Hamilton-Jacobi PDE 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
function solve(opt)
    strategy = QuadratureTraining()

    ##  DECLARATIONS
    @parameters  t x1 x2 x3 x4
    @variables   u(..)

    Dt = Differential(t)

    Dx1 = Differential(x1)
    Dx2 = Differential(x2)
    Dx3 = Differential(x3)
    Dx4 = Differential(x4)

    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

    λ = 1.0f0

    # 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
    ∇u = [Dx1(u(t,x1,x2,x3,x4)), Dx2(u(t,x1,x2,x3,x4)),Dx3(u(t,x1,x2,x3,x4)),Dx4(u(t,x1,x2,x3,x4))]

    # Equation
    eq = Dt(u(t,x1,x2,x3,x4)) + Δu - λ*sum(∇u.^2) ~ 0  #HAMILTON-JACOBI-BELLMAN EQUATION

    terminalCondition =  log((1 + x1*x1 + x2*x2 + x3*x3 + x4*x4)/2) # see PNAS paper

    bcs = [u(tmax,x1,x2,x3,x4) ~ terminalCondition]  #PNAS paper again

    ## NEURAL NETWORK
    n = 20   #neuron number

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

    discretization = PhysicsInformedNN(chain, strategy)

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

    loss = []
    initial_time = 0

    times = []

    cb = function (p,l)
        if initial_time == 0
            initial_time = time()
        end
        push!(times, time() - initial_time)
        #println("Current loss for $opt is: $l")
        push!(loss, l)
        return false
    end

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

    if opt == "both"
        res = Optimization.solve(prob, ADAM(); callback = cb, maxiters=50)
        prob = remake(prob,u0=res.minimizer)
        res = Optimization.solve(prob, BFGS(); callback = cb, maxiters=150)
    else
        res = Optimization.solve(prob, opt; callback = cb, maxiters=200)
    end

    times[1] = 0.001

    return loss, times #add numeric solution
end
solve (generic function with 1 method)
opt1 = ADAM()
opt2 = ADAM(0.005)
opt3 = ADAM(0.05)
opt4 = RMSProp()
opt5 = RMSProp(0.005)
opt6 = RMSProp(0.05)
opt7 = OptimizationOptimJL.BFGS()
opt8 = OptimizationOptimJL.LBFGS()
Optim.LBFGS{Nothing, LineSearches.InitialStatic{Float64}, LineSearches.Hage
rZhang{Float64, Base.RefValue{Bool}}, Optim.var"#19#21"}(10, LineSearches.I
nitialStatic{Float64}
  alpha: Float64 1.0
  scaled: Bool false
, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}
  delta: Float64 0.1
  sigma: Float64 0.9
  alphamax: Float64 Inf
  rho: Float64 5.0
  epsilon: Float64 1.0e-6
  gamma: Float64 0.66
  linesearchmax: Int64 50
  psi3: Float64 0.1
  display: Int64 0
  mayterminate: Base.RefValue{Bool}
, nothing, Optim.var"#19#21"(), Optim.Flat(), true)

Solve

loss_1, times_1 = solve(opt1)
loss_2, times_2 = solve(opt2)
loss_3, times_3 = solve(opt3)
loss_4, times_4 = solve(opt4)
loss_5, times_5 = solve(opt5)
loss_6, times_6 = solve(opt6)
loss_7, times_7 = solve(opt7)
loss_8, times_8 = solve(opt8)
loss_9, times_9 = solve("both")
(Any[58.50292399621151, 53.550995599346145, 48.94262125358286, 44.676860295
60858, 40.74923671579711, 37.15215726265549, 33.87735323777565, 30.91492147
5730196, 28.251805906064234, 25.87477252502405  …  0.008184063578029442, 0.
008068540595508537, 0.007900963711302498, 0.007682878947461265, 0.007412722
491813935, 0.007058565228459336, 0.00686699629408717, 0.0066651730604333015
, 0.006495476630667968, 0.006311783832459027], Any[0.001, 0.261291027069091
8, 0.5163540840148926, 0.7872140407562256, 1.0426139831542969, 1.3134720325
46997, 1.568864107131958, 1.8237080574035645, 2.0946879386901855, 2.3501670
360565186  …  123.86953210830688, 124.41660594940186, 124.93901300430298, 1
25.50444102287292, 126.06708908081055, 126.61215591430664, 127.133630037307
74, 127.67642092704773, 128.2182149887085, 128.73882603645325])

Results

p = plot([times_1, times_2, times_3, times_4, times_5, times_6, times_7, times_8, times_9], [loss_1, loss_2, loss_3, loss_4, loss_5, loss_6, loss_7, loss_8, loss_9],xlabel="time (s)", ylabel="loss", xscale=:log10, yscale=:log10, labels=["ADAM(0.001)" "ADAM(0.005)" "ADAM(0.05)" "RMSProp(0.001)" "RMSProp(0.005)" "RMSProp(0.05)" "BFGS()" "LBFGS()" "ADAM + BFGS"], legend=:bottomleft, linecolor=["#2660A4" "#4CD0F4" "#FEC32F" "#F763CD" "#44BD79" "#831894" "#A6ED18" "#980000" "#FF912B"])

p = plot([loss_1, loss_2, loss_3, loss_4, loss_5, loss_6, loss_7, loss_8, loss_9], xlabel="iterations", ylabel="loss", yscale=:log10, labels=["ADAM(0.001)" "ADAM(0.005)" "ADAM(0.05)" "RMSProp(0.001)" "RMSProp(0.005)" "RMSProp(0.05)" "BFGS()" "LBFGS()" "ADAM + BFGS"], legend=:bottomleft, linecolor=["#2660A4" "#4CD0F4" "#FEC32F" "#F763CD" "#44BD79" "#831894" "#A6ED18" "#980000" "#FF912B"])

@show loss_1[end], loss_2[end], loss_3[end], loss_4[end], loss_5[end], loss_6[end], loss_7[end], loss_8[end], loss_9[end]
(loss_1[end], loss_2[end], loss_3[end], loss_4[end], loss_5[end], loss_6[en
d], loss_7[end], loss_8[end], loss_9[end]) = (2.5835739951328267, 0.5227691
789303975, 0.6041717101099481, 1.0365058850504243, 1.4265196344795572, 2.66
56707703862663, 0.6522855476336697, 0.193396873454065, 0.006311783832459027
)
(2.5835739951328267, 0.5227691789303975, 0.6041717101099481, 1.036505885050
4243, 1.4265196344795572, 2.6656707703862663, 0.6522855476336697, 0.1933968
73454065, 0.006311783832459027)

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","hamilton_jacobi.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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