Non-Uniform Neural Operator (NUNO)

This repository is the official implementation of our ICML 2023 paper: NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Data. Here are the corresponding introductory slides.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at this https URL.

If you find this work is helpful for your research, please cite us with the following BibTeX entry:

@article{liu2023nuno,
  title={NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Data},
  author={Liu, Songming and Hao, Zhongkai and Ying, Chengyang and Su, Hang and Cheng, Ze and Zhu, Jun},
  journal={arXiv preprint arXiv:2305.18694},
  year={2023}
}

Directory Structure

This repository is organized as below:

• data (folder for training and testing data)
  • channel ((2+1)D Channel Flow)
  • elasticity (2D Elasticity)
  • heatsink (3D Heatsink)
• src (folder for scripts to run experiments)
  • channel ((2+1)D Channel Flow) Note: each Python file corresponds to a single baseline method.
    • deeponet.py (to run DeepONet)
    • ...
    • ours_nuunet.py (to run one of our methods, NU-U-Net)
  • elasticity (2D Elasticity)
    • ...
  • heatsink (3D Heatsink)
    • ...
  • kd_tree (module of K-D tree algorithm, imported by the above scripts)
• util (folder for util functions)
• assets (folder for figures and slides)
• README.md (this readme)

Getting Started

1. Install necessary dependencies

   Python 3.10.8

   matplotlib==3.6.2
   scikit-learn==1.2.0
   torch==1.13.0
   tqdm==4.64.1
   

   For most experiments, you only need to install the above four libraries.

   Some (not all) baselines may also require the following libraries:

   • torch-geometric==2.2.0 (visit this repo to install the library)
     • It is required by GraphNO. If you do no run this baseline, you can ignore this library.
   • sympy==1.11.1
     • It is required by MWNO. If you do no run this baseline, you can ignore this library.

2. Download datasets

   For 2D Elasticity, we refer to this drive.

   Download all the files in folders Interp, Meshes, Omesh, and Rmesh. Then put them into the folder data/elasticity in this repo (directly move .npy files but not persevering the original folder structure).

   For (2+1)D Channel Flow and 3D Heatsink, we refer to this drive.

   Download all the files in NUNO/ChannelFlow and NUNO/Heatsink, then put them into data/channel and data/heatsink in this repo, respectively (again, directly move .npy files but not persevering the original folder structure).

3. Run experiments

   # In the root directory of this repository
   
   # To run baselines for 2D Elasticity
   python -m src.elasticity.ours_nufno
   # or one of other baselines...
   # The file name (in src/elasticity) should be able to represent the baseline method corresponding to the script.
   python -m src.elasticity.geofno
   
   # To run scripts in (2+1)D Channel Flow
   python -m src.channel.ours_nufno
   # or one of other baselines...
   
   # To run scripts in 3D Heatsink
   python -m src.heatsink.ours_nufno
   # or one of other baselines...

Experimental Results

2D Elasticity

(2+1)D Channel Flow

Note: "oversampling ratio" is a measure for the mesh size.

3D Heatsink