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A collection of analytical benchmark functions in multiple fidelities

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MF2: Multi-Fidelity-Functions

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License: GPL v3 Project Status: Active – The project has reached a stable, usable state and is being actively developed.
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The mf2 package provides consistent, efficient and tested Python implementations of a variety of multi-fidelity benchmark functions. The goal is to simplify life for numerical optimization researchers by saving time otherwise spent reimplementing and debugging the same common functions, and enabling direct comparisons with other work using the same definitions, improving reproducibility in general.

A multi-fidelity function usually reprensents an objective which should be optimized. The term 'multi-fidelity' refers to the fact that multiple versions of the objective function exist, which differ in the accuracy to describe the real objective. A typical real-world example would be the aerodynamic efficiency of an airfoil, e.g., its drag value for a given lift value. The different fidelity levels are given by the accuracy of the evaluation method used to estimate the efficiency. Lower-fidelity versions of the objective function refer to less accurate, but simpler approximations of the objective, such as computational fluid dynamic simulations on rather coarse meshes, whereas higher fidelity levels refer to more accurate but also much more demanding evaluations such as prototype tests in wind tunnels. The hope of multi-fildelity optimization approaches is that many of the not-so-accurate but simple low-fidelity evaluations can be used to achieve improved results on the realistic high-fidelity version of the objective where only very few evaluations can be performed.

The only dependency of the mf2 package is the numpy package.

Documentation is available at


The recommended way to install mf2 is with Python's pip:

python3 -m pip install --user mf2

or alternatively using conda:

conda install -c conda-forge mf2

For the latest version, you can install directly from source:

python3 -m pip install --user

To work in your own version locally, it is best to clone the repository first, and additionally install the dev-requirements:

git clone
cd mf2
python3 -m pip install --user -e .[dev]

Example Usage

import mf2
import numpy as np

# set numpy random seed for reproducibility
# generate 5 random samples in 2D as matrix
X = np.random.random((5, 2))

# print high fidelity function values
# Out: array([36.78994906 34.3332972  50.48149005 43.0569396  35.5268224 ])

# print low fidelity function values
# Out: array([-5.8762639  -6.66852889  3.84944507 -1.56314141 -6.23242223])

For more usage examples, please refer to the full documentation on readthedocs.


Contributions to this project such as bug reports or benchmark function suggestions are more than welcome! Please refer to for more details.


The Gitter channel is the preferred way to get in touch for any other questions, comments or discussions about this package.


Was this package useful to you? Great! If this leads to a publication, we'd appreciate it if you would cite our JOSS paper:

  doi = {10.21105/joss.02049},
  url = {},
  year = {2020},
  publisher = {The Open Journal},
  volume = {5},
  number = {52},
  pages = {2049},
  author = {Sander van Rijn and Sebastian Schmitt},
  title = {MF2: A Collection of Multi-Fidelity Benchmark Functions in Python},
  journal = {Journal of Open Source Software}

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