Allen-Cahn 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, OptimizationFlux, ModelingToolkit, Optimization, OptimizationOptimJL
using Lux, Plots
import ModelingToolkit: Interval, infimum, supremum

function solve(opt)
    strategy = QuadratureTraining()

    @parameters  t x1 x2 x3 x4
    @variables   u(..)

    Dt = Differential(t)

    Dxx1 = Differential(x1)^2
    Dxx2 = Differential(x2)^2
    Dxx3 = Differential(x3)^2
    Dxx4 = Differential(x4)^2


    # Discretization
    tmax         = 1.0
    x1width      = 1.0
    x2width      = 1.0
    x3width      = 1.0
    x4width      = 1.0

    tMeshNum     = 10
    x1MeshNum    = 10
    x2MeshNum    = 10
    x3MeshNum    = 10
    x4MeshNum    = 10

    dt   = tmax/tMeshNum
    dx1  = x1width/x1MeshNum
    dx2  = x2width/x2MeshNum
    dx3  = x3width/x3MeshNum
    dx4  = x4width/x4MeshNum

    domains = [t ∈ Interval(0.0,tmax),
               x1 ∈ Interval(0.0,x1width),
               x2 ∈ Interval(0.0,x2width),
               x3 ∈ Interval(0.0,x3width),
               x4 ∈ Interval(0.0,x4width)]

    ts  = 0.0 : dt : tmax
    x1s = 0.0 : dx1 : x1width
    x2s = 0.0 : dx2 : x2width
    x3s = 0.0 : dx3 : x3width
    x4s = 0.0 : dx4 : x4width

    # Operators
    Δu = Dxx1(u(t,x1,x2,x3,x4)) + Dxx2(u(t,x1,x2,x3,x4)) + Dxx3(u(t,x1,x2,x3,x4)) + Dxx4(u(t,x1,x2,x3,x4)) # Laplacian


    # Equation
    eq = Dt(u(t,x1,x2,x3,x4)) - Δu - u(t,x1,x2,x3,x4) + u(t,x1,x2,x3,x4)*u(t,x1,x2,x3,x4)*u(t,x1,x2,x3,x4) ~ 0  #ALLEN CAHN EQUATION

    initialCondition =  1/(2 + 0.4 * (x1*x1 + x2*x2 + x3*x3 + x4*x4)) # see PNAS paper

    bcs = [u(0,x1,x2,x3,x4) ~ initialCondition]  #from literature

    ## NEURAL NETWORK
    n = 20   #neuron number

    chain = Lux.Chain(Lux.Dense(5,n,tanh),Lux.Dense(n,n,tanh),Lux.Dense(n,1))   #Neural network from OptimizationFlux library

    discretization = PhysicsInformedNN(chain, strategy)

    indvars = [t,x1,x2,x3,x4]   #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 = Optimization.solve(prob, ADAM(); callback = cb, maxiters=50)
        prob = remake(prob,u0=res.minimizer)
        res = Optimization.solve(prob, BFGS(); callback = cb, maxiters=150)
    else
        res = Optimization.solve(prob, opt; callback = cb, maxiters=200)
    end

    times[1] = 0.001

    return loss, times #add numeric solution
end

solve (generic function with 1 method)

opt1 = ADAM()
opt2 = ADAM(0.005)
opt3 = ADAM(0.05)
opt4 = RMSProp()
opt5 = RMSProp(0.005)
opt6 = RMSProp(0.05)
opt7 = OptimizationOptimJL.BFGS()
opt8 = OptimizationOptimJL.LBFGS()

Optim.LBFGS{Nothing, LineSearches.InitialStatic{Float64}, LineSearches.Hage
rZhang{Float64, Base.RefValue{Bool}}, Optim.var"#19#21"}(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"#19#21"(), 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[14.762675994935918, 12.873595824826838, 11.186137125308552, 9.69371574
4582565, 8.38438034284334, 7.243054667313744, 6.253797228761048, 5.40027067
32595716, 4.66737313077248, 4.041772447811379  …  6.760463613376724e-5, 6.7
10561714837414e-5, 6.6123812915119e-5, 6.485312649379451e-5, 6.478944105440
748e-5, 6.376089363313795e-5, 6.368960978621748e-5, 6.365205484982014e-5, 6
.342036583394033e-5, 6.260649289583194e-5], Any[0.001, 0.21753787994384766,
 0.3766789436340332, 0.5729339122772217, 0.7319860458374023, 0.928238868713
3789, 1.087172031402588, 1.2775940895080566, 1.4366068840026855, 1.62744402
88543701  …  104.80442094802856, 105.33469104766846, 105.94285988807678, 10
6.4722089767456, 106.88036704063416, 107.49105787277222, 117.47734594345093
, 127.69644403457642, 128.31730103492737, 128.87648105621338])

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([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]) = (0.2829740476772876, 0.0549191
7345326161, 0.07812628999217362, 0.06977542278104656, 0.14429402421570417, 
0.19692223805941483, 9.527436549748497, 0.0017107403068184698, 6.2606492895
83194e-5)
(0.2829740476772876, 0.05491917345326161, 0.07812628999217362, 0.0697754227
8104656, 0.14429402421570417, 0.19692223805941483, 9.527436549748497, 0.001
7107403068184698, 6.260649289583194e-5)

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","allen_cahn.jmd")

Computer Information:

Julia Version 1.8.5
Commit 17cfb8e65ea (2023-01-08 06:45 UTC)
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-13.0.1 (ORCJIT, znver2)
  Threads: 128 on 128 virtual cores
Environment:
  JULIA_CPU_THREADS = 128
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNOptimizers/Project.toml`
⌅ [b2108857] Lux v0.4.11
⌃ [961ee093] ModelingToolkit v8.18.1
⌃ [315f7962] NeuralPDE v5.0.0
⌃ [7f7a1694] Optimization v3.8.1
⌃ [253f991c] OptimizationFlux v0.1.0
⌃ [36348300] OptimizationOptimJL v0.1.2
⌃ [91a5bcdd] Plots v1.31.4
⌃ [31c91b34] SciMLBenchmarks v0.1.0
Info Packages marked with ⌃ and ⌅ have new versions available, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.

And the full manifest:

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⌅ [2913bbd2] StatsBase v0.33.20
⌃ [4c63d2b9] StatsFuns v1.0.1
⌃ [789caeaf] StochasticDiffEq v6.51.0
⌅ [7792a7ef] StrideArraysCore v0.3.15
⌃ [69024149] StringEncodings v0.3.5
⌃ [09ab397b] StructArrays v0.6.11
⌅ [d1185830] SymbolicUtils v0.19.11
⌅ [0c5d862f] Symbolics v4.10.2
  [3783bdb8] TableTraits v1.0.1
⌃ [bd369af6] Tables v1.7.0
  [62fd8b95] TensorCore v0.1.1
⌅ [8ea1fca8] TermInterface v0.2.3
⌃ [5d786b92] TerminalLoggers v0.1.0
⌃ [8290d209] ThreadingUtilities v0.5.0
⌃ [ac1d9e8a] ThreadsX v0.1.10
⌃ [a759f4b9] TimerOutputs v0.5.20
⌃ [0796e94c] Tokenize v0.5.24
⌃ [3bb67fe8] TranscodingStreams v0.9.6
⌃ [28d57a85] Transducers v0.4.73
  [a2a6695c] TreeViews v0.3.0
⌃ [d5829a12] TriangularSolve v0.1.12
⌃ [5c2747f8] URIs v1.4.0
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
⌃ [1986cc42] Unitful v1.11.0
⌅ [41fe7b60] Unzip v0.1.2
⌃ [3d5dd08c] VectorizationBase v0.21.43
  [81def892] VersionParsing v1.3.0
  [19fa3120] VertexSafeGraphs v0.2.0
⌃ [44d3d7a6] Weave v0.10.9
⌃ [ddb6d928] YAML v0.4.7
⌃ [c2297ded] ZMQ v1.2.1
⌃ [e88e6eb3] Zygote v0.6.41
⌃ [700de1a5] ZygoteRules v0.2.2
  [6e34b625] Bzip2_jll v1.0.8+0
  [83423d85] Cairo_jll v1.16.1+1
  [7bc98958] Cubature_jll v1.0.5+0
⌃ [5ae413db] EarCut_jll v2.2.3+0
⌃ [2e619515] Expat_jll v2.4.8+0
⌃ [b22a6f82] FFMPEG_jll v4.4.2+0
  [a3f928ae] Fontconfig_jll v2.13.93+0
⌃ [d7e528f0] FreeType2_jll v2.10.4+0
  [559328eb] FriBidi_jll v1.0.10+0
⌃ [0656b61e] GLFW_jll v3.3.6+0
⌅ [d2c73de3] GR_jll v0.66.0+0
  [78b55507] Gettext_jll v0.21.0+0
⌅ [f8c6e375] Git_jll v2.34.1+0
⌃ [7746bdde] Glib_jll v2.68.3+2
  [3b182d85] Graphite2_jll v1.3.14+0
  [2e76f6c2] HarfBuzz_jll v2.8.1+1
⌃ [aacddb02] JpegTurbo_jll v2.1.2+0
  [c1c5ebd0] LAME_jll v3.100.1+0
  [88015f11] LERC_jll v3.0.0+1
⌅ [dad2f222] LLVMExtra_jll v0.0.16+0
  [dd4b983a] LZO_jll v2.10.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
  [d4300ac3] Libgcrypt_jll v1.8.7+0
⌃ [7e76a0d4] Libglvnd_jll v1.3.0+3
  [7add5ba3] Libgpg_error_jll v1.42.0+0
⌃ [94ce4f54] Libiconv_jll v1.16.1+1
  [4b2f31a3] Libmount_jll v2.35.0+0
⌅ [89763e89] Libtiff_jll v4.4.0+0
  [38a345b3] Libuuid_jll v2.36.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
⌅ [458c3c95] OpenSSL_jll v1.1.17+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 v5.15.3+1
⌅ [f50d1b31] Rmath_jll v0.3.0+0
⌃ [a2964d1f] Wayland_jll v1.19.0+0
  [2381bf8a] Wayland_protocols_jll v1.25.0+0
⌃ [02c8fc9c] XML2_jll v2.9.14+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
⌃ [14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
⌃ [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 v1.5.2+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [b53b4c65] libpng_jll v1.6.38+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
⌃ [d8fb68d0] xkbcommon_jll v0.9.1+5
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.8.0
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.0
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v0.5.2+0
  [deac9b47] LibCURL_jll v7.81.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.0+0
  [14a3606d] MozillaCACerts_jll v2022.2.1
  [4536629a] OpenBLAS_jll v0.3.20+0
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.40.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+0
  [83775a58] Zlib_jll v1.2.12+3
  [8e850b90] libblastrampoline_jll v5.1.0+0
  [8e850ede] nghttp2_jll v1.41.0+1
  [3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.