Multiscale Neural Operators for Solving Time-Independent PDEs

TL;DR: We study how to solve time-independent Partial Differential Equations on large meshes and introduce a novel graph rewiring technique for this.

Multiscale Neural Operators for Solving Time-Independent PDEs

by Winfried Ripken¹ *, Lisa Coiffard¹ *, Felix Pieper¹ * and Sebastian Dziadzio².

¹ Merantix Momentum, ² Tübingen AI Center.

(*) equal contribution.

Requirement

• Run pip install -e . and pip install -r requirements.txt in root folder before using.
• Install gcloud CLI and authenticate:

gcloud auth login
gcloud auth application-default login

• The BSMS operator needs intel mkl installed, best installed via conda.
• Check data/download_data.py for downloading data

We recommend installing our repository using

pip install . -e

Training

Start training run:

python -m multiscale_operator.model.trainer --config-name={config_name}

Acknowledgement

We integrated 3 datasets:

• Darcy Flow in 2D Data courtesy under PDE Bench.
• Magnetic Field for electric motor simulations Data courtesy under Multiphysics Optimization.
• 2D Poisson Equation for Magnetostatics: Data courtesy under GNN BVP Solver.

Our operator implementations are based on the following public repositories:

• Mesh Graph Nets in Pytorch
• Bi-stride Multi-Scale GNN

Please cite the relevant publications. For our datasets:

• Darcy Flow:

@inproceedings{PDEBench2022,
author = {Takamoto, Makoto and Praditia, Timothy and Leiteritz, Raphael and MacKinlay, Dan and Alesiani, Francesco and Pflüger, Dirk and Niepert, Mathias},
title = {{PDEBench: An Extensive Benchmark for Scientific Machine Learning}},
year = {2022},
booktitle = {36th Conference on Neural Information Processing Systems (NeurIPS 2022) Track on Datasets and Benchmarks},
url = {https://arxiv.org/abs/2210.07182}
}

• Motor Dataset:

@article{botache2023enhancing,
  title={Enhancing Multi-Objective Optimization through Machine Learning-Supported Multiphysics Simulation},
  author={Botache, Diego and Decke, Jens and Ripken, Winfried and Dornipati, Abhinay and G{\"o}tz-Hahn, Franz and Ayeb, Mohamed and Sick, Bernhard},
  journal={arXiv preprint arXiv:2309.13179},
  year={2023}
}

• Magnetostatics Dataset:

@inproceedings{lotzsch2022learning,
  title={Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks},
  author={L{\"o}tzsch, Winfried and Ohler, Simon and Otterbach, Johannes},
  booktitle={ICML 2022 2nd AI for Science Workshop},
  year={2022}
}

For the benchmarked methods:

• BSMS:

@inproceedings{cao2022bi,
  title={Bi-stride multi-scale graph neural network for mesh-based physical simulation},
  author={Cao, Yadi and Chai, Menglei and Li, Minchen and Jiang, Chenfanfu},
  booktitle={International conference on machine learning},
  organization={PMLR},
  year={2023}
}

• Perceiver IO:

@inproceedings{jaegle2021perceiver,
  title={Perceiver IO: A General Architecture for Structured Inputs \& Outputs},
  author={Jaegle, Andrew and Borgeaud, Sebastian and Alayrac, Jean-Baptiste and Doersch, Carl and Ionescu, Catalin and Ding, David and Koppula, Skanda and Zoran, Daniel and Brock, Andrew and Shelhamer, Evan and others},
  booktitle={International Conference on Learning Representations},
  year={2021}
}

• Mesh Graph Nets (MGN):

@inproceedings{pfaff2020learning,
  title={Learning Mesh-Based Simulation with Graph Networks},
  author={Pfaff, Tobias and Fortunato, Meire and Sanchez-Gonzalez, Alvaro and Battaglia, Peter},
  booktitle={International Conference on Learning Representations},
  year={2020}
}