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

Error: Failed to precompile NeuralPDE [315f7962-48a3-4962-8226-d0f33b1235f0
] to "/cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937
f953/compiled/v1.10/NeuralPDE/jl_OsSE2h".

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

Error: LoadError: UndefVarError: `@parameters` not defined
in expression starting at /cache/build/exclusive-amdci3-0/julialang/scimlbe
nchmarks-dot-jl/benchmarks/PINNErrorsVsTime/diffusion_et.jmd:5

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()

Error: UndefVarError: `HCubatureJL` not defined

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 runError: UndefVarError: `minimizers` not defined

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]))

Error: UndefVarError: `strategies_short_name` not defined

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 1.10.3
Commit 0b4590a5507 (2024-04-30 10:59 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-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNErrorsVsTime/Project.toml`
  [8bb1440f] DelimitedFiles
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⌃ [e00cd5f1] IntegralsCuba v0.2.0
⌃ [c31f79ba] IntegralsCubature v0.2.0
⌅ [b2108857] Lux v0.4.14
⌅ [961ee093] ModelingToolkit v8.18.7
⌃ [315f7962] NeuralPDE v5.0.0
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⌅ [8a4e6c94] QuasiMonteCarlo v0.2.9
⌃ [31c91b34] SciMLBenchmarks v0.1.0
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`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.

And the full manifest:

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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`
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Diffusion PDE Physics-Informed Neural Network (PINN) Loss Function Error vs Time Benchmarks

Setup

Solve

Results

Appendix