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.


using NeuralPDE
using Quadrature, Cubature, Cuba
using Flux, ModelingToolkit, GalacticOptim, Optim, DiffEqFlux
using Plots
using DelimitedFiles
using QuasiMonteCarlo
import ModelingToolkit: Interval, infimum, supremum
function diffusion(strategy, minimizer, maxIters)

    @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 = FastChain(FastDense(2,10,Flux.σ),FastDense(10,10,Flux.σ),FastDense(10,1))

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

    dx_err = [0.2,0.1]

    error_strategy = GridTraining(dx_err)

    initθ = Float64.(DiffEqFlux.initial_params(chain))
    eltypeθ = eltype(initθ)
    parameterless_type_θ = DiffEqBase.parameterless_type(initθ)

    phi = NeuralPDE.get_phi(chain,parameterless_type_θ)
    derivative = NeuralPDE.get_numeric_derivative()

    _pde_loss_function = NeuralPDE.build_loss_function(eq,indvars,depvars,

    bc_indvars = NeuralPDE.get_variables(bcs,indvars,depvars)
    _bc_loss_functions = [NeuralPDE.build_loss_function(bc,indvars,depvars,
                                              bc_indvars = bc_indvar) for (bc,bc_indvar) in zip(bcs,bc_indvars)]

    train_sets = NeuralPDE.generate_training_sets(domains,dx_err,[eq],bcs,eltypeθ,indvars,depvars)
    train_domain_set, train_bound_set = train_sets

    pde_loss_functions = [NeuralPDE.get_loss_function(_pde_loss_function,

    bc_loss_functions = [NeuralPDE.get_loss_function(bc,
                                          parameterless_type_θ,error_strategy) for (train_set,bc) in zip(train_bound_set, _bc_loss_functions)]

    loss_functions =  [pde_loss_functions;bc_loss_functions]
    function loss_function_(θ,p)
        return sum(map(l->l(θ) ,loss_functions))

    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

    discretization = PhysicsInformedNN(chain,strategy; init_params =initθ)

    @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 = GalacticOptim.solve(prob, minimizer; cb = 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]
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 = 1, abstol = 1e-3, maxiters = 10, batch = 10),
              NeuralPDE.QuadratureTraining(quadrature_alg = HCubatureJL(), reltol=1, abstol=1e-5, maxiters=100, batch = 0),
              NeuralPDE.QuadratureTraining(quadrature_alg = CubatureJLh(), reltol=1, abstol=1e-5, maxiters=100),
              NeuralPDE.QuadratureTraining(quadrature_alg = CubatureJLp(), reltol=1, abstol=1e-5, maxiters=100),
              NeuralPDE.StochasticTraining(400 ; bcs_points= 50),
              NeuralPDE.QuasiRandomTraining(400 ; bcs_points= 50)]

strategies_short_name = [#"CubaCuhre",

minimizers = [ADAM(0.001),

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()
prediction = Dict()
losses_res = Dict()
Dict{Any, Any}()


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])

Starting run


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]))


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

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
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (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-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNErrorsVsTime/Project.toml`
  [8a292aeb] Cuba v2.2.0
  [667455a9] Cubature v1.5.1
  [aae7a2af] DiffEqFlux v1.44.0
  [587475ba] Flux v0.12.8
  [a75be94c] GalacticOptim v2.2.1
  [961ee093] ModelingToolkit v7.1.3
  [315f7962] NeuralPDE v4.1.0
  [429524aa] Optim v1.5.0
  [91a5bcdd] Plots v1.25.2
  [67601950] Quadrature v1.12.0
  [8a4e6c94] QuasiMonteCarlo v0.2.3
  [31c91b34] SciMLBenchmarks v0.1.0
  [8bb1440f] DelimitedFiles

And the full manifest:

      Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNErrorsVsTime/Manifest.toml`
  [621f4979] AbstractFFTs v1.0.1
  [1520ce14] AbstractTrees v0.3.4
  [79e6a3ab] Adapt v3.3.1
  [dce04be8] ArgCheck v2.1.0
  [ec485272] ArnoldiMethod v0.2.0
  [4fba245c] ArrayInterface v3.2.1
  [4c555306] ArrayLayouts v0.7.8
  [15f4f7f2] AutoHashEquals v0.2.0
  [13072b0f] AxisAlgorithms v1.0.1
  [ab4f0b2a] BFloat16s v0.2.0
  [aae01518] BandedMatrices v0.16.11
  [198e06fe] BangBang v0.3.32
  [9718e550] Baselet v0.1.1
  [e2ed5e7c] Bijections v0.1.3
  [62783981] BitTwiddlingConvenienceFunctions v0.1.1
  [8e7c35d0] BlockArrays v0.16.9
  [ffab5731] BlockBandedMatrices v0.11.1
  [fa961155] CEnum v0.4.1
  [2a0fbf3d] CPUSummary v0.1.6
  [00ebfdb7] CSTParser v3.3.0
  [052768ef] CUDA v3.5.0
  [7057c7e9] Cassette v0.3.9
  [082447d4] ChainRules v1.17.0
  [d360d2e6] ChainRulesCore v1.11.2
  [9e997f8a] ChangesOfVariables v0.1.2
  [fb6a15b2] CloseOpenIntervals v0.1.4
  [944b1d66] CodecZlib v0.7.0
  [35d6a980] ColorSchemes v3.15.0
  [3da002f7] ColorTypes v0.11.0
  [5ae59095] Colors v0.12.8
  [861a8166] Combinatorics v1.0.2
  [a80b9123] CommonMark v0.8.4
  [38540f10] CommonSolve v0.2.0
  [bbf7d656] CommonSubexpressions v0.3.0
  [34da2185] Compat v3.41.0
  [b152e2b5] CompositeTypes v0.1.2
  [a33af91c] CompositionsBase v0.1.1
  [8f4d0f93] Conda v1.6.0
  [88cd18e8] ConsoleProgressMonitor v0.1.2
  [187b0558] ConstructionBase v1.3.0
  [d38c429a] Contour v0.5.7
  [a8cc5b0e] Crayons v4.0.4
  [8a292aeb] Cuba v2.2.0
  [667455a9] Cubature v1.5.1
  [754358af] DEDataArrays v0.2.0
  [9a962f9c] DataAPI v1.9.0
  [82cc6244] DataInterpolations v3.7.0
  [864edb3b] DataStructures v0.18.11
  [e2d170a0] DataValueInterfaces v1.0.0
  [244e2a9f] DefineSingletons v0.1.1
  [b429d917] DensityInterface v0.4.0
  [2b5f629d] DiffEqBase v6.77.1
  [459566f4] DiffEqCallbacks v2.18.0
  [aae7a2af] DiffEqFlux v1.44.0
  [c894b116] DiffEqJump v8.0.0
  [77a26b50] DiffEqNoiseProcess v5.9.0
  [9fdde737] DiffEqOperators v4.35.0
  [41bf760c] DiffEqSensitivity v6.64.0
  [163ba53b] DiffResults v1.0.3
  [b552c78f] DiffRules v1.9.0
  [b4f34e82] Distances v0.10.7
  [31c24e10] Distributions v0.25.36
  [ced4e74d] DistributionsAD v0.6.34
  [ffbed154] DocStringExtensions v0.8.6
  [5b8099bc] DomainSets v0.5.9
  [7c1d4256] DynamicPolynomials v0.3.21
  [da5c29d0] EllipsisNotation v1.1.3
  [7da242da] Enzyme v0.8.0
  [d4d017d3] ExponentialUtilities v1.10.2
  [e2ba6199] ExprTools v0.1.6
  [c87230d0] FFMPEG v0.4.1
  [7a1cc6ca] FFTW v1.4.5
  [7034ab61] FastBroadcast v0.1.11
  [9aa1b823] FastClosures v0.3.2
  [1a297f60] FillArrays v0.12.7
  [6a86dc24] FiniteDiff v2.8.1
  [53c48c17] FixedPointNumbers v0.8.4
  [587475ba] Flux v0.12.8
  [59287772] Formatting v0.4.2
  [f6369f11] ForwardDiff v0.10.24
  [069b7b12] FunctionWrappers v1.1.2
  [d9f16b24] Functors v0.2.7
  [0c68f7d7] GPUArrays v8.1.2
  [61eb1bfa] GPUCompiler v0.13.10
  [28b8d3ca] GR v0.62.1
  [a75be94c] GalacticOptim v2.2.1
  [5c1252a2] GeometryBasics v0.4.1
  [d7ba0133] Git v1.2.1
  [af5da776] GlobalSensitivity v1.2.2
  [86223c79] Graphs v1.4.1
  [42e2da0e] Grisu v1.0.2
  [19dc6840] HCubature v1.5.0
  [cd3eb016] HTTP v0.9.17
  [eafb193a] Highlights v0.4.5
  [3e5b6fbb] HostCPUFeatures v0.1.5
  [0e44f5e4] Hwloc v2.0.0
  [7073ff75] IJulia v1.23.2
  [7869d1d1] IRTools v0.4.4
  [615f187c] IfElse v0.1.1
  [d25df0c9] Inflate v0.1.2
  [83e8ac13] IniFile v0.5.0
  [22cec73e] InitialValues v0.2.11
  [a98d9a8b] Interpolations v0.13.5
  [8197267c] IntervalSets v0.5.3
  [3587e190] InverseFunctions v0.1.2
  [92d709cd] IrrationalConstants v0.1.1
  [c8e1da08] IterTools v1.4.0
  [42fd0dbc] IterativeSolvers v0.9.2
  [82899510] IteratorInterfaceExtensions v1.0.0
  [692b3bcd] JLLWrappers v1.3.0
  [682c06a0] JSON v0.21.2
  [98e50ef6] JuliaFormatter v0.19.2
  [e5e0dc1b] Juno v0.8.4
  [ef3ab10e] KLU v0.2.1
  [5ab0869b] KernelDensity v0.6.3
  [ba0b0d4f] Krylov v0.7.9
  [0b1a1467] KrylovKit v0.5.3
  [929cbde3] LLVM v4.7.0
  [b964fa9f] LaTeXStrings v1.3.0
  [2ee39098] LabelledArrays v1.6.7
  [23fbe1c1] Latexify v0.15.9
  [a5e1c1ea] LatinHypercubeSampling v1.8.0
  [73f95e8e] LatticeRules v0.0.1
  [10f19ff3] LayoutPointers v0.1.4
  [5078a376] LazyArrays v0.22.4
  [d7e5e226] LazyBandedMatrices v0.7.6
  [1d6d02ad] LeftChildRightSiblingTrees v0.1.2
  [d3d80556] LineSearches v7.1.1
  [7ed4a6bd] LinearSolve v0.1.6
  [2ab3a3ac] LogExpFunctions v0.3.5
  [e6f89c97] LoggingExtras v0.4.7
  [bdcacae8] LoopVectorization v0.12.99
  [1914dd2f] MacroTools v0.5.9
  [d125e4d3] ManualMemory v0.1.6
  [a3b82374] MatrixFactorizations v0.8.5
  [739be429] MbedTLS v1.0.3
  [442fdcdd] Measures v0.3.1
  [e89f7d12] Media v0.5.0
  [e9d8d322] Metatheory v1.3.2
  [128add7d] MicroCollections v0.1.1
  [e1d29d7a] Missings v1.0.2
  [961ee093] ModelingToolkit v7.1.3
  [4886b29c] MonteCarloIntegration v0.0.3
  [46d2c3a1] MuladdMacro v0.2.2
  [102ac46a] MultivariatePolynomials v0.3.18
  [ffc61752] Mustache v1.0.12
  [d8a4904e] MutableArithmetics v0.2.22
  [d41bc354] NLSolversBase v7.8.2
  [2774e3e8] NLsolve v4.5.1
  [872c559c] NNlib v0.7.31
  [a00861dc] NNlibCUDA v0.1.11
  [77ba4419] NaNMath v0.3.5
  [315f7962] NeuralPDE v4.1.0
  [8913a72c] NonlinearSolve v0.3.12
  [d8793406] ObjectFile v0.3.7
  [6fe1bfb0] OffsetArrays v1.10.8
  [429524aa] Optim v1.5.0
  [bac558e1] OrderedCollections v1.4.1
  [1dea7af3] OrdinaryDiffEq v5.70.1
  [90014a1f] PDMats v0.11.5
  [d96e819e] Parameters v0.12.3
  [69de0a69] Parsers v2.1.2
  [ccf2f8ad] PlotThemes v2.0.1
  [995b91a9] PlotUtils v1.1.0
  [91a5bcdd] Plots v1.25.2
  [e409e4f3] PoissonRandom v0.4.0
  [f517fe37] Polyester v0.5.4
  [1d0040c9] PolyesterWeave v0.1.2
  [85a6dd25] PositiveFactorizations v0.2.4
  [d236fae5] PreallocationTools v0.2.1
  [21216c6a] Preferences v1.2.2
  [33c8b6b6] ProgressLogging v0.1.4
  [92933f4c] ProgressMeter v1.7.1
  [1fd47b50] QuadGK v2.4.2
  [67601950] Quadrature v1.12.0
  [8a4e6c94] QuasiMonteCarlo v0.2.3
  [74087812] Random123 v1.4.2
  [e6cf234a] RandomNumbers v1.5.3
  [c84ed2f1] Ratios v0.4.2
  [c1ae055f] RealDot v0.1.0
  [3cdcf5f2] RecipesBase v1.2.1
  [01d81517] RecipesPipeline v0.4.1
  [731186ca] RecursiveArrayTools v2.20.0
  [f2c3362d] RecursiveFactorization v0.2.5
  [189a3867] Reexport v1.2.2
  [42d2dcc6] Referenceables v0.1.2
  [05181044] RelocatableFolders v0.1.3
  [ae029012] Requires v1.2.0
  [ae5879a3] ResettableStacks v1.1.1
  [37e2e3b7] ReverseDiff v1.11.0
  [79098fc4] Rmath v0.7.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.3
  [3cdde19b] SIMDDualNumbers v0.1.0
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.28
  [1bc83da4] SafeTestsets v0.0.1
  [0bca4576] SciMLBase v1.22.1
  [31c91b34] SciMLBenchmarks v0.1.0
  [6c6a2e73] Scratch v1.1.0
  [efcf1570] Setfield v0.8.1
  [992d4aef] Showoff v1.0.3
  [699a6c99] SimpleTraits v0.9.4
  [ed01d8cd] Sobol v1.5.0
  [b85f4697] SoftGlobalScope v1.1.0
  [a2af1166] SortingAlgorithms v1.0.1
  [47a9eef4] SparseDiffTools v1.18.3
  [276daf66] SpecialFunctions v1.8.1
  [171d559e] SplittablesBase v0.1.14
  [860ef19b] StableRNGs v1.0.0
  [aedffcd0] Static v0.4.1
  [90137ffa] StaticArrays v1.2.13
  [82ae8749] StatsAPI v1.1.0
  [2913bbd2] StatsBase v0.33.13
  [4c63d2b9] StatsFuns v0.9.14
  [789caeaf] StochasticDiffEq v6.41.0
  [7792a7ef] StrideArraysCore v0.2.9
  [69024149] StringEncodings v0.3.5
  [09ab397b] StructArrays v0.6.3
  [53d494c1] StructIO v0.3.0
  [d1185830] SymbolicUtils v0.18.1
  [0c5d862f] Symbolics v4.2.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.6.1
  [8ea1fca8] TermInterface v0.2.3
  [5d786b92] TerminalLoggers v0.1.5
  [8290d209] ThreadingUtilities v0.4.6
  [ac1d9e8a] ThreadsX v0.1.8
  [a759f4b9] TimerOutputs v0.5.13
  [0796e94c] Tokenize v0.5.21
  [9f7883ad] Tracker v0.2.16
  [3bb67fe8] TranscodingStreams v0.9.6
  [28d57a85] Transducers v0.4.67
  [592b5752] Trapz v2.0.3
  [a2a6695c] TreeViews v0.3.0
  [d5829a12] TriangularSolve v0.1.8
  [5c2747f8] URIs v1.3.0
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.9.2
  [3d5dd08c] VectorizationBase v0.21.22
  [81def892] VersionParsing v1.2.1
  [19fa3120] VertexSafeGraphs v0.2.0
  [44d3d7a6] Weave v0.10.10
  [efce3f68] WoodburyMatrices v0.5.5
  [ddb6d928] YAML v0.4.7
  [c2297ded] ZMQ v1.2.1
  [a5390f91] ZipFile v0.9.4
  [e88e6eb3] Zygote v0.6.32
  [700de1a5] ZygoteRules v0.2.2
  [6e34b625] Bzip2_jll v1.0.6+5
  [83423d85] Cairo_jll v1.16.0+6
  [3bed1096] Cuba_jll v4.2.2+0
  [7bc98958] Cubature_jll v1.0.5+0
  [5ae413db] EarCut_jll v2.2.3+0
  [7cc45869] Enzyme_jll v0.0.25+0
  [2e619515] Expat_jll v2.2.10+0
  [b22a6f82] FFMPEG_jll v4.3.1+4
  [f5851436] FFTW_jll v3.3.10+0
  [a3f928ae] Fontconfig_jll v2.13.1+14
  [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
  [4536629a] OpenBLAS_jll
  [05823500] OpenLibm_jll
  [efcefdf7] PCRE2_jll
  [bea87d4a] SuiteSparse_jll
  [83775a58] Zlib_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll