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, ModelingToolkit, Optimization, OptimizationOptimJL
using Lux, Plots, OptimizationOptimisers
import ModelingToolkit: Interval, infimum, supremum

function solve(opt)
    strategy = QuadratureTraining()

    @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 ∈ Interval(0.0,1.0),
               y ∈ Interval(0.0,1.0)]

    # Neural network
    dim = 2 # number of dimensions
    chain = Lux.Chain(Lux.Dense(dim,16,tanh),Lux.Dense(16,16,tanh),Lux.Dense(16,1))

    discretization = PhysicsInformedNN(chain,strategy)

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

    loss = []
    initial_time = nothing

    times = []

    cb = function (p,l)
        if initial_time == nothing
            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 = Optimisers.ADAM()
opt2 = Optimisers.ADAM(0.005)
opt3 = Optimisers.ADAM(0.05)
opt4 = Optimisers.RMSProp()
opt5 = Optimisers.RMSProp(0.005)
opt6 = Optimisers.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}
  cache: Nothing nothing
, 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[5.572445847837208, 5.137474437530168, 4.731972852667706, 4.35535204707
153, 4.00694206049847, 3.6861039608996204, 3.3918956666338973, 3.1232815634
475317, 2.879133487853969, 2.6577248434611143  …  1.676446285494423e-5, 1.6
764408510121696e-5, 1.642363468579265e-5, 1.530788151534634e-5, 1.530788493
5191017e-5, 1.530789909889804e-5, 1.5255180987807334e-5, 1.5255147718388406
e-5, 1.5255147558631162e-5, 1.5255125472795477e-5], Any[0.001, 1.2061619758
605957, 2.4110240936279297, 3.597520112991333, 4.790925025939941, 6.0057621
00219727, 7.235180139541626, 8.41975712776184, 9.591458082199097, 10.783600
091934204  …  1117.2025179862976, 1195.2331612110138, 1261.2803111076355, 1
264.7305331230164, 1336.9927639961243, 1409.6710381507874, 1477.25170207023
62, 1546.1116490364075, 1617.539570093155, 1683.150109052658])

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], 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]) = (0.22502098220641686, 0.007378
732500777732, 0.0012586749226888906, 0.009016065235612157, 0.04431600859756
8415, 0.01078263612484731, 1.7623633960551587e-5, 0.0003399754609925307, 1.
5255125472795477e-5)
(0.22502098220641686, 0.007378732500777732, 0.0012586749226888906, 0.009016
065235612157, 0.044316008597568415, 0.01078263612484731, 1.7623633960551587
e-5, 0.0003399754609925307, 1.5255125472795477e-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","poisson.jmd")

Computer Information:

Julia Version 1.10.7
Commit 4976d05258e (2024-11-26 15:57 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
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNOptimizers/Project.toml`
⌃ [b2108857] Lux v1.2.3
⌃ [961ee093] ModelingToolkit v9.60.0
  [315f7962] NeuralPDE v5.17.0
  [7f7a1694] Optimization v4.0.5
  [36348300] OptimizationOptimJL v0.4.1
  [42dfb2eb] OptimizationOptimisers v0.3.7
  [91a5bcdd] Plots v1.40.9
  [31c91b34] SciMLBenchmarks v0.1.3
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNOptimizers/Manifest.toml`
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  [0c5d862f] Symbolics v6.23.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.0
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [5d786b92] TerminalLoggers v0.1.7
  [1c621080] TestItems v1.0.0
  [8290d209] ThreadingUtilities v0.5.2
  [a759f4b9] TimerOutputs v0.5.26
  [0796e94c] Tokenize v0.5.29
  [3bb67fe8] TranscodingStreams v0.11.3
  [28d57a85] Transducers v0.4.84
  [d5829a12] TriangularSolve v0.2.1
  [410a4b4d] Tricks v0.1.10
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.5.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.22.0
  [45397f5d] UnitfulLatexify v1.6.4
  [a7c27f48] Unityper v0.1.6
  [013be700] UnsafeAtomics v0.3.0
  [41fe7b60] Unzip v0.2.0
  [3d5dd08c] VectorizationBase v0.21.71
  [81def892] VersionParsing v1.3.0
  [897b6980] WeakValueDicts v0.1.0
  [44d3d7a6] Weave v0.10.12
  [d49dbf32] WeightInitializers v1.1.1
  [efce3f68] WoodburyMatrices v1.0.0
  [ddb6d928] YAML v0.4.12
  [c2297ded] ZMQ v1.4.0
⌅ [e88e6eb3] Zygote v0.6.75
⌃ [700de1a5] ZygoteRules v0.2.5
  [6e34b625] Bzip2_jll v1.0.8+4
  [83423d85] Cairo_jll v1.18.2+1
  [7bc98958] Cubature_jll v1.0.5+0
  [ee1fde0b] Dbus_jll v1.14.10+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.6.4+3
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
  [f5851436] FFTW_jll v3.3.10+3
  [a3f928ae] Fontconfig_jll v2.15.0+0
  [d7e528f0] FreeType2_jll v2.13.3+1
  [559328eb] FriBidi_jll v1.0.16+0
  [0656b61e] GLFW_jll v3.4.0+2
⌅ [d2c73de3] GR_jll v0.73.10+0
  [78b55507] Gettext_jll v0.21.0+0
  [f8c6e375] Git_jll v2.47.1+0
  [7746bdde] Glib_jll v2.82.4+0
  [3b182d85] Graphite2_jll v1.3.14+1
  [2e76f6c2] HarfBuzz_jll v8.5.0+0
  [e33a78d0] Hwloc_jll v2.11.2+3
⌅ [1d5cc7b8] IntelOpenMP_jll v2024.2.1+0
  [aacddb02] JpegTurbo_jll v3.1.1+0
  [c1c5ebd0] LAME_jll v3.100.2+0
  [88015f11] LERC_jll v4.0.1+0
  [dad2f222] LLVMExtra_jll v0.0.34+0
  [1d63c593] LLVMOpenMP_jll v18.1.7+0
  [dd4b983a] LZO_jll v2.10.3+0
  [81d17ec3] L_BFGS_B_jll v3.0.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+2
  [d4300ac3] Libgcrypt_jll v1.11.0+0
  [7e76a0d4] Libglvnd_jll v1.7.0+0
  [7add5ba3] Libgpg_error_jll v1.51.1+0
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.40.3+0
  [89763e89] Libtiff_jll v4.7.1+0
  [38a345b3] Libuuid_jll v2.40.3+0
⌅ [856f044c] MKL_jll v2024.2.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
  [458c3c95] OpenSSL_jll v3.0.15+3
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.3.3+0
  [36c8627f] Pango_jll v1.55.5+0
⌅ [30392449] Pixman_jll v0.43.4+0
⌅ [c0090381] Qt6Base_jll v6.7.1+1
  [629bc702] Qt6Declarative_jll v6.7.1+2
  [ce943373] Qt6ShaderTools_jll v6.7.1+1
  [e99dba38] Qt6Wayland_jll v6.7.1+1
  [f50d1b31] Rmath_jll v0.5.1+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.21.0+2
  [2381bf8a] Wayland_protocols_jll v1.36.0+0
  [02c8fc9c] XML2_jll v2.13.5+0
  [aed1982a] XSLT_jll v1.1.42+0
  [ffd25f8a] XZ_jll v5.6.4+0
  [f67eecfb] Xorg_libICE_jll v1.1.1+0
  [c834827a] Xorg_libSM_jll v1.2.4+0
  [4f6342f7] Xorg_libX11_jll v1.8.6+3
  [0c0b7dd1] Xorg_libXau_jll v1.0.12+0
  [935fb764] Xorg_libXcursor_jll v1.2.3+0
  [a3789734] Xorg_libXdmcp_jll v1.1.5+0
  [1082639a] Xorg_libXext_jll v1.3.6+3
  [d091e8ba] Xorg_libXfixes_jll v6.0.0+0
  [a51aa0fd] Xorg_libXi_jll v1.8.2+0
  [d1454406] Xorg_libXinerama_jll v1.1.5+0
  [ec84b674] Xorg_libXrandr_jll v1.5.4+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.11+1
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.2+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.2+1
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+0
  [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 v1.4.6+1
  [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
  [c5fb5394] Xorg_xtrans_jll v1.5.1+0
  [8f1865be] ZeroMQ_jll v4.3.5+3
  [3161d3a3] Zstd_jll v1.5.7+0
  [35ca27e7] eudev_jll v3.2.9+0
  [214eeab7] fzf_jll v0.56.3+0
  [1a1c6b14] gperf_jll v3.1.1+1
  [a4ae2306] libaom_jll v3.11.0+0
  [0ac62f75] libass_jll v0.15.2+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.3+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.45+1
  [a9144af2] libsodium_jll v1.0.20+3
  [f27f6e37] libvorbis_jll v1.3.7+2
  [009596ad] mtdev_jll v1.1.6+0
  [1317d2d5] oneTBB_jll v2021.12.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+2
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
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`