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Tensor Product Function

The tensor product function is defined as:

$f(x) = \prod_{i=1}^{d} \cos(a\pi x_i)$

where:

  • (d): Represents the dimensionality of the input vector (x).
  • (x_i): Represents the (i)-th component of the input vector.
  • (a): A constant parameter.

Package Imports

using Surrogates
using XGBoost
using Plots
using Statistics
using PrettyTables
using BenchmarkTools

## Python SMT
using PythonCall
using CondaPkg
smt = pyimport("smt.surrogate_models")
np = pyimport("numpy")
Python: <module 'numpy' from '/home/crackauc/github-runners/amdci8-1/_work/
SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Surrogates/.CondaPkg/.pixi
/envs/default/lib/python3.14/site-packages/numpy/__init__.py'>

Define the function

function tensor_product_function(x)
    a = 0.5
    return prod(cos.(a * π * x))
end
tensor_product_function (generic function with 1 method)

Define parameters for training and test data

lb = -5.0 # Lower bound of sampling range
ub = 5.0  # Upper bound of sampling range
n_train = 100 # Number of training points
n_test = 150 # Number of testing points
150

Sample training and test data points

x_train = sample(n_train, lb, ub, SobolSample())  # Sample training data points
y_train = tensor_product_function.(x_train)  # Calculate corresponding function values
x_test = sample(n_test, lb, ub, RandomSample())  # Sample larger test data set
y_test = tensor_product_function.(x_test)  # Calculate corresponding true function values
150-element Vector{Float64}:
  0.8108797175907632
 -0.42939183929405356
  0.6931399865187435
 -0.829133827824701
 -0.8631216170802395
  0.7127817352352362
  0.9549015043774433
 -0.2671508359237833
 -0.06581429589682196
  0.9177493284910282
  ⋮
  0.9819660256801712
  0.7948310869770674
  0.9958939388150962
  0.591670368092818
  0.9872894683128622
  0.40537109006317124
  0.9540657297184868
  0.1532980074054892
 -0.09451429671555064

Plot training and test points

scatter(x_train, y_train, label="Training Points", xlabel="X-axis", ylabel="Y-axis", legend=:topright)
scatter!(x_test, y_test, label="Testing Points")

Fit surrogate models

# SMT
radial_surrogate_smt = smt.RBF(d0=1.0, poly_degree=1.0, print_global=Py(false))
radial_surrogate_smt.set_training_values(np.array(x_train), np.array(y_train))
radial_surrogate_smt.train()

theta = 0.5 / max(1e-6 * abs(ub - lb), std(x_train))^2.0
kriging_surrogate_smt = smt.KRG(theta0=np.array([theta]), poly="quadratic", print_global=Py(false))
kriging_surrogate_smt.set_training_values(np.array(x_train), np.array(y_train))
kriging_surrogate_smt.train()

# Julia
xgboost_surrogate = XGBoostSurrogate(x_train, y_train, lb, ub, num_round = 10)
radial_surrogate = RadialBasis(x_train, y_train, lb, ub)
kriging_surrogate = Kriging(x_train, y_train, lb, ub)
loba_surrogate = LobachevskySurrogate(x_train, y_train, lb, ub, alpha = 2.0, n = 6)
(::Surrogates.LobachevskySurrogate{Vector{Float64}, Vector{Float64}, Float6
4, Int64, Float64, Float64, Vector{Float64}, Bool}) (generic function with 
2 methods)

Predictions on training and test data

## Training data
radial_train_pred_smt = radial_surrogate_smt.predict_values(np.array(x_train))
radial_train_pred_smt = pyconvert(Matrix{Float64}, radial_train_pred_smt)[:, 1]
kriging_train_pred_smt = kriging_surrogate_smt.predict_values(np.array(x_train))
kriging_train_pred_smt = pyconvert(Matrix{Float64}, kriging_train_pred_smt)[:, 1]
xgboost_train_pred = xgboost_surrogate.(x_train)
radial_train_pred = radial_surrogate.(x_train)
kriging_train_pred = kriging_surrogate.(x_train)
loba_train_pred = loba_surrogate.(x_train)

## Test data
radial_test_pred_smt = radial_surrogate_smt.predict_values(np.array(x_test))
radial_test_pred_smt = pyconvert(Matrix{Float64}, radial_test_pred_smt)[:, 1]
kriging_test_pred_smt = kriging_surrogate_smt.predict_values(np.array(x_test))
kriging_test_pred_smt = pyconvert(Matrix{Float64}, kriging_test_pred_smt)[:, 1]
xgboost_test_pred = xgboost_surrogate.(x_test)
radial_test_pred = radial_surrogate.(x_test)
kriging_test_pred = kriging_surrogate.(x_test)
loba_test_pred = loba_surrogate.(x_test)
150-element Vector{Float64}:
  0.8108835066814468
 -0.4294251226824539
  0.6930781912373691
 -0.8291354241161084
 -0.8631015908186067
  0.7127771609290205
  0.9548816942483735
 -0.2671508744082067
 -0.06580375073603555
  0.9177565056062115
  ⋮
  0.9819697158710072
  0.7948308679781109
  0.9958299131294224
  0.5915997924790721
  0.9873016555617158
  0.40536814590454023
  0.9540727515167781
  0.1532703816311498
 -0.09446451573860637

Define the MSE function

function calculate_mse(predictions, true_values)
    return mean((predictions .- true_values).^2)  # Calculate mean of squared errors
end
calculate_mse (generic function with 1 method)

Calculate MSE for the models

## Training MSE
mse_radial_train_smt = calculate_mse(radial_train_pred_smt, y_train)
mse_krig_train_smt = calculate_mse(kriging_train_pred_smt, y_train)
mse_xgb_train = calculate_mse(xgboost_train_pred, y_train)
mse_radial_train = calculate_mse(radial_train_pred, y_train)
mse_krig_train = calculate_mse(kriging_train_pred, y_train)
mse_loba_train = calculate_mse(loba_train_pred, y_train)

## Test MSE
mse_radial_test_smt = calculate_mse(radial_test_pred_smt, y_test)
mse_krig_test_smt = calculate_mse(kriging_test_pred_smt, y_test)
mse_xgb_test = calculate_mse(xgboost_test_pred, y_test)
mse_radial_test = calculate_mse(radial_test_pred, y_test)
mse_krig_test = calculate_mse(kriging_test_pred, y_test)
mse_loba_test = calculate_mse(loba_test_pred, y_test)
7.81987016329584e-8

Compare MSE

models = ["XGBoost", "Radial Basis", "Kriging", "Lobachevsky", "Radial Basis (SMT)", "Kriging (SMT)"]
train_mses = [mse_xgb_train, mse_radial_train, mse_krig_train, mse_loba_train, mse_radial_train_smt, mse_krig_train_smt]
test_mses = [mse_xgb_test, mse_radial_test, mse_krig_test, mse_loba_test, mse_radial_test_smt, mse_krig_test_smt]
mses = sort(collect(zip(test_mses, train_mses, models)))
pretty_table(hcat(getindex.(mses, 3), getindex.(mses, 2), getindex.(mses, 1)), column_labels=["Model", "Training MSE", "Test MSE"])
┌────────────────────┬──────────────┬─────────────┐
│              Model │ Training MSE │    Test MSE │
├────────────────────┼──────────────┼─────────────┤
│      Kriging (SMT) │  3.51665e-18 │ 7.24057e-18 │
│            Kriging │  3.50414e-14 │ 1.33384e-12 │
│ Radial Basis (SMT) │  2.34051e-14 │ 6.23002e-12 │
│        Lobachevsky │   3.7375e-19 │  7.81987e-8 │
│       Radial Basis │  1.02176e-30 │ 0.000311387 │
│            XGBoost │   0.00176894 │  0.00729346 │
└────────────────────┴──────────────┴─────────────┘

Plot predictions

xs = -5:0.01:5

radial_pred_smt = radial_surrogate_smt.predict_values(np.array(xs))
radial_pred_smt = pyconvert(Matrix{Float64}, radial_pred_smt)[:, 1]
kriging_pred_smt = kriging_surrogate_smt.predict_values(np.array(xs))
kriging_pred_smt = pyconvert(Matrix{Float64}, kriging_pred_smt)[:, 1]

plot(xs, radial_pred_smt, label="Radial Basis (SMT)", legend=:top, color=:cyan)
plot!(xs, kriging_surrogate.(xs), label="Kriging (SMT)", legend=:top, color=:magenta)
plot!(xs, tensor_product_function.(xs), label="True function", legend=:top, color=:black)
plot!(xs, xgboost_surrogate.(xs), label="XGBoost", legend=:top, color=:green)
plot!(xs, radial_surrogate.(xs), label="Radial Basis", legend=:top, color=:red)
plot!(xs, kriging_surrogate.(xs), label="Kriging", legend=:top, color=:blue)
plot!(xs, loba_surrogate.(xs), label="Lobachevsky", legend=:top, color=:purple)

Time evaluation

time_original = @belapsed tensor_product_function.(x_test)
time_radial_smt = @belapsed radial_surrogate_smt.predict_values(np.array(x_test))
time_krig_smt = @belapsed kriging_surrogate_smt.predict_values(np.array(x_test))
time_xgb = @belapsed xgboost_surrogate.(x_test)
time_radial = @belapsed radial_surrogate.(x_test)
time_krig = @belapsed kriging_surrogate.(x_test)
time_loba = @belapsed loba_surrogate.(x_test)
0.000962981

Compare time performance

times = ["XGBoost" => time_xgb, "Radial Basis" => time_radial, "Kriging" => time_krig, "Lobachevsky" => time_loba, "Radial Basis (SMT)" => time_radial_smt, "Kriging (SMT)" => time_krig_smt, "Original Function" => time_original]
sorted_times = sort(times, by=x->x[2])
pretty_table(hcat(first.(sorted_times), last.(sorted_times)), column_labels=["Model", "Time(s)"])
┌────────────────────┬─────────────┐
│              Model │     Time(s) │
├────────────────────┼─────────────┤
│  Original Function │  1.97889e-6 │
│       Radial Basis │   1.9029e-5 │
│ Radial Basis (SMT) │ 0.000349607 │
│            Kriging │ 0.000418347 │
│      Kriging (SMT) │ 0.000799343 │
│        Lobachevsky │ 0.000962981 │
│            XGBoost │      6.0856 │
└────────────────────┴─────────────┘

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/Surrogates","tensor_product.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 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
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_DEPOT_PATH = /home/crackauc/github-runners/amdci8-1/.julia
  JULIA_NUM_THREADS = auto
  JULIA_PYTHONCALL_EXE = /home/crackauc/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Surrogates/.CondaPkg/.pixi/envs/default/bin/python

Package Information:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Surrogates/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
⌃ [992eb4ea] CondaPkg v0.2.33
  [91a5bcdd] Plots v1.41.7
  [08abe8d2] PrettyTables v3.4.8
  [6099a3de] PythonCall v0.9.35
  [31c91b34] SciMLBenchmarks v0.2.1
  [10745b16] Statistics v1.11.5
  [6fc51010] Surrogates v7.10.3
⌃ [009559a3] XGBoost v2.5.2
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Surrogates/Manifest.toml`
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⌅ [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.42.0+0
  [89763e89] Libtiff_jll v4.7.3+0
  [38a345b3] Libuuid_jll v2.42.0+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [458c3c95] OpenSSL_jll v3.5.8+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.58.2+0
  [30392449] Pixman_jll v0.46.4+0
  [c0090381] Qt6Base_jll v6.10.2+2
  [629bc702] Qt6Declarative_jll v6.10.2+2
  [ce943373] Qt6ShaderTools_jll v6.10.2+1
  [6de9746b] Qt6Svg_jll v6.10.2+0
  [e99dba38] Qt6Wayland_jll v6.10.2+1
  [f50d1b31] Rmath_jll v0.5.2+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [bf13095a] XGBoost_GPU_jll v2.1.5+0
  [a5c6f535] XGBoost_jll v2.1.5+0
  [ffd25f8a] XZ_jll v5.8.4+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
  [a51aa0fd] Xorg_libXi_jll v1.8.4+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.134+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.58+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [f8abcde7] micromamba_jll v2.3.1+0
  [009596ad] mtdev_jll v1.1.7+0
  [4d7b5844] pixi_jll v0.76.2+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [9abbd945] Profile v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
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`