Table of contents

• General info
• Method
• Application
• Contents
• Get started
• Contact

General info

This Git repository contains codes for the 'Deep transfer operator learning for partial differential equations under conditional shift' paper published in Nature Machine Intelligence.

Authors: Somdatta Goswami, Katiana Kontolati, Michael D. Shields, George Em Karniadakis

Method

The key idea behind transfer learning is learning machines that leverage knowledge gained from one task to improve accuracy and generalization in another different but related task.

In physics and engineering we often need the accurate solution of a PDE solved on heterogeneous but subtly correlated domains, i.e., there exists a conditional distribution mismatch.

In our latest work, we propose a novel framework which exploits information from pre-trained (source) deep neural operators (DeepONets), for fast and accurate task-specific partial differential equation (PDE) learning (target).

The key ingredients of this approach are:

• the extraction of domain-invariant features,
• the embedding of conditional distributions on a reproducing kernel Hilbert space (RKHS) and
• the self-adaptive learning of penalizing parameters which allows adaptation between source and target domains.

Application

As presented in the Table below, we demonstrate the capabilities of our approach on three classes of PDE problems where domains and physical parameters have significant differences.

We transfer information from the trained source model (DeepONet) to the target model (TL-DeepONet) and finetune it via the hybrid loss function, which allows for efficient multi-task operator learning under various distribution mismatch scenarios.

Contents

• data_generation/ - contains script to generate labeled data for all three PDE applications. For the python scripts, packages in requirements.txt need to be installed. For the Matlab scripts, the PDEToolbox is required.

• TL{1-8}/ - contains python scripts and notebooks for implementing the proposed transfer learning approach for certain applications in the paper. As a first step, the source_model.py script needs to be executed. The optimized source model parameters are saved inside a local folder Variables/. Finally, the target_model.py script can be used for fine-tuning of the target model for each target PDE task.

Get started

1. Create an Anaconda Python 3.7 virtual environment:

conda create -n tl-deeponet python==3.7
conda activate tl-deeponet

2. Clone the repo:

To clone and use this repository, run the following terminal commands:

git clone https://github.com/katiana22/TL-DeepONet.git

3. Install dependencies:

cd TL-DeepONet
pip install -r requirements.txt

Citation

If you find this GitHub repository useful for your work, please consider citing this work:

@article{goswami2022deep,
  title={Deep transfer operator learning for partial differential equations under conditional shift},
  author={Goswami, Somdatta and Kontolati, Katiana and Shields, Michael D and Karniadakis, George Em},
  journal={Nature Machine Intelligence},
  pages={1--10},
  year={2022},
  publisher={Nature Publishing Group}
}

Contact

For more information or questions please contact us at:

• kontolati@jhu.edu
• somdatta_goswami@brown.edu