Berger's Equation 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, OptimizationFlux, ModelingToolkit, Optimization, OptimizationOptimJL
using Lux, Plots
import ModelingToolkit: Interval, infimum, supremum
# Physical and numerical parameters (fixed)
nu = 0.07
nx = 10001 #101
x_max = 2.0 * pi
dx = x_max / (nx - 1.0)
nt = 2 #10
dt = dx * nu
t_max = dt * nt

# Analytic function
analytic_sol_func(t, x) = -2*nu*(-(-8*t + 2*x)*exp(-(-4*t + x)^2/(4*nu*(t + 1)))/
                          (4*nu*(t + 1)) - (-8*t + 2*x - 12.5663706143592)*
                          exp(-(-4*t + x - 6.28318530717959)^2/(4*nu*(t + 1)))/
                          (4*nu*(t + 1)))/(exp(-(-4*t + x - 6.28318530717959)^2/
                          (4*nu*(t + 1))) + exp(-(-4*t + x)^2/(4*nu*(t + 1)))) + 4
analytic_sol_func (generic function with 1 method)
function burgers(strategy, minimizer)

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

    eq = Dt(u(x, t)) + u(x, t) * Dx(u(x, t)) ~ nu * Dxx(u(x, t))

    bcs = [u(x, 0.0) ~ analytic_sol_func(x, 0.0),
        u(0.0, t) ~ u(x_max, t)]

    domains = [x ∈ Interval(0.0, x_max),
        t ∈ Interval(0.0, t_max)]

    chain = Lux.Chain(Lux.Dense(2, 16, tanh), Lux.Dense(16, 16, tanh), Lux.Dense(16, 1))
    discretization = PhysicsInformedNN(chain, strategy)

    indvars = [x, t]   #physically independent variables
    depvars = [u]      #dependent (target) variable

    dim = length(domains)

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

    dx_err = 0.00005

    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)
        return prob_.f.f(θ, nothing)
    end

    cb = function (p, l)

        timeCounter = 0.0
        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)
        append!(error, loss_function__(p))
        #println(length(losses), " Current loss is: ", l, " uniform error is, ", loss_function__(p))

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

        return false
    end

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

    discretization = NeuralPDE.PhysicsInformedNN(chain, strategy)
    prob = NeuralPDE.discretize(pde_system, discretization)

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


    if minimizer == "both"
        res = Optimization.solve(prob, ADAM(); callback=cb, maxiters=5)
        prob = remake(prob, u0=res.minimizer)
        res = Optimization.solve(prob, BFGS(); callback=cb, maxiters=15)
    else
        res = Optimization.solve(prob, minimizer; callback=cb, maxiters=500)
    end

    phi = discretization.phi

    params = res.minimizer

    return [error, params, times, losses]
end
burgers (generic function with 1 method)

Solve

# Settings:
#maxIters = [(0,0,0,0,0,0,20000),(300,300,300,300,300,300,300)] #iters

strategies = [NeuralPDE.QuadratureTraining()]

strategies_short_name = ["QuadratureTraining"]

minimizers = [ADAM(),
              ADAM(0.000005),
              ADAM(0.0005),
              RMSProp(),
              RMSProp(0.00005),
              RMSProp(0.05),
              BFGS(),
              LBFGS()]


minimizers_short_name = ["ADAM",
                         "ADAM(0.000005)",
                         "ADAM(0.0005)",
                         "RMS",
                         "RMS(0.00005)",
                         "RMS(0.05)",
                         "BFGS",
                         "LBFGS"]
8-element Vector{String}:
 "ADAM"
 "ADAM(0.000005)"
 "ADAM(0.0005)"
 "RMS"
 "RMS(0.00005)"
 "RMS(0.05)"
 "BFGS"
 "LBFGS"
# Run models
error_res =  Dict()
params_res = Dict()
times = Dict()
losses_res = Dict()

print("Starting run \n")


for min in 1:length(minimizers) # minimizer
      for strat in 1:length(strategies) # strategy
            #println(string(strategies_short_name[1], "  ", minimizers_short_name[min]))
            res = burgers(strategies[strat], minimizers[min])
            push!(error_res, string(strat,min)     => res[1])
            push!(params_res, string(strat,min) => res[2])
            push!(times, string(strat,min)        => res[3])
            push!(losses_res, string(strat,min)        => res[4])
      end
end
Starting run

Results

#PLOT ERROR VS ITER: to compare to compare between minimizers, keeping the same strategy (easily adjustable to compare between strategies)
error_iter = Plots.plot(1:length(error_res["11"]), error_res["11"], yaxis=:log10, title = string("Burger error vs iter"), ylabel = "Error", label = string(minimizers_short_name[1]), ylims = (0.0001,1))
plot!(error_iter, 1:length(error_res["12"]), error_res["12"], yaxis=:log10, label = string(minimizers_short_name[2]))
plot!(error_iter, 1:length(error_res["13"]), error_res["13"], yaxis=:log10, label = string(minimizers_short_name[3]))
plot!(error_iter, 1:length(error_res["14"]), error_res["14"], yaxis=:log10, label = string(minimizers_short_name[4]))
plot!(error_iter, 1:length(error_res["15"]), error_res["15"], yaxis=:log10, label = string(minimizers_short_name[5]))
plot!(error_iter, 1:length(error_res["16"]), error_res["16"], yaxis=:log10, label = string(minimizers_short_name[6]))
plot!(error_iter, 1:length(error_res["17"]), error_res["17"], yaxis=:log10, label = string(minimizers_short_name[7]))
plot!(error_iter, 1:length(error_res["18"]), error_res["18"], yaxis=:log10, label = string(minimizers_short_name[8]))

Plots.plot(error_iter)

#Use after having modified the analysis setting correctly --> Error vs iter: to compare different strategies, keeping the same minimizer
#error_iter = Plots.plot(1:length(error_res["11"]), error_res["11"], yaxis=:log10, title = string("Burger error vs iter"), ylabel = "Error", label = string(strategies_short_name[1]), ylims = (0.0001,1))
#plot!(error_iter, 1:length(error_res["21"]), error_res["21"], yaxis=:log10, label = string(strategies_short_name[2]))
#plot!(error_iter, 1:length(error_res["31"]), error_res["31"], yaxis=:log10, label = string(strategies_short_name[3]))
#plot!(error_iter, 1:length(error_res["41"]), error_res["41"], yaxis=:log10, label = string(strategies_short_name[4]))
#plot!(error_iter, 1:length(error_res["51"]), error_res["51"], yaxis=:log10, label = string(strategies_short_name[5]))
#plot!(error_iter, 1:length(error_res["61"]), error_res["61"], yaxis=:log10, label = string(strategies_short_name[6]))
#plot!(error_iter, 1:length(error_res["71"]), error_res["71"], yaxis=:log10, label = string(strategies_short_name[7]))
#PLOT ERROR VS TIME: to compare to compare between minimizers, keeping the same strategy
error_time = plot(times["11"], error_res["11"], yaxis=:log10, label = string(minimizers_short_name[1]),title = string("Burger error vs time"), ylabel = "Error", size = (1500,500))
plot!(error_time, times["12"], error_res["12"], yaxis=:log10, label = string(minimizers_short_name[2]))
plot!(error_time, times["13"], error_res["13"], yaxis=:log10, label = string(minimizers_short_name[3]))
plot!(error_time, times["14"], error_res["14"], yaxis=:log10, label = string(minimizers_short_name[4]))
plot!(error_time, times["15"], error_res["15"], yaxis=:log10, label = string(minimizers_short_name[5]))
plot!(error_time, times["16"], error_res["16"], yaxis=:log10, label = string(minimizers_short_name[6]))
plot!(error_time, times["17"], error_res["17"], yaxis=:log10, label = string(minimizers_short_name[7]))
plot!(error_time, times["18"], error_res["18"], yaxis=:log10, label = string(minimizers_short_name[7]))

Plots.plot(error_time)

#Use after having modified the analysis setting correctly --> Error vs time: to compare different strategies, keeping the same minimizer
#error_time = plot(times["11"], error_res["11"], yaxis=:log10, label = string(strategies_short_name[1]),title = string("Burger error vs time"), ylabel = "Error", size = (1500,500))
#plot!(error_time, times["21"], error_res["21"], yaxis=:log10, label = string(strategies_short_name[2]))
#plot!(error_time, times["31"], error_res["31"], yaxis=:log10, label = string(strategies_short_name[3]))
#plot!(error_time, times["41"], error_res["41"], yaxis=:log10, label = string(strategies_short_name[4]))
#plot!(error_time, times["51"], error_res["51"], yaxis=:log10, label = string(strategies_short_name[5]))
#plot!(error_time, times["61"], error_res["61"], yaxis=:log10, label = string(strategies_short_name[6]))
#plot!(error_time, times["71"], error_res["71"], yaxis=:log10, label = string(strategies_short_name[7]))

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","burgers_equation.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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNOptimizers/Project.toml`
  [b2108857] Lux v0.4.11
  [961ee093] ModelingToolkit v8.18.1
  [315f7962] NeuralPDE v5.0.0
  [7f7a1694] Optimization v3.8.1
  [253f991c] OptimizationFlux v0.1.0
  [36348300] OptimizationOptimJL v0.1.2
  [91a5bcdd] Plots v1.31.4
  [31c91b34] SciMLBenchmarks v0.1.0

And the full manifest:

      Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNOptimizers/Manifest.toml`
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