Nernst-Planck 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
t_ref = 1.0       # s
x_ref = 0.38      # dm
C_ref = 0.16      # mol/dm^3
Phi_ref = 1.0     # V

epsilon = 78.5    # K
F = 96485.3415    # A s mol^-1
R = 831.0         # kg dm^2 s^-2 K^-1 mol^-1
T = 298.0         # K

z_Na = 1.0        # non-dim
z_Cl = -1.0       # non-dim

D_Na = 0.89e-7    # dm^2 s^−1
D_Cl = 1.36e-7    # dm^2 s^−1

u_Na = D_Na * abs(z_Na) * F / (R * T)
u_Cl = D_Cl * abs(z_Cl) * F / (R * T)

t_max = 0.01 / t_ref    # non-dim
x_max = 0.38 / x_ref    # non-dim
Na_0 = 0.16 / C_ref     # non-dim
Cl_0 = 0.16 / C_ref     # non-dim
Phi_0 = 4.0 / Phi_ref   # non-dim

Na_anode = 0.0            # non-dim
Na_cathode = 2.0 * Na_0   # non-dim
Cl_anode = 1.37 * Cl_0    # non-dim
Cl_cathode = 0.0          # non-dim

Pe_Na = x_ref^2 / ( t_ref * D_Na )  # non-dim
Pe_Cl = x_ref^2 / ( t_ref * D_Cl )  # non-dim

M_Na = x_ref^2 / ( t_ref * Phi_ref * u_Na )  # non-dim
M_Cl = x_ref^2 / ( t_ref * Phi_ref * u_Cl )  # non-dim

Po_1 = (epsilon * Phi_ref) / (F * x_ref * C_ref)  # non-dim

dx = 0.01 # non-dim
0.01
function solve(opt)
    strategy = QuadratureTraining()

    @parameters t,x
    @variables Phi(..),Na(..),Cl(..)
    Dt = Differential(t)
    Dx = Differential(x)
    Dxx = Differential(x)^2

    eqs = [
            ( Dxx(Phi(t,x)) ~ ( 1.0 / Po_1 ) *
                              ( z_Na * Na(t,x) + z_Cl * Cl(t,x) ) )
            ,
            ( Dt(Na(t,x)) ~ ( 1.0 / Pe_Na ) * Dxx(Na(t,x))
                          +   z_Na / ( abs(z_Na) * M_Na )
                          * ( Dx(Na(t,x)) * Dx(Phi(t,x)) + Na(t,x) * Dxx(Phi(t,x)) ) )
            ,
            ( Dt(Cl(t,x)) ~ ( 1.0 / Pe_Cl ) * Dxx(Cl(t,x))
                          +   z_Cl / ( abs(z_Cl) * M_Cl )
                          * ( Dx(Cl(t,x)) * Dx(Phi(t,x)) + Cl(t,x) * Dxx(Phi(t,x)) ) )
          ]

    bcs = [
            Phi(t,0.0) ~ Phi_0,
            Phi(t,x_max) ~ 0.0
            ,
            Na(0.0,x) ~ Na_0,
            Na(t,0.0) ~ Na_anode,
            Na(t,x_max) ~ Na_cathode
            ,
            Cl(0.0,x) ~ Cl_0,
            Cl(t,0.0) ~ Cl_anode,
            Cl(t,x_max) ~ Cl_cathode
          ]

    # Space and time domains ###################################################

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

    # Neural network, Discretization ###########################################

    dim = length(domains)
    output = length(eqs)
    neurons = 16
    chain1 = Lux.Chain( Lux.Dense(dim, neurons, tanh),
                        Lux.Dense(neurons, neurons, tanh),
                        Lux.Dense(neurons, neurons, tanh),
                        Lux.Dense(neurons, 1))
    chain2 = Lux.Chain( Lux.Dense(dim, neurons, tanh),
                        Lux.Dense(neurons, neurons, tanh),
                        Lux.Dense(neurons, neurons, tanh),
                        Lux.Dense(neurons, 1))
    chain3 = Lux.Chain( Lux.Dense(dim, neurons, tanh),
                        Lux.Dense(neurons, neurons, tanh),
                        Lux.Dense(neurons, neurons, tanh),
                        Lux.Dense(neurons, 1))

    discretization = PhysicsInformedNN([chain1, chain2, chain3], strategy)

    indvars = [t, x]   #phisically independent variables
    depvars = [Phi, Na, Cl]       #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(eqs, 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[20102.35444399695, 13212.57211623435, 7983.143309624558, 4361.57879621
4006, 2226.5416058990436, 1365.4960819974544, 1462.2027212859969, 2120.3018
64363247, 2938.1097337272768, 3602.1932894065326  …  44.512631590182096, 44
.51267569382645, 44.512719056981084, 44.512763405498816, 44.512807461458785
, 44.51285153610368, 44.51289437358338, 44.51293661718919, 44.5129766217081
84, 44.51301890679737], Any[0.001, 0.06683707237243652, 0.16351914405822754
, 0.23041200637817383, 0.29721498489379883, 0.39128708839416504, 0.45774912
83416748, 0.5507221221923828, 0.6178159713745117, 0.6844780445098877  …  37
1.53962302207947, 375.92034101486206, 379.9217131137848, 384.0860869884491,
 388.1792540550232, 392.51245617866516, 396.84283995628357, 401.16160607337
95, 405.18982696533203, 409.20486402511597])

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[2:end]], 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]) = (45.726912337143084, 43.602064
192774634, 43.3510959633791, 156.25733150836902, 151.0043607643847, 247.430
34976989645, 43.98026907309236, 44.00327143519049, 44.51301890679737)
(45.726912337143084, 43.602064192774634, 43.3510959633791, 156.257331508369
02, 151.0043607643847, 247.43034976989645, 43.98026907309236, 44.0032714351
9049, 44.51301890679737)

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","1d_poisson_nernst_planck.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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