Thermal Fluid Symbolic Jacobian Scaling
This is a 1D advection-diffusion-source PDE that uses a second order upwind scheme.
Jacobian construction uses the current Symbolics.sparsejacobian implementation, with derivative caches cleared before each sample. CSE is a build_function code-generation option, so only the code-generation and generated-function measurements compare CSE off and on.
using Pkg
# Rev fixes precompilation https://github.com/hzgzh/XSteam.jl/pull/2
Pkg.add(Pkg.PackageSpec(;name="XSteam", rev="f2a1c589054cfd6bba307985a3a534b6f5a1863b"))
using ModelingToolkit, Symbolics, XSteam, Polynomials, CairoMakie, PrettyTables
using SparseArrays, Chairmarks, Statistics
using ModelingToolkit: t_nounits as t, D_nounits as D
using SymbolicIndexingInterface: default_valuesSetup Julia Code
# o o o o o o o < heat capacitors
# | | | | | | | < heat conductors
# o o o o o o o
# | | | | | | |
#Source -> o--o--o--o--o--o--o -> Sink
# advection diff source PDE
m_flow_source(t) = 2.75
T_source(t) = (t > 12 * 3600) * 56.0 + 12.0
# @register_symbolic m_flow_source(t)
# @register_symbolic T_source(t)
#build polynomial liquid-water property only dependent on Temperature
p_l = 5 #bar
T_vec = collect(1:1:150);
@generated kin_visc_T(t) = :(Base.evalpoly(t, $(fit(T_vec, my_pT.(p_l, T_vec) ./ rho_pT.(p_l, T_vec), 5).coeffs...,)))
@generated lambda_T(t) = :(Base.evalpoly(t, $(fit(T_vec, tc_pT.(p_l, T_vec), 3).coeffs...,)))
@generated Pr_T(t) = :(Base.evalpoly(t, $(fit(T_vec, 1e3 * Cp_pT.(p_l, T_vec) .* my_pT.(p_l, T_vec) ./ tc_pT.(p_l, T_vec), 5).coeffs...,)))
@generated rho_T(t) = :(Base.evalpoly(t, $(fit(T_vec, rho_pT.(p_l, T_vec), 4).coeffs...,)))
@generated rhocp_T(t) = :(Base.evalpoly(t, $(fit(T_vec, 1000 * rho_pT.(p_l, T_vec) .* Cp_pT.(p_l, T_vec), 5).coeffs...,)))
# @register_symbolic kin_visc_T(t)
# @register_symbolic lambda_T(t)
# @register_symbolic Pr_T(t)
# @register_symbolic rho_T(t)
# @register_symbolic rhocp_T(t)
@connector function FluidPort(; name, p=101325.0, m=0.0, T=0.0)
sts = @variables p(t) = p m(t) = m [connect = Flow] T(t) = T [connect = Stream]
ODESystem(Equation[], t, sts, []; name=name)
end
@connector function VectorHeatPort(; name, N=100, T0=0.0, Q0=0.0)
sts = @variables (T(t))[1:N] = fill(T0, N) (Q(t))[1:N] = fill(Q0, N) [connect = Flow]
ODESystem(Equation[], t, [T; Q], []; name=name)
end
@register_symbolic Dxx_coeff(u, d, T)
#Taylor-aris dispersion model
function Dxx_coeff(u, d, T)
Re = abs(u) * d / kin_visc_T(T) + 0.1
if Re < 1000.0
(d^2 / 4) * u^2 / 48 / 0.14e-6
else
d * u * (1.17e9 * Re^(-2.5) + 0.41)
end
end
@register_symbolic Nusselt(Re, Pr, f)
#Nusselt number model
function Nusselt(Re, Pr, f)
if Re <= 2300.0
3.66
elseif Re <= 3100.0
3.5239 * (Re / 1000)^4 - 45.158 * (Re / 1000)^3 + 212.13 * (Re / 1000)^2 - 427.45 * (Re / 1000) + 316.08
else
f / 8 * ((Re - 1000) * Pr) / (1 + 12.7 * (f / 8)^(1 / 2) * (Pr^(2 / 3) - 1))
end
end
# @register_symbolic Churchill_f(Re, epsilon, d)
#Darcy weisbach friction factor
function Churchill_f(Re, epsilon, d)
theta_1 = (-2.457 * log(((7 / Re)^0.9) + (0.27 * (epsilon / d))))^16
theta_2 = (37530 / Re)^16
8 * ((((8 / Re)^12) + (1 / ((theta_1 + theta_2)^1.5)))^(1 / 12))
end
function FluidRegion(; name, L=1.0, dn=0.05, N=100, T0=0.0,
lumped_T=50, diffusion=true, e=1e-4)
@named inlet = FluidPort()
@named outlet = FluidPort()
@named heatport = VectorHeatPort(N=N)
dx = L / N
c = [-1 / 8, -3 / 8, -3 / 8] # advection stencil coefficients
A = pi * dn^2 / 4
p = @parameters C_shift = 0.0 Rw = 0.0 # stuff for latter
@variables begin
(T(t))[1:N] = fill(T0, N)
Twall(t)[1:N] = fill(T0, N)
(S(t))[1:N] = fill(T0, N)
(C(t))[1:N] = fill(1.0, N)
u(t) = 1e-6
Re(t) = 1000.0
Dxx(t) = 0.0
Pr(t) = 1.0
alpha(t) = 1.0
f(t) = 1.0
end
sts = vcat(T, Twall, S, C, Num[u], Num[Re], Num[Dxx], Num[Pr], Num[alpha], Num[f])
eqs = Equation[
Re ~ 0.1 + dn * abs(u) / kin_visc_T(lumped_T)
Pr ~ Pr_T(lumped_T)
f ~ Churchill_f(Re, e, dn) #Darcy-weisbach
alpha ~ Nusselt(Re, Pr, f) * lambda_T(lumped_T) / dn
Dxx ~ diffusion * Dxx_coeff(u, dn, lumped_T)
inlet.m ~ -outlet.m
inlet.p ~ outlet.p
inlet.T ~ instream(inlet.T)
outlet.T ~ T[N]
u ~ inlet.m / rho_T(inlet.T) / A
[C[i] ~ dx * A * rhocp_T(T[i]) for i in 1:N]
[S[i] ~ heatport.Q[i] for i in 1:N]
[Twall[i] ~ heatport.T[i] for i in 1:N]
#source term
[S[i] ~ (1 / (1 / (alpha * dn * pi * dx) + abs(Rw / 1000))) * (Twall[i] - T[i]) for i in 1:N]
#second order upwind + diffusion + source
D(T[1]) ~ u / dx * (inlet.T - T[1]) + Dxx * (T[2] - T[1]) / dx^2 + S[1] / (C[1] - C_shift)
D(T[2]) ~ u / dx * (c[1] * inlet.T - sum(c) * T[1] + c[2] * T[2] + c[3] * T[3]) + Dxx * (T[1] - 2 * T[2] + T[3]) / dx^2 + S[2] / (C[2] - C_shift)
[D(T[i]) ~ u / dx * (c[1] * T[i-2] - sum(c) * T[i-1] + c[2] * T[i] + c[3] * T[i+1]) + Dxx * (T[i-1] - 2 * T[i] + T[i+1]) / dx^2 + S[i] / (C[i] - C_shift) for i in 3:N-1]
D(T[N]) ~ u / dx * (T[N-1] - T[N]) + Dxx * (T[N-1] - T[N]) / dx^2 + S[N] / (C[N] - C_shift)
]
ODESystem(eqs, t, sts, p; systems=[inlet, outlet, heatport], name=name)
end
# @register_symbolic Cn_circular_wall_inner(d, D, cp, ρ)
function Cn_circular_wall_inner(d, D, cp, ρ)
C = pi / 4 * (D^2 - d^2) * cp * ρ
return C / 2
end
# @register_symbolic Cn_circular_wall_outer(d, D, cp, ρ)
function Cn_circular_wall_outer(d, D, cp, ρ)
C = pi / 4 * (D^2 - d^2) * cp * ρ
return C / 2
end
# @register_symbolic Ke_circular_wall(d, D, λ)
function Ke_circular_wall(d, D, λ)
2 * pi * λ / log(D / d)
end
function CircularWallFEM(; name, L=100, N=10, d=0.05, t_layer=[0.002],
λ=[50], cp=[500], ρ=[7850], T0=0.0)
@named inner_heatport = VectorHeatPort(N=N)
@named outer_heatport = VectorHeatPort(N=N)
dx = L / N
Ne = length(t_layer)
Nn = Ne + 1
dn = vcat(d, d .+ 2.0 .* cumsum(t_layer))
Cn = zeros(Nn)
Cn[1:Ne] += Cn_circular_wall_inner.(dn[1:Ne], dn[2:Nn], cp, ρ) .* dx
Cn[2:Nn] += Cn_circular_wall_outer.(dn[1:Ne], dn[2:Nn], cp, ρ) .* dx
p = @parameters C_shift = 0.0
Ke = Ke_circular_wall.(dn[1:Ne], dn[2:Nn], λ) .* dx
@variables begin
(Tn(t))[1:N, 1:Nn] = fill(T0, N, Nn)
(Qe(t))[1:N, 1:Ne] = fill(T0, N, Ne)
end
sts = [vec(Tn); vec(Qe)]
e0 = Equation[inner_heatport.T[i] ~ Tn[i, 1] for i in 1:N]
e1 = Equation[outer_heatport.T[i] ~ Tn[i, Nn] for i in 1:N]
e2 = Equation[Qe[i, j] ~ Ke[j] * (-Tn[i, j+1] + Tn[i, j]) for i in 1:N for j in 1:Ne]
e3 = Equation[D(Tn[i, 1]) * (Cn[1] + C_shift) ~ inner_heatport.Q[i] - Qe[i, 1] for i in 1:N]
e4 = Equation[D(Tn[i, j]) * Cn[j] ~ Qe[i, j-1] - Qe[i, j] for i in 1:N for j in 2:Nn-1]
e5 = Equation[D(Tn[i, Nn]) * Cn[Nn] ~ Qe[i, Ne] + outer_heatport.Q[i] for i in 1:N]
eqs = vcat(e0, e1, e2, e3, e4, e5)
ODESystem(eqs, t, sts, p; systems=[inner_heatport, outer_heatport], name=name)
end
function CylindricalSurfaceConvection(; name, L=100, N=100, d=1.0, α=5.0)
dx = L / N
S = pi * d * dx
@named heatport = VectorHeatPort(N=N)
sts = @variables Tenv(t) = 0.0
eqs = [
Tenv ~ 18.0
[heatport.Q[i] ~ α * S * (heatport.T[i] - Tenv) for i in 1:N]
]
ODESystem(eqs, t, sts, []; systems=[heatport], name=name)
end
function PreinsulatedPipe(; name, L=100.0, N=100.0, dn=0.05, T0=0.0, t_layer=[0.004, 0.013],
λ=[50, 0.04], cp=[500, 1200], ρ=[7800, 40], α=5.0,
e=1e-4, lumped_T=50, diffusion=true)
@named inlet = FluidPort()
@named outlet = FluidPort()
@named fluid_region = FluidRegion(L=L, N=N, dn=dn, e=e, lumped_T=lumped_T, diffusion=diffusion)
@named shell = CircularWallFEM(L=L, N=N, d=dn, t_layer=t_layer, λ=λ, cp=cp, ρ=ρ)
@named surfconv = CylindricalSurfaceConvection(L=L, N=N, d=dn + 2.0 * sum(t_layer), α=α)
systems = [fluid_region, shell, inlet, outlet, surfconv]
eqs = [
connect(fluid_region.inlet, inlet)
connect(fluid_region.outlet, outlet)
connect(fluid_region.heatport, shell.inner_heatport)
connect(shell.outer_heatport, surfconv.heatport)
]
ODESystem(eqs, t, [], []; systems=systems, name=name)
end
function Source(; name, p_feed=100000)
@named outlet = FluidPort()
sts = @variables m_flow(t) = 1e-6
eqs = [
m_flow ~ m_flow_source(t)
outlet.m ~ -m_flow
outlet.p ~ p_feed
outlet.T ~ T_source(t)
]
compose(ODESystem(eqs, t, sts, []; name=name), [outlet])
end
function Sink(; name)
@named inlet = FluidPort()
eqs = [
inlet.T ~ instream(inlet.T)
]
compose(ODESystem(eqs, t, [], []; name=name), [inlet])
end
function TestBenchPreinsulated(; name, L=1.0, dn=0.05, t_layer=[0.0056, 0.013], N=100, diffusion=true, lumped_T=20)
@named pipe = PreinsulatedPipe(L=L, dn=dn, N=N, diffusion=diffusion, t_layer=t_layer, lumped_T=lumped_T)
@named source = Source()
@named sink = Sink()
subs = [source, pipe, sink]
eqs = [
connect(source.outlet, pipe.inlet)
connect(pipe.outlet, sink.inlet)
]
compose(ODESystem(eqs, t, [], []; name=name), subs)
end
function call(fn, args...)
fn(args...)
endcall (generic function with 1 method)function run_and_time_construction!(jacobian_times, jacobian_gctimes, jacobian_allocs, build_times, functions, i, N)
@mtkbuild sys = TestBenchPreinsulated(L=470, N=N, dn=0.3127, t_layer=[0.0056, 0.058])
rhs = [eq.rhs for eq in full_equations(sys)]
dvs = unknowns(sys)
@info "Built system"
jac_result = @be (Symbolics.clear_derivative_caches!(); Symbolics.sparsejacobian(rhs, dvs))
jacobian_times[i] = mean(x -> x.time, jac_result.samples)
jacobian_gctimes[i] = mean(x -> x.time * x.gc_fraction, jac_result.samples)
jacobian_allocs[i] = mean(x -> x.bytes, jac_result.samples)
@info "Jacobian benchmark" jacobian_times[i] jacobian_gctimes[i] jacobian_allocs[i]
Symbolics.clear_derivative_caches!()
jac = Symbolics.sparsejacobian(rhs, dvs)
ps = parameters(sys)
defs = default_values(sys)
u0 = Float64[Symbolics.value(Symbolics.fixpoint_sub(v, defs)) for v in dvs]
p = Float64[Symbolics.value(Symbolics.fixpoint_sub(v, defs)) for v in ps]
t0 = 0.0
buffer_nocse = similar(jac, Float64)
buffer_nocse.nzval .= 0.0
buffer_cse = similar(jac, Float64)
buffer_cse.nzval .= 0.0
f_jac_nocse = eval(build_function(jac, dvs, ps, t; iip_config = (false, true), expression = Val{true}, cse = false)[2])
functions[1][i] = let buffer_nocse = buffer_nocse, u0 = u0, p = p, t0 = t0, f_jac_nocse = f_jac_nocse
function nocse()
f_jac_nocse(buffer_nocse, u0, p, t0)
buffer_nocse
end
end
@info "No CSE build_function result"
build_result_nocse = @be build_function(jac, dvs, ps, t; iip_config = (false, true), expression = Val{true}, cse = false)
@info "No CSE build_function benchmark"
build_times[1][i] = mean(x -> x.time, build_result_nocse.samples)
@info "build_function time" build_times[1][i]
f_jac_cse = eval(build_function(jac, dvs, ps, t; iip_config = (false, true), expression = Val{true}, cse = true)[2])
functions[2][i] = let buffer_cse = buffer_cse, u0 = u0, p = p, t0 = t0, f_jac_cse = f_jac_cse
function nocse()
f_jac_cse(buffer_cse, u0, p, t0)
buffer_cse
end
end
@info "CSE build_function result"
build_result_cse = @be build_function(jac, dvs, ps, t; iip_config = (false, true), expression = Val{true}, cse = true)
@info "CSE build_function benchmark"
build_times[2][i] = mean(x -> x.time, build_result_cse.samples)
@info "build_function time" build_times[2][i]
return nothing
end
function run_and_time_call!(functions, first_call_times, second_call_times, i)
fnocse = functions[1][i]
fcse = functions[2][i]
first_call_result_nocse = @timed fnocse()
first_call_times[1][i] = first_call_result_nocse.time
@info "First call time" first_call_times[1][i]
second_call_result_nocse = @be fnocse()
second_call_times[1][i] = mean(x -> x.time, second_call_result_nocse.samples)
@info "Runtime" second_call_times[1][i]
first_call_result_cse = @timed fcse()
first_call_times[2][i] = first_call_result_cse.time
@info "First call time" first_call_times[2][i]
second_call_result_cse = @be fcse()
second_call_times[2][i] = mean(x -> x.time, second_call_result_cse.samples)
@info "Runtime" second_call_times[2][i]
endrun_and_time_call! (generic function with 1 method)N = [5, 10, 20, 40, 80, 160, 320];
jacobian_times = zeros(Float64, length(N))
functions = [Vector{Any}(undef, length(N)), Vector{Any}(undef, length(N))]
jacobian_gctimes = similar(jacobian_times)
jacobian_allocs = similar(jacobian_times)
# [without_cse_times, with_cse_times]
build_times = [similar(jacobian_times), similar(jacobian_times)]
first_call_times = copy.(build_times)
second_call_times = copy.(build_times)2-element Vector{Vector{Float64}}:
[6.92835182230646e-310, 6.92835182230804e-310, 6.92835182230963e-310, 6.92
83518223112e-310, 6.9283518223128e-310, 6.92835182231437e-310, 6.9283518223
1595e-310]
[6.92835182230646e-310, 6.92835182230804e-310, 6.92835182230963e-310, 6.92
83518223112e-310, 6.9283518223128e-310, 6.92835182231437e-310, 6.9283518223
1595e-310]Timings
Chairmarks.DEFAULTS.seconds = 15.0
# compile
run_and_time_construction!(jacobian_times, jacobian_gctimes, jacobian_allocs, build_times, functions, 1, 5)
run_and_time_call!(functions, first_call_times, second_call_times, 1)
for (i, n) in enumerate(N)
@info i n
@time run_and_time_construction!(jacobian_times, jacobian_gctimes, jacobian_allocs, build_times, functions, i, n)
end
for (i, n) in enumerate(N)
@info i n
run_and_time_call!(functions, first_call_times, second_call_times, i)
end45.231103 seconds (203.26 M allocations: 6.585 GiB, 2.58% gc time, 0.13% c
ompilation time: 100% of which was recompilation)
45.453229 seconds (211.15 M allocations: 6.751 GiB, 3.14% gc time, 0.15% c
ompilation time: 100% of which was recompilation)
45.839694 seconds (220.09 M allocations: 7.138 GiB, 3.96% gc time, 0.16% c
ompilation time: 100% of which was recompilation)
46.756698 seconds (217.77 M allocations: 6.924 GiB, 2.95% gc time, 0.16% c
ompilation time: 98% of which was recompilation)
48.519410 seconds (222.90 M allocations: 7.040 GiB, 3.96% gc time, 0.18% c
ompilation time: 81% of which was recompilation)
52.272713 seconds (212.67 M allocations: 6.754 GiB, 4.83% gc time, 0.21% c
ompilation time: 58% of which was recompilation)
59.853380 seconds (218.37 M allocations: 6.958 GiB, 5.54% gc time, 0.14% c
ompilation time: 77% of which was recompilation)Results
tabledata = hcat(N, jacobian_times, jacobian_gctimes, jacobian_allocs, build_times..., first_call_times..., second_call_times...)
header = ["N", "Jacobian time", "Jacobian GC time", "Jacobian allocated memory (B)", "`build_function` time (no CSE)", "`build_function` time (CSE)", "First call time (no CSE)", "First call time (CSE)", "Second call time (no CSE)", "Second call time (CSE)"]
pretty_table(tabledata; column_labels = header, backend = :html)<table> <thead> <tr class = "columnLabelRow"> <th style = "font-weight: bold; text-align: right;">N</th> <th style = "font-weight: bold; text-align: right;">Jacobian time</th> <th style = "font-weight: bold; text-align: right;">Jacobian GC time</th> <th style = "font-weight: bold; text-align: right;">Jacobian allocated memory (B)</th> <th style = "font-weight: bold; text-align: right;">build_function time (no CSE)</th> <th style = "font-weight: bold; text-align: right;">build_function time (CSE)</th> <th style = "font-weight: bold; text-align: right;">First call time (no CSE)</th> <th style = "font-weight: bold; text-align: right;">First call time (CSE)</th> <th style = "font-weight: bold; text-align: right;">Second call time (no CSE)</th> <th style = "font-weight: bold; text-align: right;">Second call time (CSE)</th> </tr> </thead> <tbody> <tr class = "dataRow"> <td style = "text-align: right;">5.0</td> <td style = "text-align: right;">0.0109349</td> <td style = "text-align: right;">0.000262055</td> <td style = "text-align: right;">8.73628e5</td> <td style = "text-align: right;">0.00402182</td> <td style = "text-align: right;">0.00237691</td> <td style = "text-align: right;">0.571618</td> <td style = "text-align: right;">0.0863633</td> <td style = "text-align: right;">9.33388e-6</td> <td style = "text-align: right;">5.28871e-7</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">10.0</td> <td style = "text-align: right;">0.0194138</td> <td style = "text-align: right;">0.000856578</td> <td style = "text-align: right;">1.82033e6</td> <td style = "text-align: right;">0.0081464</td> <td style = "text-align: right;">0.00425534</td> <td style = "text-align: right;">1.02409</td> <td style = "text-align: right;">0.15882</td> <td style = "text-align: right;">1.87698e-5</td> <td style = "text-align: right;">6.38838e-7</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">20.0</td> <td style = "text-align: right;">0.0358756</td> <td style = "text-align: right;">0.00215719</td> <td style = "text-align: right;">3.71175e6</td> <td style = "text-align: right;">0.0163975</td> <td style = "text-align: right;">0.00766582</td> <td style = "text-align: right;">2.19914</td> <td style = "text-align: right;">0.305905</td> <td style = "text-align: right;">3.7454e-5</td> <td style = "text-align: right;">8.46064e-7</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">40.0</td> <td style = "text-align: right;">0.0703216</td> <td style = "text-align: right;">0.00212831</td> <td style = "text-align: right;">7.53228e6</td> <td style = "text-align: right;">0.0342729</td> <td style = "text-align: right;">0.0154525</td> <td style = "text-align: right;">4.81715</td> <td style = "text-align: right;">0.726228</td> <td style = "text-align: right;">7.49532e-5</td> <td style = "text-align: right;">1.35691e-6</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">80.0</td> <td style = "text-align: right;">0.140111</td> <td style = "text-align: right;">0.00880595</td> <td style = "text-align: right;">1.5118e7</td> <td style = "text-align: right;">0.0677967</td> <td style = "text-align: right;">0.031004</td> <td style = "text-align: right;">10.297</td> <td style = "text-align: right;">2.5602</td> <td style = "text-align: right;">0.00015144</td> <td style = "text-align: right;">2.20118e-6</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">160.0</td> <td style = "text-align: right;">0.295958</td> <td style = "text-align: right;">0.020103</td> <td style = "text-align: right;">3.04712e7</td> <td style = "text-align: right;">0.146142</td> <td style = "text-align: right;">0.0703415</td> <td style = "text-align: right;">23.6005</td> <td style = "text-align: right;">3.94802</td> <td style = "text-align: right;">0.000304967</td> <td style = "text-align: right;">4.26125e-6</td> </tr> <tr class = "dataRow"> <td style = "text-align: right;">320.0</td> <td style = "text-align: right;">0.588658</td> <td style = "text-align: right;">0.0406329</td> <td style = "text-align: right;">6.08459e7</td> <td style = "text-align: right;">0.306286</td> <td style = "text-align: right;">0.148011</td> <td style = "text-align: right;">55.6124</td> <td style = "text-align: right;">9.16718</td> <td style = "text-align: right;">0.000611285</td> <td style = "text-align: right;">8.21093e-6</td> </tr> </tbody> </table>
f = Figure(size = (750, 400))
titles = [
"Jacobian symbolic computation", "Jacobian symbolic computation", "Code generation",
"Numerical function compilation", "Numerical function evaluation"]
labels = ["Time (seconds)", "Allocated memory (bytes)",
"Time (seconds)", "Time (seconds)", "Time (seconds)"]
times = [jacobian_times, jacobian_allocs, build_times, first_call_times, second_call_times]
axes = Axis[]
for i in 1:2
label = labels[i]
data = times[i]
ax = Axis(f[1, i], xscale = log10, yscale = log10, xlabel = "model size",
xlabelsize = 10, ylabel = label, ylabelsize = 10, xticks = N,
title = titles[i], titlesize = 12, xticklabelsize = 10, yticklabelsize = 10)
push!(axes, ax)
scatterlines!(ax, N, data)
end
axes2 = Axis[]
# make equal y-axis unit length
mn3, mx3 = extrema(reduce(vcat, times[3]))
xn3 = log10(mx3 / mn3)
mn4, mx4 = extrema(reduce(vcat, times[4]))
xn4 = log10(mx4 / mn4)
mn5, mx5 = extrema(reduce(vcat, times[5]))
xn5 = log10(mx5 / mn5)
xn = max(xn3, xn4, xn5)
xn += 0.2
hxn = xn / 2
hxn3 = (log10(mx3) + log10(mn3)) / 2
hxn4 = (log10(mx4) + log10(mn4)) / 2
hxn5 = (log10(mx5) + log10(mn5)) / 2
ylims = [(exp10(hxn3 - hxn), exp10(hxn3 + hxn)), (exp10(hxn4 - hxn), exp10(hxn4 + hxn)),
(exp10(hxn5 - hxn), exp10(hxn5 + hxn))]
for i in 1:3
ir = i + 2
label = labels[ir]
data = times[ir]
ax = Axis(f[2, i], xscale = log10, yscale = log10, xlabel = "model size",
xlabelsize = 10, ylabel = label, ylabelsize = 10, xticks = N,
title = titles[ir], titlesize = 12, xticklabelsize = 10, yticklabelsize = 10)
ylims!(ax, ylims[i]...)
push!(axes2, ax)
scatterlines!(ax, N, data[1], label = "without CSE")
scatterlines!(ax, N, data[2], label = "with CSE")
end
Legend(f[1, 3], axes2[1], "Code generation", tellwidth = false, labelsize = 12, titlesize = 15)
save("thermal_fluid.pdf", f)
f
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/Symbolics","ThermalFluid.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/Symbolics/Project.toml`
[6e4b80f9] BenchmarkTools v1.8.0
⌃ [13f3f980] CairoMakie v0.15.13
[479239e8] Catalyst v16.4.0
[0ca39b1e] Chairmarks v1.3.1
[864edb3b] DataStructures v0.19.6
⌃ [7ed4a6bd] LinearSolve v5.14.0
⌃ [961ee093] ModelingToolkit v11.40.0
⌅ [bac558e1] OrderedCollections v1.8.2 [loaded: v2.0.1]
[1dea7af3] OrdinaryDiffEq v7.8.1
[91a5bcdd] Plots v1.41.7
⌃ [f27b6e38] Polynomials v4.1.1
[08abe8d2] PrettyTables v3.4.8
[b4db0fb7] ReactionNetworkImporters v1.5.0
⌃ [31c91b34] SciMLBenchmarks v0.1.3 [loaded: v0.2.0]
⌃ [10745b16] Statistics v1.11.1
[2efcf032] SymbolicIndexingInterface v0.3.55
⌅ [d1185830] SymbolicUtils v4.45.0
⌃ [0c5d862f] Symbolics v7.38.0
⌅ [a759f4b9] TimerOutputs v0.5.29
[95ff35a0] XSteam v0.3.0 `https://github.com/hzgzh/XSteam.jl.git#f2a1c58`
[37e2e46d] LinearAlgebra v1.12.0
[2f01184e] SparseArrays v1.12.0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`And the full manifest:
Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Symbolics/Manifest.toml`
[47edcb42] ADTypes v1.24.0
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[b2a6c25c] BinaryHeaps v1.1.0
⌃ [caf10ac8] BipartiteGraphs v0.1.12
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⌅ [861a8166] Combinatorics v1.0.2
[38540f10] CommonSolve v0.2.14
[bbf7d656] CommonSubexpressions v0.3.1
⌃ [f70d9fcc] CommonWorldInvalidations v1.2.0
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[a33af91c] CompositionsBase v0.1.2
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⌃ [2b5f629d] DiffEqBase v7.18.2
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⌃ [7c1d4256] DynamicPolynomials v0.6.7
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[1fa38f19] Format v1.3.7
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⌅ [eafb193a] Highlights v0.5.3
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⌃ [10745b16] Statistics v1.11.1
[82ae8749] StatsAPI v1.8.0
[2913bbd2] StatsBase v0.34.13
[4c63d2b9] StatsFuns v2.2.1
[69024149] StringEncodings v0.3.7
⌅ [892a3eda] StringManipulation v0.5.0
[09ab397b] StructArrays v0.7.3
[2efcf032] SymbolicIndexingInterface v0.3.55
⌃ [19f23fe9] SymbolicLimits v1.2.0
⌅ [d1185830] SymbolicUtils v4.45.0
⌃ [0c5d862f] Symbolics v7.38.0
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.14.0
[ed4db957] TaskLocalValues v0.1.3
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v2.0.0
[1c621080] TestItems v1.1.0
[731e570b] TiffImages v0.11.9
⌅ [a759f4b9] TimerOutputs v0.5.29
[3bb67fe8] TranscodingStreams v0.11.3
[410a4b4d] Tricks v0.1.13
[981d1d27] TriplotBase v0.1.0
[781d530d] TruncatedStacktraces v1.4.0
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
⌃ [1986cc42] Unitful v1.28.0
[41fe7b60] Unzip v0.2.0
[81def892] VersionParsing v1.3.0
[d30d5f5c] WeakCacheSets v0.1.0
[44d3d7a6] Weave v0.10.12
[e3aaa7dc] WebP v0.1.3
[efce3f68] WoodburyMatrices v1.1.0
[95ff35a0] XSteam v0.3.0 `https://github.com/hzgzh/XSteam.jl.git#f2a1c58`
[ddb6d928] YAML v0.4.16
[c2297ded] ZMQ v1.5.1
[6e34b625] Bzip2_jll v1.0.9+0
[4e9b3aee] CRlibm_jll v1.0.1+0
[83423d85] Cairo_jll v1.18.7+0
[a38c48d9] CoreMath_jll v0.1.0+0
[ee1fde0b] Dbus_jll v1.16.2+0
⌅ [5ae413db] EarCut_jll v2.2.4+0
[2702e6a9] EpollShim_jll v0.0.20230411+1
[2e619515] Expat_jll v2.8.3+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
[a3f928ae] Fontconfig_jll v2.17.1+0
[d7e528f0] FreeType2_jll v2.14.3+1
[559328eb] FriBidi_jll v1.0.17+0
[0656b61e] GLFW_jll v3.5.1+0
[d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
[61579ee1] Ghostscript_jll v9.55.1+0
⌅ [59f7168a] Giflib_jll v5.2.3+0
[020c3dae] Git_LFS_jll v3.7.1+0
[f8c6e375] Git_jll v2.55.0+0
[7746bdde] Glib_jll v2.88.3+0
[3b182d85] Graphite2_jll v1.3.16+0
[2e76f6c2] HarfBuzz_jll v100.14003.0+0
[905a6f67] Imath_jll v3.2.2+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.2.0+1
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.1.0+0
[1d63c593] LLVMOpenMP_jll v22.1.7+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[856f044c] MKL_jll v2025.2.0+0
[e7412a2a] Ogg_jll v1.3.6+0
[6cdc7f73] OpenBLASConsistentFPCSR_jll v0.3.34+0
[18a262bb] OpenEXR_jll v3.4.14+0
[9bd350c2] OpenSSH_jll v10.5.1+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.58.2+0
[30392449] Pixman_jll v0.46.4+0
[c0090381] Qt6Base_jll v6.10.2+2
[629bc702] Qt6Declarative_jll v6.10.2+2
[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[f50d1b31] Rmath_jll v0.5.2+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.3+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.13+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[a51aa0fd] Xorg_libXi_jll v1.8.4+0
[d1454406] Xorg_libXinerama_jll v1.1.7+0
[ec84b674] Xorg_libXrandr_jll v1.5.6+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.2.0+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
[35661453] Xorg_xkbcomp_jll v1.4.7+0
[33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[8f1865be] ZeroMQ_jll v4.3.6+0
[3161d3a3] Zstd_jll v1.5.7+1
[35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
[9a68df92] isoband_jll v0.2.3+0
[a4ae2306] libaom_jll v3.14.1+0
[0ac62f75] libass_jll v0.17.5+0
[1183f4f0] libdecor_jll v0.2.2+0
[8e53e030] libdrm_jll v2.4.134+0
[2db6ffa8] libevdev_jll v1.13.4+0
[f638f0a6] libfdk_aac_jll v2.0.4+0
[36db933b] libinput_jll v1.28.1+0
[b53b4c65] libpng_jll v1.6.58+0
[075b6546] libsixel_jll v1.10.5+0
[a9144af2] libsodium_jll v1.0.21+0
[9a156e7d] libva_jll v2.23.0+0
[f27f6e37] libvorbis_jll v1.3.8+0
[c5f90fcd] libwebp_jll v1.6.0+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.13.0+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[8bf52ea8] CRC32c v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.12.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.12.1
[de0858da] Printf v1.11.0
[9abbd945] Profile v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization v1.11.0
[1a1011a3] SharedArrays v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.12.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.3.1+2
[deac9b47] LibCURL_jll v8.15.0+0
[e37daf67] LibGit2_jll v1.9.0+0
[29816b5a] LibSSH2_jll v1.11.3+1
[14a3606d] MozillaCACerts_jll v2025.11.4
[4536629a] OpenBLAS_jll v0.3.29+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.6+0
[efcefdf7] PCRE2_jll v10.44.0+1
[bea87d4a] SuiteSparse_jll v7.8.3+2
[83775a58] Zlib_jll v1.3.1+2
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.64.0+1
[3f19e933] p7zip_jll v17.7.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`