Convolutional Neural Operators for robust and accurate learning of PDEs

• This repository is the official implementation of the paper Convolutional Neural Operators for robust and accurate learning of PDEs (see https://arxiv.org/pdf/2302.01178.pdf)
• The paper was presented at NeurIPS 2023
• Representative PDE Benchmarks (RPB) are available at https://zenodo.org/records/10406879
• Read our blog about CNOs at https://link.medium.com/Mht8Th5OhFb
• This repository also covers CNO codes used in the paper Poseidon: Efficient Foundation Models for PDEs (see github page).

The CNO is tested on a novel set of benchmarks, termed as Representative PDE Benchmarks (RPB). The CNO is either on-par or outperformed the tested baselines on all the benchmarks, both when testing in-distribution as well as in out-of-distribution testing.

Relative median L¹ test errors, for both in- and out-of-distribution testing, for different benchmarks and models..

We assess the test errors of the CNO and other baselines at different testing resolutions notably, for the Navier-Stokes equations benchmarks. We observe that in this case, the CNO is the only model that demonstrates approximate error invariance with respect to test resolution.

The CNO model has almost constant testing error across different resolutions (Navier-Stokes).

Code Instructions:

1. The original CNO code from NeurIPS 2023 is located in the folder CNO2d_original_version

   • All the instructions for this version can be found in the readme.md file in the folder
   • The code is more complex to configure compared to the vanilla CNO code (see below)

2. Vanilla CNO2d and CNO1d versions are located in the folders CNO2d_vanilla_torch_version and CNO1d_vanilla_torch_version

   • All the instructions for these versions can be found in the readme.md files in the folders
   • The models are termed as "vanilla CNO" as the interpolation filters cannot be manually designed
   • The codes do not utilize the CUDA kernel, making them simple to configure

3. Codes for Time-Dependent CNO2d are located in the folder - CNO2d_time_dependent_&_foundation_model

   • The Time-Dependent CNO code is used in the paper Poseidon: Efficient Foundation Models for PDEs
   • All the instructions for the Time-Dependent CNO can be found in the readme.md file in the folder

4. Codes for CNO - Foundation Model are located in the folder - CNO2d_time_dependent_&_foundation_model

   • The CNO - Foundation Model (109M) code is used in the paper Poseidon: Efficient Foundation Models for PDEs
   • All the instructions for the CNO - Foundation Model can be found in the readme.md file in the folder
   • Codes for finetuning the CNO-FM are also located in the folder.

5. Codes for the other baselines are located in the folder _OtherModels

Datasets

We cover instances of the Poisson, Wave, Navier-Stokes, Allen-Cahn, Transport and Compressible Euler equations and Darcy flow. Data can be downloaded from https://zenodo.org/records/10406879 (~2.4GB).

Alternatively, run the script download_data.py which downloads all required data into the appropriate folder (it requires 'wget' to be installed on your system).

python3 download_data.py

Poseidon: Efficient Foundation Models for PDEs

We also provide all datasets used in the paper Poseidon: Efficient Foundation Models for PDEs on the 🤗 Hub. You can download them from the respective collections:

• 🤗 Hub – Pretraining Datasets
• 🤗 Hub – Downstream Tasks

Please also visit Poseidon github page.

Citation

If you use our models, code, or datasets, please consider citing our paper:

@misc{CNO,
      title={Convolutional Neural Operators for robust and accurate learning of PDEs}, 
      author={Bogdan Raonić and Roberto Molinaro and Tim De Ryck and Tobias Rohner and Francesca Bartolucci and Rima Alaifari and Siddhartha Mishra and Emmanuel de Bézenac},
      year={2023},
      eprint={2302.01178},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}