Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators

The data and code for the paper Solving forward and inverse partial differential equation problems on unknown manifolds via physics-informed neural operators, SIAM Journal on Scientific Computing, 48 (1), C136–C163, 2026.

Data

The datasets are generated from the MATLAB code in the data folder. The repository includes three approaches for approximating differential operators on unknown manifolds:

• Diffusion Maps
• Radial Basis Functions
• Generalized Moving Least Squares

The full datasets are available on OneDrive.

Code

The code for solving the forward and inverse problems on unknown manifolds can be found in the src folder.

• Second-order linear elliptic PDE on torus
• Second-order linear elliptic PDE on semi-torus with Dirichlet boundary conditions
• Nonlinear PDE on torus
• Second-order linear elliptic PDE on Bunny
• Application to solving Bayesian inverse problems

For physics-informed training with observational data, consistency between the data and the governing PDE is essential. We therefore provide a validation procedure to check whether the generated data is consistent with the PDE (see example).

Cite this work

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

@article{jiao2026manifold,
  author  = {Jiao, Anran and Yan, Qile and Harlim, John and Lu, Lu},
  title   = {Solving forward and inverse partial differential equation problems on unknown manifolds via physics-informed neural operators},
  journal = {SIAM Journal on Scientific Computing},
  volume  = {48},
  number  = {1},
  pages   = {C136-C163},
  year    = {2026},
  doi     = {https://doi.org/10.1137/24M1675254}
}

Questions

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