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
Kriging
Custom Kriging provided with Stheno
Neural Network
Support Vector Machine
Random Forest
Second Order Polynomial
Inverse Distance

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
Custom Kriging (via Stheno)

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)

135.03840648224897

Reproducibility

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

Status `~/work/Surrogates.jl/Surrogates.jl/docs/Project.toml`
  [99985d1d] AbstractGPs v0.5.23
  [e30172f5] Documenter v1.9.0
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  [91a5bcdd] Plots v1.40.11
  [1fd47b50] QuadGK v2.11.2
  [6fc51010] Surrogates v6.11.0 `~/work/Surrogates.jl/Surrogates.jl`
  [78aa1720] SurrogatesAbstractGPs v0.1.0 `~/work/Surrogates.jl/Surrogates.jl/lib/SurrogatesAbstractGPs`
  [4f55584b] SurrogatesFlux v0.1.1 `~/work/Surrogates.jl/Surrogates.jl/lib/SurrogatesFlux`
  [778709c9] SurrogatesMOE v0.1.4 `~/work/Surrogates.jl/Surrogates.jl/lib/SurrogatesMOE`
  [50679fc6] SurrogatesPolyChaos v0.1.0 `~/work/Surrogates.jl/Surrogates.jl/lib/SurrogatesPolyChaos`
  [3fee2672] SurrogatesRandomForest v0.1.1 `~/work/Surrogates.jl/Surrogates.jl/lib/SurrogatesRandomForest`
  [eb0c118b] SurrogatesSVM v0.1.0 `~/work/Surrogates.jl/Surrogates.jl/lib/SurrogatesSVM`
  [009559a3] XGBoost v2.5.1
  [e88e6eb3] Zygote v0.7.4

and using this machine and Julia version.

Julia Version 1.11.4
Commit 8561cc3d68d (2025-03-10 11:36 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
Threads: 1 default, 0 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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  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf 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 v0.8.1+4
  [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`

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