Allen-Cahn 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 Code

using NeuralPDE
using Integrals, IntegralsCubature, IntegralsCuba
using OptimizationFlux, ModelingToolkit, Optimization, OptimizationOptimJL
using Lux, Plots
using DelimitedFiles
using QuasiMonteCarlo
import ModelingToolkit: Interval, infimum, supremum

function allen_cahn(strategy, minimizer, maxIters)

    @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

    n = 10   #neuron number
    chain = Lux.Chain(Lux.Dense(5, n, tanh), Lux.Dense(n, n, tanh), Lux.Dense(n, 1))   #Neural network from OptimizationFlux library

    indvars = [t, x1, x2, x3, x4]   #phisically independent variables
    depvars = [u(t, x1, x2, x3, x4)]       #dependent (target) variable

    dim = length(domains)

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

    dx_err = 0.2

    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)

    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

        #if (ctime/10^9 > time) #if I exceed the limit time I stop the training
        #    return true #Stop the minimizer and continue from line 142

        return false

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

    discretization = NeuralPDE.PhysicsInformedNN(chain, strategy)
    prob = NeuralPDE.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 = [ts, x1s, x2s, x3s, x4s]

    u_predict = [reshape([first(phi([t, x1, x2, x3, x4], res.minimizer)) for x1 in x1s for x2 in x2s for x3 in x3s for x4 in x4s], (length(x1s), length(x2s), length(x3s), length(x4s))) for t in ts]  #matrix of model's prediction

    return [error, params, domain, times, losses]
allen_cahn (generic function with 1 method)
maxIters = [(1,1,1,1,1,1,1000),(1,1,1,1,300,300,300)] #iters for ADAM/LBFGS
# maxIters = [(1,1,1,1,1,1,10),(1,1,1,3,3,3,3)] #iters for ADAM/LBFGS

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

strategies_short_name = ["CubaCuhre",

minimizers = [ADAM(0.005),BFGS()]
minimizers_short_name = ["ADAM","BFGS"]

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


## Convergence

for min =1:length(minimizers) # minimizer
      for strat=1:length(strategies) # strategy
            # println(string(strategies_short_name[strat], "  ", minimizers_short_name[min]))
            res = allen_cahn(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!(losses_res, string(strat,min)        => res[5])


print("\n Plotting error vs times")
#Plotting the first strategy with the first minimizer out from the loop to initialize the canvas
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 = (0,10))#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["71"], error_res["71"], yaxis=:log10, label = string(strategies_short_name[7], " + " , 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, label = string(strategies_short_name[6], " + " , minimizers_short_name[2]))
plot!(error, times["72"], error_res["72"], yaxis=:log10, title = string("Allen Cahn convergence ADAM/LBFGS"), ylabel = "log(error)",xlabel = "t", label = string(strategies_short_name[7], " + " , minimizers_short_name[2]))
Plotting error vs times


These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization

To locally run this benchmark, do the following commands:

using SciMLBenchmarks

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
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
  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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNErrorsVsTime/Project.toml`
  [de52edbc] Integrals v3.1.1
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  [8a4e6c94] QuasiMonteCarlo v0.2.9
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  [8bb1440f] DelimitedFiles

And the full manifest:

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  [f50d1b31] Rmath_jll v0.3.0+0
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  [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
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  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
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  [d1454406] Xorg_libXinerama_jll v1.1.4+4
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  [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
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  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
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  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
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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