# Berger's Equation 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

using NeuralPDE, Flux, ModelingToolkit, GalacticOptim, Optim, DiffEqFlux
using Plots
# Physical and numerical parameters (fixed)
nu = 0.07
nx = 10001 #101
x_max = 2.0 * pi
dx = x_max / (nx - 1.0)
nt = 2 #10
dt = dx * nu
t_max = dt * nt

# Analytic function
analytic_sol_func(t, x) = -2*nu*(-(-8*t + 2*x)*exp(-(-4*t + x)^2/(4*nu*(t + 1)))/
(4*nu*(t + 1)) - (-8*t + 2*x - 12.5663706143592)*
exp(-(-4*t + x - 6.28318530717959)^2/(4*nu*(t + 1)))/
(4*nu*(t + 1)))/(exp(-(-4*t + x - 6.28318530717959)^2/
(4*nu*(t + 1))) + exp(-(-4*t + x)^2/(4*nu*(t + 1)))) + 4
analytic_sol_func (generic function with 1 method)
function burgers(strategy, minimizer)

@parameters x t
@variables u(..)
Dt = Differential(t)
Dx = Differential(x)
Dxx = Differential(x)^2

eq = Dt(u(x, t)) + u(x, t) * Dx(u(x, t))  ~ nu * Dxx(u(x, t))

bcs = [u(x,0.0) ~ analytic_sol_func(x, 0.0),
u(0.0,t) ~ u(x_max,t)]

domains = [x ∈ IntervalDomain(0.0, x_max),
t ∈ IntervalDomain(0.0, t_max)]

chain = FastChain(FastDense(2,16,Flux.σ),FastDense(16,16,Flux.σ),FastDense(16,1))
discretization = PhysicsInformedNN(chain, strategy)

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

dim = length(domains)

losses = []
error = []
times = []

dx_err = 0.00005

error_strategy = GridTraining(dx_err)

initθ = Float64.(DiffEqFlux.initial_params(chain))
eltypeθ = eltype(initθ)
parameterless_type_θ = DiffEqBase.parameterless_type(initθ)
phi = NeuralPDE.get_phi(chain,parameterless_type_θ)
derivative = NeuralPDE.get_numeric_derivative()

_pde_loss_function = NeuralPDE.build_loss_function(eq,indvars,depvars,
phi,derivative,nothing,chain,initθ,error_strategy)

bc_indvars = NeuralPDE.get_variables(bcs,indvars,depvars)
_bc_loss_functions = [NeuralPDE.build_loss_function(bc,indvars,depvars,
phi,derivative,nothing,chain,initθ,error_strategy,
bc_indvars = bc_indvar) for (bc,bc_indvar) in zip(bcs,bc_indvars)]

train_sets = NeuralPDE.generate_training_sets(domains,dx_err,[eq],bcs,eltypeθ,indvars,depvars)
train_domain_set, train_bound_set = train_sets

pde_loss_function = NeuralPDE.get_loss_function(_pde_loss_function,
train_domain_set[1],eltypeθ,
parameterless_type_θ,error_strategy)

bc_loss_functions = [NeuralPDE.get_loss_function(loss,set,
eltypeθ, parameterless_type_θ,
error_strategy) for (loss, set) in zip(_bc_loss_functions,train_bound_set)]

loss_functions = [pde_loss_function; bc_loss_functions]
loss_function__ = θ -> sum(map(l->l(θ) ,loss_functions))

function loss_function_(θ,p)
return loss_function__(θ)
end

cb = function (p,l)

timeCounter = 0.0
deltaT_s = time_ns() #Start a clock when the callback begins, this will evaluate questo misurerà anche il calcolo degli uniform error

ctime = time_ns() - startTime - timeCounter #This variable is the time to use for the time benchmark plot
append!(times, ctime/10^9) #Conversion nanosec to seconds
append!(losses, l)
append!(error, loss_function__(p))
#println(length(losses), " Current loss is: ", l, " uniform error is, ", loss_function__(p))

timeCounter = timeCounter + time_ns() - deltaT_s #timeCounter sums all delays due to the callback functions of the previous iterations

return false
end

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

discretization = NeuralPDE.PhysicsInformedNN(chain, strategy, init_params =initθ)
prob = NeuralPDE.discretize(pde_system,discretization)

startTime = time_ns() #Fix initial time (t=0) before starting the training

if minimizer == "both"
res = GalacticOptim.solve(prob, ADAM(); cb = cb, maxiters=5)
prob = remake(prob,u0=res.minimizer)
res = GalacticOptim.solve(prob, BFGS(); cb = cb, maxiters=15)
else
res = GalacticOptim.solve(prob, minimizer; cb = cb, maxiters=500)
end

phi = discretization.phi

params = res.minimizer

return [error, params, times, losses]
end
burgers (generic function with 1 method)

## Solve

# Settings:
#maxIters = [(0,0,0,0,0,0,20000),(300,300,300,300,300,300,300)] #iters

RMSProp(),
RMSProp(0.00005),
RMSProp(0.05),
BFGS(),
LBFGS()]

"RMS",
"RMS(0.00005)",
"RMS(0.05)",
"BFGS",
"LBFGS"]
8-element Vector{String}:
"RMS"
"RMS(0.00005)"
"RMS(0.05)"
"BFGS"
"LBFGS"
# Run models
error_res =  Dict()
params_res = Dict()
times = Dict()
losses_res = Dict()

print("Starting run \n")

for min in 1:length(minimizers) # minimizer
for strat in 1:length(strategies) # strategy
#println(string(strategies_short_name[1], "  ", minimizers_short_name[min]))
res = burgers(strategies[strat], minimizers[min])
push!(error_res, string(strat,min)     => res[1])
push!(params_res, string(strat,min) => res[2])
push!(times, string(strat,min)        => res[3])
push!(losses_res, string(strat,min)        => res[4])
end
end
Starting run

## Results

#PLOT ERROR VS ITER: to compare to compare between minimizers, keeping the same strategy (easily adjustable to compare between strategies)
error_iter = Plots.plot(1:length(error_res["11"]), error_res["11"], yaxis=:log10, title = string("Burger error vs iter"), ylabel = "Error", label = string(minimizers_short_name[1]), ylims = (0.0001,1))
plot!(error_iter, 1:length(error_res["12"]), error_res["12"], yaxis=:log10, label = string(minimizers_short_name[2]))
plot!(error_iter, 1:length(error_res["13"]), error_res["13"], yaxis=:log10, label = string(minimizers_short_name[3]))
plot!(error_iter, 1:length(error_res["14"]), error_res["14"], yaxis=:log10, label = string(minimizers_short_name[4]))
plot!(error_iter, 1:length(error_res["15"]), error_res["15"], yaxis=:log10, label = string(minimizers_short_name[5]))
plot!(error_iter, 1:length(error_res["16"]), error_res["16"], yaxis=:log10, label = string(minimizers_short_name[6]))
plot!(error_iter, 1:length(error_res["17"]), error_res["17"], yaxis=:log10, label = string(minimizers_short_name[7]))
plot!(error_iter, 1:length(error_res["18"]), error_res["18"], yaxis=:log10, label = string(minimizers_short_name[8]))

Plots.plot(error_iter)

#Use after having modified the analysis setting correctly --> Error vs iter: to compare different strategies, keeping the same minimizer
#error_iter = Plots.plot(1:length(error_res["11"]), error_res["11"], yaxis=:log10, title = string("Burger error vs iter"), ylabel = "Error", label = string(strategies_short_name[1]), ylims = (0.0001,1))
#plot!(error_iter, 1:length(error_res["21"]), error_res["21"], yaxis=:log10, label = string(strategies_short_name[2]))
#plot!(error_iter, 1:length(error_res["31"]), error_res["31"], yaxis=:log10, label = string(strategies_short_name[3]))
#plot!(error_iter, 1:length(error_res["41"]), error_res["41"], yaxis=:log10, label = string(strategies_short_name[4]))
#plot!(error_iter, 1:length(error_res["51"]), error_res["51"], yaxis=:log10, label = string(strategies_short_name[5]))
#plot!(error_iter, 1:length(error_res["61"]), error_res["61"], yaxis=:log10, label = string(strategies_short_name[6]))
#plot!(error_iter, 1:length(error_res["71"]), error_res["71"], yaxis=:log10, label = string(strategies_short_name[7]))
#PLOT ERROR VS TIME: to compare to compare between minimizers, keeping the same strategy
error_time = plot(times["11"], error_res["11"], yaxis=:log10, label = string(minimizers_short_name[1]),title = string("Burger error vs time"), ylabel = "Error", size = (1500,500))
plot!(error_time, times["12"], error_res["12"], yaxis=:log10, label = string(minimizers_short_name[2]))
plot!(error_time, times["13"], error_res["13"], yaxis=:log10, label = string(minimizers_short_name[3]))
plot!(error_time, times["14"], error_res["14"], yaxis=:log10, label = string(minimizers_short_name[4]))
plot!(error_time, times["15"], error_res["15"], yaxis=:log10, label = string(minimizers_short_name[5]))
plot!(error_time, times["16"], error_res["16"], yaxis=:log10, label = string(minimizers_short_name[6]))
plot!(error_time, times["17"], error_res["17"], yaxis=:log10, label = string(minimizers_short_name[7]))
plot!(error_time, times["18"], error_res["18"], yaxis=:log10, label = string(minimizers_short_name[7]))

Plots.plot(error_time)

#Use after having modified the analysis setting correctly --> Error vs time: to compare different strategies, keeping the same minimizer
#error_time = plot(times["11"], error_res["11"], yaxis=:log10, label = string(strategies_short_name[1]),title = string("Burger error vs time"), ylabel = "Error", size = (1500,500))
#plot!(error_time, times["21"], error_res["21"], yaxis=:log10, label = string(strategies_short_name[2]))
#plot!(error_time, times["31"], error_res["31"], yaxis=:log10, label = string(strategies_short_name[3]))
#plot!(error_time, times["41"], error_res["41"], yaxis=:log10, label = string(strategies_short_name[4]))
#plot!(error_time, times["51"], error_res["51"], yaxis=:log10, label = string(strategies_short_name[5]))
#plot!(error_time, times["61"], error_res["61"], yaxis=:log10, label = string(strategies_short_name[6]))
#plot!(error_time, times["71"], error_res["71"], yaxis=:log10, label = string(strategies_short_name[7]))

## 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","burgers_equation.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
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[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:

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[2e619515] Expat_jll v2.2.10+0
[b22a6f82] FFMPEG_jll v4.3.1+4
[f5851436] FFTW_jll v3.3.10+0
[a3f928ae] Fontconfig_jll v2.13.1+14
[d7e528f0] FreeType2_jll v2.10.1+5
[559328eb] FriBidi_jll v1.0.10+0
[0656b61e] GLFW_jll v3.3.5+1
[d2c73de3] GR_jll v0.58.1+0
[78b55507] Gettext_jll v0.20.1+7
[f8c6e375] Git_jll v2.31.0+0
[7746bdde] Glib_jll v2.59.0+4
[e33a78d0] Hwloc_jll v2.5.0+0
[1d5cc7b8] IntelOpenMP_jll v2018.0.3+2
[aacddb02] JpegTurbo_jll v2.1.0+0
[c1c5ebd0] LAME_jll v3.100.1+0
[dd4b983a] LZO_jll v2.10.1+0
[dd192d2f] LibVPX_jll v1.10.0+0
[e9f186c6] Libffi_jll v3.2.2+1
[d4300ac3] Libgcrypt_jll v1.8.7+0
[7e76a0d4] Libglvnd_jll v1.3.0+3
[94ce4f54] Libiconv_jll v1.16.1+1
[4b2f31a3] Libmount_jll v2.35.0+0
[89763e89] Libtiff_jll v4.3.0+0
[38a345b3] Libuuid_jll v2.36.0+0
[856f044c] MKL_jll v2021.1.1+2
[e7412a2a] Ogg_jll v1.3.5+0
[458c3c95] OpenSSL_jll v1.1.10+0
[efe28fd5] OpenSpecFun_jll v0.5.5+0
[91d4177d] Opus_jll v1.3.2+0
[2f80f16e] PCRE_jll v8.44.0+0
[30392449] Pixman_jll v0.40.1+0
[ea2cea3b] Qt5Base_jll v5.15.2+0
[f50d1b31] Rmath_jll v0.3.0+0
[a2964d1f] Wayland_jll v1.19.0+0
[2381bf8a] Wayland_protocols_jll v1.23.0+0
[02c8fc9c] XML2_jll v2.9.12+0
[aed1982a] XSLT_jll v1.1.34+0
[4f6342f7] Xorg_libX11_jll v1.6.9+4
[0c0b7dd1] Xorg_libXau_jll v1.0.9+4
[935fb764] Xorg_libXcursor_jll v1.2.0+4
[a3789734] Xorg_libXdmcp_jll v1.1.3+4
[1082639a] Xorg_libXext_jll v1.3.4+4
[d091e8ba] Xorg_libXfixes_jll v5.0.3+4
[a51aa0fd] Xorg_libXi_jll v1.7.10+4
[d1454406] Xorg_libXinerama_jll v1.1.4+4
[ec84b674] Xorg_libXrandr_jll v1.5.2+4
[ea2f1a96] Xorg_libXrender_jll v0.9.10+4
[c7cfdc94] Xorg_libxcb_jll v1.13.0+3
[cc61e674] Xorg_libxkbfile_jll v1.1.0+4
[12413925] Xorg_xcb_util_image_jll v0.4.0+1
[2def613f] Xorg_xcb_util_jll v0.4.0+1
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
[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