Audi and pyaudi

Audi (not the car, rather from latin: “listen!”) is an open source, header only, C++ library (exposed to python in the pyaudi package) that implements the differential algebra of Taylor truncated polynomials and a few algorithms useful for its applications (Differential Intelligence, automatic differentiation, Taylor Models, etc.)

When used for automated differentiation only, other automated differentiation codes may be more efficient on some targeted application, Audi was built to be fast and efficient across all application ranges (low/high order of derivation, one variable, many variables, sparse/dense cases).

Audi is internally powered by the truncated polynomial multiplication algorithm of the open source project obake some details of which are described in:

Biscani, Francesco. Parallel sparse polynomial multiplication on modern hardware architectures. Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. ACM, 2012.

Biscani, Francesco. Multiplication of sparse Laurent polynomials and Poisson series on modern hardware architectures. arXiv preprint arXiv:1004.4548 (2010).

Note

Audi is thread-safe and, when possible, makes use of obake fine-grained parallelization of the truncated polynomial multiplication. The benefits of this fine grained parallelization are well visible for many variables and high differentiation orders.

Important

Citing pyaudi

If you use pyaudi in your research, please cite:

Izzo, Dario, Francesco Biscani, and Sean Cowan. “pyaudi: A truncated Taylor polynomial algebra toolbox for differentiable intelligence, automatic differentiation, and verified integration applications.” Journal of Open Source Software 11.124 (2026): 9905.

@article{izzo2026pyaudi,
  title={pyaudi: A truncated Taylor polynomial algebra toolbox for differentiable intelligence, automatic differentiation, and verified integration applications.},
  author={Izzo, Dario and Biscani, Francesco and Cowan, Sean},
  journal={Journal of Open Source Software},
  volume={11},
  number={124},
  pages={9905},
  year={2026}
}

Audi is open source (GPL3) and its code available in github

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Contents:

• Quick start
  • Python
  • C++
• Installation guide
  • Packages
  • From source
  • Python bindings
• Documentation
  • Documentation (C++ header only library)
  • Documentation (Python package)
• Theory
  • The algebra of truncated polynomials
  • Elementary functions
  • Inverse functions
  • Practical Examples (to be done by hand)
  • Taylor models

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Basic Python Tutorials:

• The very basics
  • Importing stuff
  • Define some variables (gduals)
  • Create a function of these variables
  • Extracting the derivatives
  • Changing the point
  • Encapsulating f in a function call
• Understanding gduals and floats
  • Importing stuff
  • Arithmetic between floats and gduals
  • Promoting a gdual
• Understanding Taylor models
  • Importing stuff
  • Defining function
  • Taylor polynomial (with gduals)
  • Taylor model
• Using vectorized doubles
  • Importing stuff
  • The test case
  • Operating on doubles
  • Operating on vectorized doubles
  • A deeper look into the speedup

Advanced Python Tutorials:

• Differential Intelligence
  • Importing stuff
  • Controller representation and “simulator”
  • The Euler Propagation
  • Studying the effects of the weights on the behavior
  • Check that we learned the correct map
• Training Artificial Neural Networks
  • Definition of parameters
  • Neural network as a gdual expression
  • Loss function
  • Training (Gradient descent)
• High Order Taylor Maps I
  • We perform the numerical integration using floats (the standard way)
  • We perform the numerical integration using gduals (to get a HOTM)
  • We visualize the HOTM
  • How much faster is now to evaluate the Map rather than perform a new numerical integration?
• High Order Taylor Maps II
  • We perform the numerical integration using floats (the standard way)
  • We perform the numerical integration using gduals (to get a HOTM)
  • We visualize the HOTM
  • How much faster is now to evaluate the Map rather than perform a new numerical integration?
• Map (series) inversion
  • Importing stuff
  • The Double Pendulum end-point
  • The Kepler’s Equation

Taylor Model Tutorials:

• Taylor models of univariate functions
  • Importing stuff
  • Exponential
  • Sine wave
  • 6-th order power-sine
• Taylor models of multivariate functions
  • Importing stuff
  • 3-dimensional complex (real-valued) function
• Exploration of Volterra ODE system
  • Importing stuff
  • Integration function
  • Problem definition
  • Numerical propagation
  • Taylor model propagation
• Exploration of Kepler dynamics with a Neural network
  • Importing stuff
  • Define Neural Network stuff
  • Integration function
  • Define problem
  • Numerical propagation
  • Taylor model propagation

Indices and tables

• Index

• Module Index

• Search Page