Hamilton-Jacobi 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 Integrals, IntegralsCubature, IntegralsCuba
using OptimizationFlux, ModelingToolkit, Optimization, OptimizationOptimJL
using Lux, Plots
using DelimitedFiles
using QuasiMonteCarlo
import ModelingToolkit: Interval, infimum, supremumfunction hamilton_jacobi(strategy, minimizer, maxIters)
## 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 = 10 #neuron number
chain = Lux.Chain(Lux.Dense(5, n, tanh), Lux.Dense(n, n, tanh), Lux.Dense(n, 1)) #Neural network from OptimizationFlux library
indvars = [t, x1, x2, x3, x4] #phisically independent variables
depvars = [u(t, x1, x2, x3, x4)] #dependent (target) variable
dim = length(domains)
losses = []
error = []
times = []
dx_err = 0.2
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
@named pde_system = PDESystem(eq, bcs, domains, indvars, depvars)
discretization = NeuralPDE.PhysicsInformedNN(chain, strategy)
prob = NeuralPDE.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 = [ts, x1s, x2s, x3s, x4s]
u_predict = [reshape([first(phi([t, x1, x2, x3, x4], res.minimizer)) for x1 in x1s for x2 in x2s for x3 in x3s for x4 in x4s], (length(x1s), length(x2s), length(x3s), length(x4s))) for t in ts] #matrix of model's prediction
return [error, params, domain, times, losses]
end
maxIters = [(1,1,1,1000,1000,1000,1000),(1,1,1,300,300,300,300)] #iters for ADAM/LBFGS
# maxIters = [(1,1,1,1,1,2,2),(1,1,1,3,3,3,3)] #iters for ADAM/LBFGS
strategies = [NeuralPDE.QuadratureTraining(quadrature_alg = CubaCuhre(), reltol = 1e-4, abstol = 1e-4, maxiters = 100),
NeuralPDE.QuadratureTraining(quadrature_alg = HCubatureJL(), reltol = 1e-4, abstol = 1e-4, maxiters = 100, batch = 0),
NeuralPDE.QuadratureTraining(quadrature_alg = CubatureJLh(), reltol = 1e-4, abstol = 1e-4, maxiters = 100),
NeuralPDE.QuadratureTraining(quadrature_alg = CubatureJLp(), reltol = 1e-4, abstol = 1e-4, maxiters = 100),
NeuralPDE.GridTraining(0.2),
NeuralPDE.StochasticTraining(400 ; bcs_points= 50),
NeuralPDE.QuasiRandomTraining(400 ; bcs_points= 50)]
strategies_short_name = ["CubaCuhre",
"HCubatureJL",
"CubatureJLh",
"CubatureJLp",
"GridTraining",
"StochasticTraining",
"QuasiRandomTraining"]
minimizers = [ADAM(0.005),
#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()
losses_res = Dict()Dict{Any, Any}()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 = hamilton_jacobi(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!(losses_res, string(strat,min) => res[5])
end
endStarting run#Plotting the first strategy with the first minimizer out from the loop to initialize the canvas
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 = (0,10))#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["71"], error_res["71"], yaxis=:log10, label = string(strategies_short_name[7], " + " , 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, label = string(strategies_short_name[6], " + " , minimizers_short_name[2]))
plot!(error, times["72"], error_res["72"], yaxis=:log10, title = string("Hamilton Jacobi convergence ADAM/LBFGS"), ylabel = "log(error)",xlabel = "t", label = string(strategies_short_name[7], " + " , minimizers_short_name[2]))
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/PINNErrorsVsTime","hamilton_jacobi_et.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-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/PINNErrorsVsTime/Project.toml`
[de52edbc] Integrals v3.1.1
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[8290d209] ThreadingUtilities v0.5.0
[ac1d9e8a] ThreadsX v0.1.10
[a759f4b9] TimerOutputs v0.5.20
[0796e94c] Tokenize v0.5.24
[3bb67fe8] TranscodingStreams v0.9.6
[28d57a85] Transducers v0.4.73
[a2a6695c] TreeViews v0.3.0
[d5829a12] TriangularSolve v0.1.13
[5c2747f8] URIs v1.4.0
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
[1986cc42] Unitful v1.11.0
[41fe7b60] Unzip v0.1.2
[3d5dd08c] VectorizationBase v0.21.44
[81def892] VersionParsing v1.3.0
[19fa3120] VertexSafeGraphs v0.2.0
[44d3d7a6] Weave v0.10.9
[ddb6d928] YAML v0.4.7
[c2297ded] ZMQ v1.2.1
[e88e6eb3] Zygote v0.6.43
[700de1a5] ZygoteRules v0.2.2
[6e34b625] Bzip2_jll v1.0.8+0
[83423d85] Cairo_jll v1.16.1+1
[3bed1096] Cuba_jll v4.2.2+1
[7bc98958] Cubature_jll v1.0.5+0
[5ae413db] EarCut_jll v2.2.3+0
[2e619515] Expat_jll v2.4.8+0
[b22a6f82] FFMPEG_jll v4.4.2+0
[a3f928ae] Fontconfig_jll v2.13.93+0
[d7e528f0] FreeType2_jll v2.10.4+0
[559328eb] FriBidi_jll v1.0.10+0
[0656b61e] GLFW_jll v3.3.8+0
[d2c73de3] GR_jll v0.66.0+0
[78b55507] Gettext_jll v0.21.0+0
[f8c6e375] Git_jll v2.34.1+0
[7746bdde] Glib_jll v2.68.3+2
[3b182d85] Graphite2_jll v1.3.14+0
[2e76f6c2] HarfBuzz_jll v2.8.1+1
[aacddb02] JpegTurbo_jll v2.1.2+0
[c1c5ebd0] LAME_jll v3.100.1+0
[88015f11] LERC_jll v3.0.0+1
[dad2f222] LLVMExtra_jll v0.0.16+0
[dd4b983a] LZO_jll v2.10.1+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 v4.4.0+0
[38a345b3] Libuuid_jll v2.36.0+0
[e7412a2a] Ogg_jll v1.3.5+1
[458c3c95] OpenSSL_jll v1.1.17+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 v5.15.3+1
[f50d1b31] Rmath_jll v0.3.0+0
[a2964d1f] Wayland_jll v1.19.0+0
[2381bf8a] Wayland_protocols_jll v1.25.0+0
[02c8fc9c] XML2_jll v2.9.14+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 v1.5.2+0
[a4ae2306] libaom_jll v3.4.0+0
[0ac62f75] libass_jll v0.15.1+0
[f638f0a6] libfdk_aac_jll v2.0.2+0
[b53b4c65] libpng_jll v1.6.38+0
[a9144af2] libsodium_jll v1.0.20+0
[f27f6e37] libvorbis_jll v1.3.7+1
[1270edf5] x264_jll v2021.5.5+0
[dfaa095f] x265_jll v3.5.0+0
[d8fb68d0] xkbcommon_jll v1.4.1+0
[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
[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
[8e850b90] libblastrampoline_jll
[8e850ede] nghttp2_jll
[3f19e933] p7zip_jll