Hamilton-Jacobi 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

using NeuralPDE, Flux, ModelingToolkit, GalacticOptim, Optim, DiffEqFlux
using Quadrature,Cubature,Cuba
using Plots

function solve(opt)
    strategy = QuadratureTraining()

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

    Dt = Differential(t)

    Dx1 = Differential(x1)
    Dx2 = Differential(x2)
    Dx3 = Differential(x3)
    Dx4 = Differential(x4)

    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 ∈ IntervalDomain(0.0,tmax),
               x1 ∈ IntervalDomain(0.0,x1width),
               x2 ∈ IntervalDomain(0.0,x2width),
               x3 ∈ IntervalDomain(0.0,x3width),
               x4 ∈ IntervalDomain(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

    λ = 1.0f0

    # 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
    ∇u = [Dx1(u(t,x1,x2,x3,x4)), Dx2(u(t,x1,x2,x3,x4)),Dx3(u(t,x1,x2,x3,x4)),Dx4(u(t,x1,x2,x3,x4))]

    # Equation
    eq = Dt(u(t,x1,x2,x3,x4)) + Δu - λ*sum(∇u.^2) ~ 0  #HAMILTON-JACOBI-BELLMAN EQUATION

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

    bcs = [u(tmax,x1,x2,x3,x4) ~ terminalCondition]  #PNAS paper again

    ## NEURAL NETWORK
    n = 20   #neuron number

    chain = FastChain(FastDense(5,n,Flux.σ),FastDense(n,n,Flux.σ),FastDense(n,1))   #Neural network from Flux 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 = 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()
opt2 = ADAM(0.005)
opt3 = ADAM(0.05)
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[4.6311245606119105, 4.334876740410525, 4.11716075045588, 3.97634028837
64947, 3.9059338801869083, 3.8924795572470976, 3.9154761155965288, 3.951518
5894622316, 3.9810640887572695, 3.9930851186309124  …  0.05393840819153126,
 0.05315386820064365, 0.052232464764056186, 0.05133600786695876, 0.05025714
106037393, 0.04910740521631241, 0.047479776465887545, 0.046098982871024444,
 0.04531884273027065, 0.04468593140441521], Any[0.001, 0.4334990978240967, 
0.8686490058898926, 1.297950029373169, 1.7013370990753174, 2.12850308418273
93, 2.5570430755615234, 2.985930919647217, 3.388416051864624, 3.81581091880
79834  …  167.46812796592712, 168.30243396759033, 169.1603798866272, 170.00
156688690186, 170.87405109405518, 171.70947694778442, 172.56969904899597, 1
73.40300011634827, 174.2604899406433, 175.09343791007996])

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]) = (3.5055717714070016, 1.1094561
619123842, 0.29068409280506924, 2.30461132406756, 1.4825022887172918, 1.107
205671482202, 0.013031727570267063, 0.270705125928886, 0.04468593140441521)
(3.5055717714070016, 1.1094561619123842, 0.29068409280506924, 2.30461132406
756, 1.4825022887172918, 1.107205671482202, 0.013031727570267063, 0.2707051
25928886, 0.04468593140441521)

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","hamilton_jacobi.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
  [67601950] Quadrature v1.12.0
  [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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  [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
  [dad2f222] LLVMExtra_jll v0.0.13+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
  [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 v4.3.0+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
  [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 v1.5.0+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
  [0dad84c5] ArgTools
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [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