian",
"QuadratureAdjoint(autodiff = true, autojacvec = false)", false),
("Gauss AD-Jacobian",
"GaussAdjoint(autodiff = true, autojacvec = false)", false),
("GaussKronrod AD-Jacobian",
"GaussKronrodAdjoint(autodiff = true, autojacvec = false)", false),
]
adjoint_ad_mem = map(adjoint_ad_configs) do (name, sensalg_str, needs_jac)
results = map(mem_sizes) do n
f_expr = needs_jac ? "ODEFunction(bfun, jac = brusselator_jac)" : "bfun"
result = run_memory_benchmark(n, """
sensalg = $(sensalg_str)
f = $(f_expr)
solver = Rodas5(autodiff = AutoFiniteDiff())
diffeq_sen_l2(f, b_u0, tspan, b_p, bt, solver; sensalg = sensalg, tols...)
t = @elapsed diffeq_sen_l2(f, b_u0, tspan, b_p, bt, solver;
sensalg = sensalg, tols...)
""")
@show name, n, result
result
end
(name = name, results = results)
end(name, n, result) = ("Interp user-Jacobian", 2, (rss_before = 1054.25, rss_
after = 1269.0, delta_mib = 214.75, timing = 0.006790763))
(name, n, result) = ("Interp user-Jacobian", 4, (rss_before = 1058.74609375
, rss_after = 1240.6484375, delta_mib = 181.90234375, timing = 0.109615458)
)
(name, n, result) = ("Interp user-Jacobian", 6, (rss_before = 1062.1796875,
rss_after = 1243.41796875, delta_mib = 181.23828125, timing = 0.992203019)
)
(name, n, result) = ("Interp user-Jacobian", 8, (rss_before = 1063.8515625,
rss_after = 1245.82421875, delta_mib = 181.97265625, timing = 5.488170208)
)
(name, n, result) = ("Interp user-Jacobian", 10, (rss_before = 1049.984375,
rss_after = 1260.26953125, delta_mib = 210.28515625, timing = 21.858526666
))
(name, n, result) = ("Interp user-Jacobian", 12, (rss_before = 1058.2226562
5, rss_after = 1295.46484375, delta_mib = 237.2421875, timing = 64.91960673
7))
(name, n, result) = ("Interp AD-Jacobian", 2, (rss_before = 1061.01953125,
rss_after = 1228.64453125, delta_mib = 167.625, timing = 0.003785809))
(name, n, result) = ("Interp AD-Jacobian", 4, (rss_before = 1061.0625, rss_
after = 1277.60546875, delta_mib = 216.54296875, timing = 0.055551443))
(name, n, result) = ("Interp AD-Jacobian", 6, (rss_before = 1052.67578125,
rss_after = 1268.65625, delta_mib = 215.98046875, timing = 0.402205139))
(name, n, result) = ("Interp AD-Jacobian", 8, (rss_before = 1052.28515625,
rss_after = 1261.29296875, delta_mib = 209.0078125, timing = 2.255278622))
(name, n, result) = ("Interp AD-Jacobian", 10, (rss_before = 1053.1328125,
rss_after = 1263.99609375, delta_mib = 210.86328125, timing = 9.046674812))
(name, n, result) = ("Interp AD-Jacobian", 12, (rss_before = 1057.3046875,
rss_after = 1289.20703125, delta_mib = 231.90234375, timing = 29.726450905)
)
(name, n, result) = ("Quad user-Jacobian", 2, (rss_before = 1057.21875, rss
_after = 1305.8671875, delta_mib = 248.6484375, timing = 0.002723985))
(name, n, result) = ("Quad user-Jacobian", 4, (rss_before = 1060.97265625,
rss_after = 1299.16015625, delta_mib = 238.1875, timing = 0.010001066))
(name, n, result) = ("Quad user-Jacobian", 6, (rss_before = 1069.65625, rss
_after = 1332.31640625, delta_mib = 262.66015625, timing = 0.040655362))
(name, n, result) = ("Quad user-Jacobian", 8, (rss_before = 1056.4296875, r
ss_after = 1299.27734375, delta_mib = 242.84765625, timing = 0.128827367))
(name, n, result) = ("Quad user-Jacobian", 10, (rss_before = 1057.3828125,
rss_after = 1301.59375, delta_mib = 244.2109375, timing = 0.342561216))
(name, n, result) = ("Quad user-Jacobian", 12, (rss_before = 1058.96484375,
rss_after = 1329.375, delta_mib = 270.41015625, timing = 0.877061737))
(name, n, result) = ("Quad AD-Jacobian", 2, (rss_before = 1072.125, rss_aft
er = 1323.4375, delta_mib = 251.3125, timing = 0.002383447))
(name, n, result) = ("Quad AD-Jacobian", 4, (rss_before = 1046.640625, rss_
after = 1328.3671875, delta_mib = 281.7265625, timing = 0.011314111))
(name, n, result) = ("Quad AD-Jacobian", 6, (rss_before = 1066.20703125, rs
s_after = 1321.20703125, delta_mib = 255.0, timing = 0.076300865))
(name, n, result) = ("Quad AD-Jacobian", 8, (rss_before = 1057.453125, rss_
after = 1310.921875, delta_mib = 253.46875, timing = 0.345494046))
(name, n, result) = ("Quad AD-Jacobian", 10, (rss_before = 1059.203125, rss
_after = 1318.80859375, delta_mib = 259.60546875, timing = 1.270096463))
(name, n, result) = ("Quad AD-Jacobian", 12, (rss_before = 1061.97265625, r
ss_after = 1329.015625, delta_mib = 267.04296875, timing = 3.939367725))
(name, n, result) = ("Gauss AD-Jacobian", 2, (rss_before = 1062.859375, rss
_after = 1271.6640625, delta_mib = 208.8046875, timing = 0.002463167))
(name, n, result) = ("Gauss AD-Jacobian", 4, (rss_before = 1055.6953125, rs
s_after = 1272.1953125, delta_mib = 216.5, timing = 0.012260795))
(name, n, result) = ("Gauss AD-Jacobian", 6, (rss_before = 1053.86328125, r
ss_after = 1251.92578125, delta_mib = 198.0625, timing = 0.079090919))
(name, n, result) = ("Gauss AD-Jacobian", 8, (rss_before = 1055.5390625, rs
s_after = 1264.19921875, delta_mib = 208.66015625, timing = 0.496952761))
(name, n, result) = ("Gauss AD-Jacobian", 10, (rss_before = 1052.53125, rss
_after = 1261.86328125, delta_mib = 209.33203125, timing = 1.400832268))
(name, n, result) = ("Gauss AD-Jacobian", 12, (rss_before = 1041.90234375,
rss_after = 1294.0078125, delta_mib = 252.10546875, timing = 4.687952525))
(name, n, result) = ("GaussKronrod AD-Jacobian", 2, (rss_before = 1056.3359
375, rss_after = 1271.94140625, delta_mib = 215.60546875, timing = 0.003018
444))
(name, n, result) = ("GaussKronrod AD-Jacobian", 4, (rss_before = 1053.2226
5625, rss_after = 1273.80859375, delta_mib = 220.5859375, timing = 0.017698
026))
(name, n, result) = ("GaussKronrod AD-Jacobian", 6, (rss_before = 1061.5820
3125, rss_after = 1254.75390625, delta_mib = 193.171875, timing = 0.1139098
01))
(name, n, result) = ("GaussKronrod AD-Jacobian", 8, (rss_before = 1054.5156
25, rss_after = 1269.67578125, delta_mib = 215.16015625, timing = 0.6437434
75))
(name, n, result) = ("GaussKronrod AD-Jacobian", 10, (rss_before = 1057.890
625, rss_after = 1288.76953125, delta_mib = 230.87890625, timing = 1.851769
374))
(name, n, result) = ("GaussKronrod AD-Jacobian", 12, (rss_before = 1074.300
78125, rss_after = 1286.1328125, delta_mib = 211.83203125, timing = 5.89324
909))
6-element Vector{@NamedTuple{name::String, results::Vector{@NamedTuple{rss_
before::Float64, rss_after::Float64, delta_mib::Float64, timing::Float64}}}
}:
(name = "Interp user-Jacobian", results = [(rss_before = 1054.25, rss_afte
r = 1269.0, delta_mib = 214.75, timing = 0.006790763), (rss_before = 1058.7
4609375, rss_after = 1240.6484375, delta_mib = 181.90234375, timing = 0.109
615458), (rss_before = 1062.1796875, rss_after = 1243.41796875, delta_mib =
181.23828125, timing = 0.992203019), (rss_before = 1063.8515625, rss_after
= 1245.82421875, delta_mib = 181.97265625, timing = 5.488170208), (rss_bef
ore = 1049.984375, rss_after = 1260.26953125, delta_mib = 210.28515625, tim
ing = 21.858526666), (rss_before = 1058.22265625, rss_after = 1295.46484375
, delta_mib = 237.2421875, timing = 64.919606737)])
(name = "Interp AD-Jacobian", results = [(rss_before = 1061.01953125, rss_
after = 1228.64453125, delta_mib = 167.625, timing = 0.003785809), (rss_bef
ore = 1061.0625, rss_after = 1277.60546875, delta_mib = 216.54296875, timin
g = 0.055551443), (rss_before = 1052.67578125, rss_after = 1268.65625, delt
a_mib = 215.98046875, timing = 0.402205139), (rss_before = 1052.28515625, r
ss_after = 1261.29296875, delta_mib = 209.0078125, timing = 2.255278622), (
rss_before = 1053.1328125, rss_after = 1263.99609375, delta_mib = 210.86328
125, timing = 9.046674812), (rss_before = 1057.3046875, rss_after = 1289.20
703125, delta_mib = 231.90234375, timing = 29.726450905)])
(name = "Quad user-Jacobian", results = [(rss_before = 1057.21875, rss_aft
er = 1305.8671875, delta_mib = 248.6484375, timing = 0.002723985), (rss_bef
ore = 1060.97265625, rss_after = 1299.16015625, delta_mib = 238.1875, timin
g = 0.010001066), (rss_before = 1069.65625, rss_after = 1332.31640625, delt
a_mib = 262.66015625, timing = 0.040655362), (rss_before = 1056.4296875, rs
s_after = 1299.27734375, delta_mib = 242.84765625, timing = 0.128827367), (
rss_before = 1057.3828125, rss_after = 1301.59375, delta_mib = 244.2109375,
timing = 0.342561216), (rss_before = 1058.96484375, rss_after = 1329.375,
delta_mib = 270.41015625, timing = 0.877061737)])
(name = "Quad AD-Jacobian", results = [(rss_before = 1072.125, rss_after =
1323.4375, delta_mib = 251.3125, timing = 0.002383447), (rss_before = 1046
.640625, rss_after = 1328.3671875, delta_mib = 281.7265625, timing = 0.0113
14111), (rss_before = 1066.20703125, rss_after = 1321.20703125, delta_mib =
255.0, timing = 0.076300865), (rss_before = 1057.453125, rss_after = 1310.
921875, delta_mib = 253.46875, timing = 0.345494046), (rss_before = 1059.20
3125, rss_after = 1318.80859375, delta_mib = 259.60546875, timing = 1.27009
6463), (rss_before = 1061.97265625, rss_after = 1329.015625, delta_mib = 26
7.04296875, timing = 3.939367725)])
(name = "Gauss AD-Jacobian", results = [(rss_before = 1062.859375, rss_aft
er = 1271.6640625, delta_mib = 208.8046875, timing = 0.002463167), (rss_bef
ore = 1055.6953125, rss_after = 1272.1953125, delta_mib = 216.5, timing = 0
.012260795), (rss_before = 1053.86328125, rss_after = 1251.92578125, delta_
mib = 198.0625, timing = 0.079090919), (rss_before = 1055.5390625, rss_afte
r = 1264.19921875, delta_mib = 208.66015625, timing = 0.496952761), (rss_be
fore = 1052.53125, rss_after = 1261.86328125, delta_mib = 209.33203125, tim
ing = 1.400832268), (rss_before = 1041.90234375, rss_after = 1294.0078125,
delta_mib = 252.10546875, timing = 4.687952525)])
(name = "GaussKronrod AD-Jacobian", results = [(rss_before = 1056.3359375,
rss_after = 1271.94140625, delta_mib = 215.60546875, timing = 0.003018444)
, (rss_before = 1053.22265625, rss_after = 1273.80859375, delta_mib = 220.5
859375, timing = 0.017698026), (rss_before = 1061.58203125, rss_after = 125
4.75390625, delta_mib = 193.171875, timing = 0.113909801), (rss_before = 10
54.515625, rss_after = 1269.67578125, delta_mib = 215.16015625, timing = 0.
643743475), (rss_before = 1057.890625, rss_after = 1288.76953125, delta_mib
= 230.87890625, timing = 1.851769374), (rss_before = 1074.30078125, rss_af
ter = 1286.1328125, delta_mib = 211.83203125, timing = 5.89324909)])mem_params = n_to_param.(mem_sizes)
plt_mem1 = plot(title = "Brusselator Sensitivity Memory Scaling");
plot!(plt_mem1, mem_params, [r.delta_mib for r in forwarddiff_mem],
lab = "Forward-Mode DSAAD", lw = lw, marksize = ms,
linestyle = :auto, marker = :auto);
plot!(plt_mem1, mem_params, [r.delta_mib for r in numdiff_mem],
lab = "Numerical Differentiation", lw = lw, marksize = ms,
linestyle = :auto, marker = :auto);
for entry in adjoint_ad_mem
plot!(plt_mem1, mem_params, [r.delta_mib for r in entry.results],
lab = entry.name, lw = lw, marksize = ms,
linestyle = :auto, marker = :auto)
end
xaxis!(plt_mem1, "Number of Parameters", :log10);
yaxis!(plt_mem1, "Memory (MiB)");
plot!(plt_mem1, 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 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 UTC)
Build Info:
Official https://julialang.org release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LLVM: libLLVM-18.1.7 (ORCJIT, znver2)
GC: Built with stock GC
Threads: 128 default, 1 interactive, 128 GC (on 128 virtual cores)
Environment:
JULIA_NUM_THREADS = auto
Package Information:
Status `~/github-runners/amdci3-1/_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 v0.13.203
[6a86dc24] FiniteDiff v2.33.0
[f6369f11] ForwardDiff v1.4.6
⌃ [7ed4a6bd] LinearSolve v5.17.3
⌃ [da2b9cff] Mooncake v0.5.56
[1dea7af3] OrdinaryDiffEq v7.8.1
[43230ef6] OrdinaryDiffEqRosenbrock v2.7.3
[65888b18] ParameterizedFunctions v5.27.0
[91a5bcdd] Plots v1.41.7
[08abe8d2] PrettyTables v3.4.8
[37e2e3b7] ReverseDiff v1.17.0
[31c91b34] SciMLBenchmarks v0.2.1 [loaded: `/home/crackauc/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/src/SciMLBenchmarks.jl` (v0.2.1) expected `/home/crackauc/.julia/packages/SciMLBenchmarks/ceJyd/src/SciMLBenchmarks.jl` (v0.2.1)]
⌃ [1ed8b502] SciMLSensitivity v7.119.7
[90137ffa] StaticArrays v1.9.20
[c3572dad] Sundials v6.7.1
[9f7883ad] Tracker v0.2.39
[e88e6eb3] Zygote v0.7.13
[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/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/AutomaticDifferentiation/Manifest.toml`
[47edcb42] ADTypes v1.24.0
[14f7f29c] AMD v0.5.4
[621f4979] AbstractFFTs v1.5.0
[6e696c72] AbstractPlutoDingetjes v1.4.1
[1520ce14] AbstractTrees v0.4.5
[7d9f7c33] Accessors v0.1.45
[79e6a3ab] Adapt v4.7.0
[66dad0bd] AliasTables v1.1.3
[9b6a8646] AllocCheck v0.2.6
[ec485272] ArnoldiMethod v0.4.0
⌃ [4fba245c] ArrayInterface v7.30.1
[4c555306] ArrayLayouts v1.12.2
[a9b6321e] Atomix v1.2.1
[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 v0.1.14
[8e7c35d0] BlockArrays v1.10.0
[70df07ce] BracketingNonlinearSolve v1.12.7
[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 v1.2.2
[34da2185] Compat v4.18.1
[b152e2b5] CompositeTypes v0.1.4
[a33af91c] CompositionsBase v0.1.2
[2569d6c7] ConcreteStructs v0.2.8
[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 v7.21.1
[459566f4] DiffEqCallbacks v4.19.4
[77a26b50] DiffEqNoiseProcess v5.36.3
[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
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⌅ [61eb1bfa] GPUCompiler v1.23.0
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⌅ [eafb193a] Highlights v0.5.3
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[431bcebd] SciMLPublic v1.3.0
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⌅ [e9f186c6] Libffi_jll v3.4.7+0
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[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
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[b27032c2] LibCURL v0.6.4
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[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.12.0
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[e66e0078] CompilerSupportLibraries_jll v1.3.1+2
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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`