Fourier Neural Mappings

Fourier Neural Mappings (FNMs) generalize Fourier Neural Operators (FNOs) by allowing the input space and/or the output space to be finite-dimensional (instead of both being purely infinite-dimensional function spaces as with FNO). This is especially relevant for surrogate modeling tasks in uncertainty quantification, inverse problems, and design optimization, where a finite number of parameters or quantities of interest (QoIs) characterize the inputs and/or outputs.

In particular, FNMs are able to accommodate

• nonlinear functions (V2V): Fourier Neural Networks mapping vectors to vectors (going through a latent function space in between);
• nonlinear functionals (F2V): Fourier Neural Functionals mapping functions to vectors (a.k.a. nonlinear encoders);
• nonlinear decoders (V2F): Fourier Neural Decoders mapping vectors to functions; and of course
• nonlinear operators (F2F): Fourier Neural Operators mapping functions to functions.

In fourier-neural-mappings, the network layers in all four types of mappings above are efficiently implemented (via FFT) in Fourier space in a function-space consistent way. The code defaults to running on GPU, if one is available.

Installation

The command

conda env create -f Project.yml

creates an environment called fno. PyTorch will be installed in this step.

Activate the environment with

conda activate fno

and deactivate with

conda deactivate

The advection-diffusion example requires additional packages to generate and process the data; please refer to the README instructions within that directory for more details.

Data

The data may be downloaded at , which contains three *.zip files:

1. advection_diffusion: train and test sets for KLE dimension 2, 20, 1000.
2. airfoil: deformation map (X,Y) coordinates, pressure field, and control nodes.
3. homogenization: V2V, F2V, V2F, and F2F formats.

The data are stored as PyTorch *.pt files, *.npy arrays, or pickle *.pkl files.

Huang, D. Z., Nelsen, N. H., & Trautner, M. (2024). An operator learning perspective on parameter-to-observable maps [Data set]. CaltechDATA. https://doi.org/10.22002/r5ga1-55d06. Feb. 12, 2024.

References

The main reference that explains the Fourier Neural Mappings framework is the paper ``An operator learning perspective on parameter-to-observable maps'' by Daniel Zhengyu Huang, Nicholas H. Nelsen, and Margaret Trautner. Other relevant references include:

• Fourier Neural Operator for Parametric Partial Differential Equations
• Fourier Neural Operator with Learned Deformations for PDEs on General Geometries
• Learning Homogenization for Elliptic Operators

Citing

If you use fourier-neural-mappings in an academic paper, please cite the main reference ``An operator learning perspective on parameter-to-observable maps'' as follows:

@article{huang2024fnm,
  title={An operator learning perspective on parameter-to-observable maps},
  author={Huang, Daniel Zhengyu and Nelsen, Nicholas H and Trautner, Margaret},
  journal={arXiv preprint arXiv:2402.06031},
  year={2024}
}

Contribute

You are welcome to submit an issue for any questions related to fourier-neural-mappings or to contribute to the code by submitting pull requests.

Acknowledgements

The FNO implementation in fourier-neural-mappings is adapted from the original implementation by Nikola Kovachki and Zongyi Li. The data generation code for the advection-diffusion example was provided by Zachary Morrow. The matplotlib formatting used to produce figures is adapted from the PyApprox package by John Jakeman.