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One-shot learning for solution operators of partial differential equations

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# One-shot learning for solution operators of partial differential equations

The data and code for the paper [One-shot learning for solution operators of partial differential equations](https://www.nature.com/articles/s41467-025-63076-z), *Nature Communications*, 16, 8386, 2025.

## Data

The datasets in the study are generated directly from the code in [data](data) folder. For the following experiments, the data can be found on OneDrive:
- [Data for 2D nonlinear Poisson with a circle cutout](https://yaleedu-my.sharepoint.com/:f:/g/personal/lu_lu_yale_edu/EsGbouMHSuJCtJKKiRTYFxoBJVs4el4iL1xUSvufzN7otA?e=GMcNYd) generated by COMSOL

## Code

The code for training the local solution operators, FPI, LOINN and cLOINN approaches can be found in [src](src). Each folder within this directory is named according to the PDEs. Files ending in "_G.py" is the code for training the local solution operators. Files ending with "FPI", "cLOINN", or "LOINN" correspond to each approach. A Jupyter Notebook example for the nonlinear diffusion–reaction equation is provided here: 📘[Example_nonlinear_diffusion-reaction_equation.ipynb](src/Example_nonlinear_diffusion-reaction_equation.ipynb).

- [1D Poisson equation](src/1d_poisson/)
- [Linear diffusion equation](src/linear_diffusion/)
- [nonlinear diffusion-reaction equation](src/nonlinear_diffusion/) and [testing on diverse testing functions](src/nonlinear_diffusion_training_selection/)
- [Advection equation](src/advection/)
- [2D nonlinear Poisson equation](src/2d_poisson_nonlinear/)
- [2D nonlinear Poisson equation in a square domain with a circle cutout of radius 0.2](src/2d_nonlinear_poisson_circle/) and [radius 0.18](src/2d_nonlinear_poisson_smaller_circle/)
- [Diffusion-reaction system in porous media](src/diffusion_reaction_porous_media/)
- [Application in spatial infection spread through heterogeneous populations](src/1d_SIR_model/)

## Cite this work

If you use this data or code for academic research, you are encouraged to cite the following paper:

```
@article{jiao2025one,
  author  = {Jiao, Anran and He, Haiyang and Ranade, Rishikesh and Pathak, Jay and Lu, Lu},
  title   = {One-shot learning for solution operators of partial differential equations},
  journal = {Nature Communications},
  volume  = {16},
  pages   = {8386},
  year    = {2025},
  doi     = {https://doi.org/10.1038/s41467-025-63076-z}
}
```

## Questions

To get help on how to use the data or code, simply open an issue in the GitHub "Issues" section.

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