Function operators are ubiquitous in mathematics and physics: They are used to describe dynamics of physical systems, such as the Navier-Stokes equations in fluid dynamics. As solutions of these systems are functions, it is natural to transfer the concept of function mapping into machine learning.

Checkout the Documentation page to learn more about operator learning in continuiti. If you prefer to jump into code directly, take a look at our collection of Examples.

continuiti implements the following neural operator architectures:

DeepONet [Lu21L]
Fourier Neural Operators [Kov23N]
BelNet [Zha23B]
and more to come…

To see what new methods, features and improvements are coming, check out the issues on GitHub.

References

[Lu21L]

Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, George Em Karniadakis.

Mar 2021

It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous …

[Kov23N]

Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs, Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar.

2023

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation …

[Zha23B]

BelNet: basis enhanced learning, a mesh-free neural operator, Zecheng Zhang, Leung Wing Tat, Hayden Schaeffer.

Aug 2023

Operator learning trains a neural network to map functions to functions. An ideal operator learning framework should be mesh-free in the sense that the training does not require a particular choice of discretization for the input functions, allows for the input and output functions to be on different domains, and is able to have different grids between samples. We propose a mesh-free neural …

References

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