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, Nature Communications, 16, 8386, 2025.

Data

The datasets in the study are generated directly from the code in data folder. For the following experiments, the data can be found on OneDrive:

• Data for 2D nonlinear Poisson with a circle cutout generated by COMSOL

Code

The code for training the local solution operators, FPI, LOINN and cLOINN approaches can be found in 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.

• 1D Poisson equation
• Linear diffusion equation
• nonlinear diffusion-reaction equation and testing on diverse testing functions
• Advection equation
• 2D nonlinear Poisson equation
• 2D nonlinear Poisson equation in a square domain with a circle cutout of radius 0.2 and radius 0.18
• Diffusion-reaction system in porous media
• Application in spatial infection spread through heterogeneous populations

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.