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)165.22756662384006Reproducibility
The documentation of this SciML package was built using these direct dependencies,
Status `~/work/Surrogates.jl/Surrogates.jl/docs/Project.toml`
[99985d1d] AbstractGPs v0.5.24
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⌃ [587475ba] Flux v0.16.10
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Info Packages marked with ⌃ have new versions available and may be upgradable.and using this machine and Julia version.
Julia Version 1.13.0
Commit d1c37793dd2 (2026-09-09 19:00 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-20.1.8 (ORCJIT, znver3)
GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)
Environment:
JULIA_DEBUG = DocumenterA more complete overview of all dependencies and their versions is also provided.
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[458c3c95] OpenSSL_jll v3.5.6+0
[efcefdf7] PCRE2_jll v10.46.0+0
[bea87d4a] SuiteSparse_jll v7.10.1+0
[83775a58] Zlib_jll v1.3.1+2
[3161d3a3] Zstd_jll v1.5.7+1
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.67.1+0
[3f19e933] p7zip_jll v17.8.2+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.