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Neural-Operator Element Method (NOEM)

The data and code for the paper Neural-operator element method: Efficient and scalable finite element method enabled by reusable neural operators.

Code

This repository contains reference implementations for the Neural-Operator Element Method (NOEM)—a hybrid approach that combines the Finite Element Method (FEM) with reusable neural operators for efficient numerical simulations of partial differential equations (PDEs).

Setup

Install dependencies (Python + pip):

python -m pip install -r requirements.txt

Directory layout

• pedagogical_example/: Pedagogical examples from the Methods section.
• multiscale_1d_problem/: 1D multiscale problems (Results section).
• heat_transfer/: Heat transfer example (Results section).
• darcy_flow/: Darcy flow example (Results section).
• convexity_test/, uq_test/: Additional test cases (Supplementary Information).

Running experiments

Most experiments are driven by a run.py script. Many scripts rely on files referenced by relative paths, so it is recommended to cd into the corresponding folder first and then run python run.py.

Examples (from the repository root):

cd pedagogical_example/quadratic_coefficient
python run.py

Available entry points:

• pedagogical_example/quadratic_coefficient/run.py
• pedagogical_example/random_coefficient_functions/run.py
• multiscale_1d_problem/multiscale_coefficient/run.py
• multiscale_1d_problem/multiscale_source_term/run.py
• multiscale_1d_problem/multiscale_coefficient/results_fig_d/run.py
• heat_transfer/run.py
• darcy_flow/run.py
• convexity_test/run.py
• uq_test/run.py

Cite this work

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

@article{ouyang2026neural,
  author  = {Ouyang, Weihang and Shin, Yeonjong and Liu, Si-Wei and Lu, Lu},
  title   = {Neural-operator element method: Efficient and scalable finite element method enabled by reusable neural operators},
  journal = {Nature Computational Science},
  volume  = {6},
  pages   = {417–429},
  year    = {2026},
  doi     = {https://doi.org/10.1038/s43588-026-00974-2}
}

Questions

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