Surrogates.jl: Surrogate models and optimization for scientific machine learning

A surrogate model is an approximation method that mimics the behavior of a computationally expensive simulation. In more mathematical terms: suppose we are attempting to optimize a function $\; f(p)$, but each calculation of $\; f$ is very expensive. It may be the case that we need to solve a PDE for each point or use advanced numerical linear algebra machinery, which is usually costly. The idea is then to develop a surrogate model $\; g$ which approximates $\; f$ by training on previous data collected from evaluations of $\; f$. The construction of a surrogate model can be seen as a three-step process:

Sample selection
Construction of the surrogate model
Surrogate optimization

The sampling methods are super important for the behavior of the surrogate. Sampling can be done through QuasiMonteCarlo.jl, all the functions available there can be used in Surrogates.jl.

The available surrogates are:

Linear
Radial Basis
Wendland
Inverse Distance
Second Order Polynomial
Lobachevsky
Earth (multivariate adaptive regression splines)
Kriging
Gradient Enhanced Kriging (GEK)
GEKPLS
KPLS and KPLSK
Variable Fidelity
Custom Kriging (via AbstractGPs)
Neural Network (via Flux)
Gradient Enhanced Neural Network (via Flux)
Support Vector Machine (via LIBSVM)
Gradient Boosted Trees (via XGBoost)
Polynomial Chaos (via PolyChaos)
Mixture of Experts (via GaussianMixtures)

After the surrogate is built, we need to optimize it with respect to some objective function. That is, simultaneously looking for a minimum and sampling the most unknown region. The available optimization methods are:

Stochastic RBF (SRBF)
Lower confidence-bound strategy (LCBS)
Expected improvement (EI)
Dynamic coordinate search (DYCORS)

Multi-output Surrogates

In certain situations, the function being modeled may have a multi-dimensional output space. In such a case, the surrogate models can take advantage of correlations between the observed output variables to obtain more accurate predictions.

When constructing the original surrogate, each element of the passed y vector should itself be a vector. For example, the following y are all valid.

using Surrogates
using StaticArrays

x = sample(5, [0.0; 0.0], [1.0; 1.0], SobolSample())
f_static = (x) -> StaticVector(x[1], log(x[2]*x[1]))
f = (x) -> [x, log(x)/2]

y = f_static.(x)
y = f.(x)

Currently, the following are implemented as multi-output surrogates:

Radial Basis
Neural Network (via Flux)
Second Order Polynomial
Inverse Distance
Lobachevsky
Custom Kriging (via AbstractGPs)

Gradients

The surrogates implemented here are all automatically differentiable via Zygote. Because of this property, surrogates are useful models for processes which aren't explicitly differentiable, and can be used as layers in, for instance, Flux models.

Installation

Surrogates is registered in the Julia General Registry. In the REPL:

using Pkg
Pkg.add("Surrogates")

Contributing

Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.
See the SciML Style Guide for common coding practices and other style decisions.
There are a few community forums:
- The #diffeq-bridged and #sciml-bridged channels in the Julia Slack
- The #diffeq-bridged and #sciml-bridged channels in the Julia Zulip
- On the Julia Discourse forums
- See also SciML Community page

Quick example

using Surrogates
num_samples = 10
lb = 0.0
ub = 10.0

#Sampling
x = sample(num_samples, lb, ub, SobolSample())
f = x -> log(x) * x^2 + x^3
y = f.(x)

#Creating surrogate
alpha = 2.0
n = 6
my_lobachevsky = LobachevskySurrogate(x, y, lb, ub, alpha = alpha, n = n)

#Approximating value at 5.0
value = my_lobachevsky(5.0)

#Adding more data points
surrogate_optimize!(f, SRBF(), lb, ub, my_lobachevsky, RandomSample())

#New approximation
value = my_lobachevsky(5.0)

165.22756662384006

Reproducibility

The documentation of this SciML package was built using these direct dependencies,

Status `~/work/Surrogates.jl/Surrogates.jl/docs/Project.toml`
  [99985d1d] AbstractGPs v0.5.25
  [e30172f5] Documenter v1.19.0
  [587475ba] Flux v0.16.15
  [f6369f11] ForwardDiff v1.4.7
  [cc18c42c] GaussianMixtures v0.3.14
  [961ee093] ModelingToolkit v11.46.1
  [1dea7af3] OrdinaryDiffEq v7.8.1
  [91a5bcdd] Plots v1.41.7
  [8d666b04] PolyChaos v2.2.0
  [1fd47b50] QuadGK v2.11.3
  [6fc51010] Surrogates v8.0.0 `~/work/Surrogates.jl/Surrogates.jl`
  [009559a3] XGBoost v2.5.5
  [e88e6eb3] Zygote v0.7.15

and using this machine and Julia version.

Julia Version 1.13.3
Commit 67ac47b9cd8 (2026-10-25 15:28 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × Intel(R) Xeon(R) 6975P-C
  WORD_SIZE: 64
  LLVM: libLLVM-20.1.8 (ORCJIT, graniterapids)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_DEBUG = Documenter

A more complete overview of all dependencies and their versions is also provided.

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  [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 v1.8.14+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 v1.8.5+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.13+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 v1.5.0+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.48.0+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.15.3+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 v1.6.59+1
  [9a156e7d] libva_jll v2.24.1+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2023.1.0+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 v1.7.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v1.0.0
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.13.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.13.0
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v1.0.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.13.0
  [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 v1.5.5+2
  [deac9b47] LibCURL_jll v8.22.0+0
  [e37daf67] LibGit2_jll v1.9.7+0
  [29816b5a] LibSSH2_jll v1.11.104+0
  [14a3606d] MozillaCACerts_jll v2026.10.20
  [4536629a] OpenBLAS_jll v0.3.30+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.8+0
  [efcefdf7] PCRE2_jll v10.47.0+0
  [bea87d4a] SuiteSparse_jll v7.10.1+0
  [83775a58] Zlib_jll v1.3.2+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.70.0+0
  [3f19e933] p7zip_jll v17.8.3+0
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`

You can also download the manifest file and the project file.