# Poisson PDE Physics-Informed Neural Network (PINN) Optimizer Benchmarks

Adapted from NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations. Uses the NeuralPDE.jl library from the SciML Scientific Machine Learning Open Source Organization for the implementation of physics-informed neural networks (PINNs) and other science-guided AI techniques.

## Setup Code

using NeuralPDE, Flux, ModelingToolkit, GalacticOptim, Optim, DiffEqFlux
using Plots
function solve(opt)

@parameters x y
@variables u(..)
Dxx = Differential(x)^2
Dyy = Differential(y)^2

# 2D PDE
eq  = Dxx(u(x,y)) + Dyy(u(x,y)) ~ -sin(pi*x)*sin(pi*y)

# Boundary conditions
bcs = [u(0,y) ~ 0.f0, u(1,y) ~ -sin(pi*1)*sin(pi*y),
u(x,0) ~ 0.f0, u(x,1) ~ -sin(pi*x)*sin(pi*1)]
# Space and time domains
domains = [x ∈ IntervalDomain(0.0,1.0),
y ∈ IntervalDomain(0.0,1.0)]

# Neural network
dim = 2 # number of dimensions
chain = FastChain(FastDense(dim,16,Flux.σ),FastDense(16,16,Flux.σ),FastDense(16,1))

discretization = PhysicsInformedNN(chain,strategy)

indvars = [x, y]   #phisically independent variables
depvars = [u]       #dependent (target) variable

loss = []
initial_time = 0

times = []

cb = function (p,l)
if initial_time == 0
initial_time = time()
end
push!(times, time() - initial_time)
#println("Current loss for $opt is:$l")
push!(loss, l)
return false
end

@named pde_system = PDESystem(eq, bcs, domains, indvars, depvars)
prob = discretize(pde_system, discretization)

if opt == "both"
res = GalacticOptim.solve(prob, ADAM(); cb = cb, maxiters=50)
prob = remake(prob,u0=res.minimizer)
res = GalacticOptim.solve(prob, BFGS(); cb = cb, maxiters=150)
else
res = GalacticOptim.solve(prob, opt; cb = cb, maxiters=200)
end

times[1] = 0.001

return loss, times #add numeric solution
end
solve (generic function with 1 method)
opt1 = ADAM()
opt4 = RMSProp()
opt5 = RMSProp(0.005)
opt6 = RMSProp(0.05)
opt7 = Optim.BFGS()
opt8 = Optim.LBFGS()
Optim.LBFGS{Nothing, LineSearches.InitialStatic{Float64}, LineSearches.Hage
rZhang{Float64, Base.RefValue{Bool}}, Optim.var"#17#19"}(10, LineSearches.I
nitialStatic{Float64}
alpha: Float64 1.0
scaled: Bool false
, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}
delta: Float64 0.1
sigma: Float64 0.9
alphamax: Float64 Inf
rho: Float64 5.0
epsilon: Float64 1.0e-6
gamma: Float64 0.66
linesearchmax: Int64 50
psi3: Float64 0.1
display: Int64 0
mayterminate: Base.RefValue{Bool}
, nothing, Optim.var"#17#19"(), Optim.Flat(), true)

## Solve

loss_1, times_1 = solve(opt1)
loss_2, times_2 = solve(opt2)
loss_3, times_3 = solve(opt3)
loss_4, times_4 = solve(opt4)
loss_5, times_5 = solve(opt5)
loss_6, times_6 = solve(opt6)
loss_7, times_7 = solve(opt7)
loss_8, times_8 = solve(opt8)
loss_9, times_9 = solve("both")
(Any[104.2684735341356, 101.07773885593438, 97.94397489905161, 94.867988074
16685, 91.85047362340062, 88.89199402025474, 85.9930383485208, 83.154064923
7131, 80.37550967081319, 77.65774472404973  …  0.0003188404442962145, 0.000
2753817434812261, 0.00025846908100736715, 0.00022924711159700477, 0.0002119
7987118970702, 0.0001984253703549455, 0.00019508461948719214, 0.00019226966
042200556, 0.00018558273727754988, 0.00017566174663987923], Any[0.001, 0.01
857614517211914, 0.03719305992126465, 0.055773019790649414, 0.0743460655212
4023, 0.09290003776550293, 0.1114511489868164, 0.1299901008605957, 0.202987
1940612793, 0.22084498405456543  …  8.621284008026123, 8.6758131980896, 8.7
12090015411377, 8.74848198890686, 8.78489899635315, 8.821274042129517, 8.85
7802152633667, 8.894743204116821, 8.950502157211304, 9.006330013275146])

## Results

p = plot([times_1, times_2, times_3, times_4, times_5, times_6, times_7, times_8, times_9], [loss_1, loss_2, loss_3, loss_4, loss_5, loss_6, loss_7, loss_8, loss_9],xlabel="time (s)", ylabel="loss", xscale=:log10, yscale=:log10, labels=["ADAM(0.001)" "ADAM(0.005)" "ADAM(0.05)" "RMSProp(0.001)" "RMSProp(0.005)" "RMSProp(0.05)" "BFGS()" "LBFGS()" "ADAM + BFGS"], legend=:bottomleft, linecolor=["#2660A4" "#4CD0F4" "#FEC32F" "#F763CD" "#44BD79" "#831894" "#A6ED18" "#980000" "#FF912B"])

p = plot(1:201, [loss_1, loss_2, loss_3, loss_4, loss_5, loss_6, loss_7, loss_8, loss_9[2:end]], xlabel="iterations", ylabel="loss", yscale=:log10, labels=["ADAM(0.001)" "ADAM(0.005)" "ADAM(0.05)" "RMSProp(0.001)" "RMSProp(0.005)" "RMSProp(0.05)" "BFGS()" "LBFGS()" "ADAM + BFGS"], legend=:bottomleft, linecolor=["#2660A4" "#4CD0F4" "#FEC32F" "#F763CD" "#44BD79" "#831894" "#A6ED18" "#980000" "#FF912B"])

@show loss_1[end], loss_2[end], loss_3[end], loss_4[end], loss_5[end], loss_6[end], loss_7[end], loss_8[end], loss_9[end]
(loss_1[end], loss_2[end], loss_3[end], loss_4[end], loss_5[end], loss_6[en
d], loss_7[end], loss_8[end], loss_9[end]) = (3.732579140083826, 3.83871068
57725737, 0.0734829146939961, 3.8387786739370595, 2.3763009329942824, 0.090
72786483935653, 0.00010257991924182217, 0.02300612987128965, 0.000175661746
63987923)
(3.732579140083826, 3.8387106857725737, 0.0734829146939961, 3.8387786739370
595, 2.3763009329942824, 0.09072786483935653, 0.00010257991924182217, 0.023
00612987128965, 0.00017566174663987923)

## 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/PINNOptimizers","poisson.jmd")

Computer Information:

Julia Version 1.6.5
Commit 9058264a69 (2021-12-19 12:30 UTC)
Platform Info:
OS: Linux (x86_64-pc-linux-gnu)
CPU: AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
Environment:
BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953


Package Information:

      Status /cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNOptimizers/Project.toml
[8a292aeb] Cuba v2.2.0
[667455a9] Cubature v1.5.1
[aae7a2af] DiffEqFlux v1.44.0
[587475ba] Flux v0.12.8
[a75be94c] GalacticOptim v2.2.0
[961ee093] ModelingToolkit v6.7.1
[315f7962] NeuralPDE v4.0.1
[429524aa] Optim v1.5.0
[91a5bcdd] Plots v1.24.2
[31c91b34] SciMLBenchmarks v0.1.0

And the full manifest:

      Status /cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNOptimizers/Manifest.toml
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[052768ef] CUDA v3.5.0
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[a51aa0fd] Xorg_libXi_jll v1.7.10+4
[d1454406] Xorg_libXinerama_jll v1.1.4+4
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[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
[35661453] Xorg_xkbcomp_jll v1.4.2+4
[33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
[c5fb5394] Xorg_xtrans_jll v1.4.0+3
[8f1865be] ZeroMQ_jll v4.3.4+0
[3161d3a3] Zstd_jll v1.5.0+0
[0ac62f75] libass_jll v0.14.0+4
[f638f0a6] libfdk_aac_jll v0.1.6+4
[b53b4c65] libpng_jll v1.6.38+0
[a9144af2] libsodium_jll v1.0.20+0
[f27f6e37] libvorbis_jll v1.3.7+0
[1270edf5] x264_jll v2020.7.14+2
[dfaa095f] x265_jll v3.0.0+3
[d8fb68d0] xkbcommon_jll v0.9.1+5
[56f22d72] Artifacts
[2a0f44e3] Base64
[8bb1440f] DelimitedFiles
[8ba89e20] Distributed
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[ca575930] NetworkOptions
[44cfe95a] Pkg
[de0858da] Printf
[9abbd945] Profile
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays
[10745b16] Statistics
[4607b0f0] SuiteSparse
[fa267f1f] TOML
[a4e569a6] Tar
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll
[deac9b47] LibCURL_jll
[29816b5a] LibSSH2_jll
[c8ffd9c3] MbedTLS_jll
[14a3606d] MozillaCACerts_jll
[05823500] OpenLibm_jll
[efcefdf7] PCRE2_jll
[83775a58] Zlib_jll
[8e850ede] nghttp2_jll
[3f19e933] p7zip_jll