Diffusion PDE Physics-Informed Neural Network (PINN) Loss Function Error vs Time 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
using Integrals, IntegralsCubature, IntegralsCuba
using OptimizationFlux, ModelingToolkit, Optimization, OptimizationOptimJL
using Lux, Plots
using DelimitedFiles
using QuasiMonteCarlo
import ModelingToolkit: Interval, infimum, supremum
function diffusion(strategy, minimizer, maxIters)

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

    eq = Dt(u(x,t)) - Dxx(u(x,t)) ~ -exp(-t) * (sin(pi * x) - pi^2 * sin(pi * x))

    bcs = [u(x,0) ~ sin(pi*x),
           u(-1,t) ~ 0.,
           u(1,t) ~ 0.]

    domains = [x ∈ Interval(-1.0,1.0),
               t ∈ Interval(0.0,1.0)]

    dx = 0.2; dt = 0.1
    xs,ts = [infimum(domain.domain):dx/10:supremum(domain.domain) for (dx,domain) in zip([dx,dt],domains)]

    indvars = [x,t]
    depvars = [u(x,t)]

    chain = Lux.Chain(Lux.Dense(2,10,tanh),Lux.Dense(10,10,tanh),Lux.Dense(10,1))

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

    dx_err = [0.2,0.1]

    error_strategy = GridTraining(dx_err)

    discretization_ = PhysicsInformedNN(chain,error_strategy)
    @named pde_system_ = PDESystem(eq, bcs, domains, indvars, depvars)
    prob_ = discretize(pde_system_, discretization_)

    function loss_function_(θ, p)
        return prob_.f.f(θ, nothing)
    end
    
    cb_ = function (p,l)
        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)
        loss_ = loss_function_(p,nothing)
        append!(error, loss_)

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

        return false
    end

    discretization = PhysicsInformedNN(chain,strategy)

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


    timeCounter = 0.0
    startTime = time_ns() #Fix initial time (t=0) before starting the training

    res = Optimization.solve(prob, minimizer; callback=cb_, maxiters=maxIters)
    phi = discretization.phi

    params = res.minimizer

    # Model prediction
    domain = [x,t]

    u_predict = reshape([first(phi([x,t],res.minimizer)) for x in xs for t in ts],(length(xs),length(ts)))

    return [error, params, domain, times, u_predict, losses]
end
diffusion (generic function with 1 method)
maxIters = [(5000,5000,5000,5000,5000,5000),(300,300,300,300,300,300)] #iters for ADAM/LBFGS
# maxIters = [(5,5,5,5,5,5),(3,3,3,3,3,3)] #iters for ADAM/LBFGS

strategies = [#NeuralPDE.QuadratureTraining(quadrature_alg = CubaCuhre(), reltol = 1e-4, abstol = 1e-3, maxiters = 10, batch = 10),
              NeuralPDE.QuadratureTraining(quadrature_alg = HCubatureJL(), reltol = 1e-4, abstol=1e-5, maxiters=100, batch = 0),
              NeuralPDE.QuadratureTraining(quadrature_alg = CubatureJLh(), reltol = 1e-4, abstol=1e-5, maxiters=100),
              NeuralPDE.QuadratureTraining(quadrature_alg = CubatureJLp(), reltol = 1e-4, abstol=1e-5, maxiters=100),
              NeuralPDE.GridTraining([0.2,0.1]),
              NeuralPDE.StochasticTraining(400 ; bcs_points= 50),
              NeuralPDE.QuasiRandomTraining(400 ; bcs_points= 50)]

strategies_short_name = [#"CubaCuhre",
                        "HCubatureJL",
                        "CubatureJLh",
                        "CubatureJLp",
                        #"CubaVegas",
                        #"CubaSUAVE"]
                        "GridTraining",
                        "StochasticTraining",
                        "QuasiRandomTraining"]

minimizers = [ADAM(0.001),
              #BFGS()]
              LBFGS()]


minimizers_short_name = ["ADAM",
                         "LBFGS"]
                        # "BFGS"]


# Run models
error_res =  Dict()
domains = Dict()
params_res = Dict()  #to use same params for the next run
times = Dict()
prediction = Dict()
losses_res = Dict()
Dict{Any, Any}()

Solve

print("Starting run")
## Convergence

for min =1:length(minimizers) # minimizer
      for strat=1:length(strategies) # strategy
            # println(string(strategies_short_name[strat], "  ", minimizers_short_name[min]))
            res = diffusion(strategies[strat], minimizers[min], maxIters[min][strat])
            push!(error_res, string(strat,min)     => res[1])
            push!(params_res, string(strat,min) => res[2])
            push!(domains, string(strat,min)        => res[3])
            push!(times, string(strat,min)        => res[4])
            push!(prediction, string(strat,min)        => res[5])
            push!(losses_res, string(strat,min)        => res[6])

      end
end
Starting run

Results

current_label = string(strategies_short_name[1], " + " , minimizers_short_name[1])
error = Plots.plot(times["11"], error_res["11"], yaxis=:log10, label = current_label)#, xlims = (0,100))#legend = true)#, size=(1200,700))
plot!(error, times["21"], error_res["21"], yaxis=:log10, label = string(strategies_short_name[2], " + " , minimizers_short_name[1]))
plot!(error, times["31"], error_res["31"], yaxis=:log10, label = string(strategies_short_name[3], " + " , minimizers_short_name[1]))
plot!(error, times["41"], error_res["41"], yaxis=:log10, label = string(strategies_short_name[4], " + " , minimizers_short_name[1]))
plot!(error, times["51"], error_res["51"], yaxis=:log10, label = string(strategies_short_name[5], " + " , minimizers_short_name[1]))
plot!(error, times["61"], error_res["61"], yaxis=:log10, label = string(strategies_short_name[6], " + " , minimizers_short_name[1]))


plot!(error, times["12"], error_res["12"], yaxis=:log10, label = string(strategies_short_name[1], " + " , minimizers_short_name[2]))
plot!(error, times["22"], error_res["22"], yaxis=:log10, label = string(strategies_short_name[2], " + " , minimizers_short_name[2]))
plot!(error, times["32"], error_res["32"], yaxis=:log10, label = string(strategies_short_name[3], " + " , minimizers_short_name[2]))
plot!(error, times["42"], error_res["42"], yaxis=:log10, label = string(strategies_short_name[4], " + " , minimizers_short_name[2]))
plot!(error, times["52"], error_res["52"], yaxis=:log10, label = string(strategies_short_name[5], " + " , minimizers_short_name[2]))
plot!(error, times["62"], error_res["62"], yaxis=:log10, title = string("Diffusion convergence ADAM/LBFGS"), ylabel = "log(error)",xlabel = "t", label = string(strategies_short_name[6], " + " , minimizers_short_name[2]))

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/PINNErrorsVsTime","diffusion_et.jmd")

Computer Information:

Julia Version 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
Environment:
  JULIA_CPU_THREADS = 128
  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/PINNErrorsVsTime/Project.toml`
  [de52edbc] Integrals v3.1.1
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  [b2108857] Lux v0.4.14
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  [8a4e6c94] QuasiMonteCarlo v0.2.9
  [31c91b34] SciMLBenchmarks v0.1.0
  [8bb1440f] DelimitedFiles

And the full manifest:

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  [b22a6f82] FFMPEG_jll v4.4.2+0
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  [559328eb] FriBidi_jll v1.0.10+0
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  [d2c73de3] GR_jll v0.66.0+0
  [78b55507] Gettext_jll v0.21.0+0
  [f8c6e375] Git_jll v2.34.1+0
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  [dd4b983a] LZO_jll v2.10.1+0
  [e9f186c6] Libffi_jll v3.2.2+1
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  [7e76a0d4] Libglvnd_jll v1.3.0+3
  [7add5ba3] Libgpg_error_jll v1.42.0+0
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  [38a345b3] Libuuid_jll v2.36.0+0
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  [458c3c95] OpenSSL_jll v1.1.17+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
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  [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
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  [a4ae2306] libaom_jll v3.4.0+0
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  [f638f0a6] libfdk_aac_jll v2.0.2+0
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  [a9144af2] libsodium_jll v1.0.20+0
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  [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
  [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
  [4536629a] OpenBLAS_jll
  [05823500] OpenLibm_jll
  [efcefdf7] PCRE2_jll
  [bea87d4a] SuiteSparse_jll
  [83775a58] Zlib_jll
  [8e850b90] libblastrampoline_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll