PDEBench

The code repository for the NeurIPS 2022 paper PDEBench: An Extensive Benchmark for Scientific Machine Learning

🎉 SimTech Best Paper Award 2023 🎊

PDEBench provides a diverse and comprehensive set of benchmarks for scientific machine learning, including challenging and realistic physical problems. This repository consists of the code used to generate the datasets, to upload and download the datasets from the data repository, as well as to train and evaluate different machine learning models as baselines. PDEBench features a much wider range of PDEs than existing benchmarks and includes realistic and difficult problems (both forward and inverse), larger ready-to-use datasets comprising various initial and boundary conditions, and PDE parameters. Moreover, PDEBench was created to make the source code extensible and we invite active participation from the SciML community to improve and extend the benchmark.

Created and maintained by Makoto Takamoto <[email protected], [email protected]>, Timothy Praditia <[email protected]>, Raphael Leiteritz, Dan MacKinlay, Francesco Alesiani, Dirk Pflüger, and Mathias Niepert.

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Datasets and Pretrained Models

We also provide datasets and pretrained machine learning models.

PDEBench Datasets: https://darus.uni-stuttgart.de/dataset.xhtml?persistentId=doi:10.18419/darus-2986

PDEBench Pre-Trained Models: https://darus.uni-stuttgart.de/dataset.xhtml?persistentId=doi:10.18419/darus-2987

DOIs

Installation

Using pip

Locally:

pip install --upgrade pip wheel
pip install .

From PyPI:

pip install pdebench

To include dependencies for data generation:

pip install "pdebench[datagen310]"
pip install ".[datagen310]" # locally

or

pip install "pdebench[datagen39]"
pip install ".[datagen39]" # locally

GPU Support

For GPU support there are additional platform-specific instructions:

For PyTorch, the latest version we support is v1.13.1 see previous-versions/#linux - CUDA 11.7.

For JAX, which is approximately 6 times faster for simulations than PyTorch in our tests, see jax#pip-installation-gpu-cuda-installed-via-pip

Installation using conda:

If you like you can also install dependencies using anaconda, we suggest to use mambaforge as a distribution. Otherwise you may have to enable the conda-forge channel for the following commands.

Starting from a fresh environment:

conda create -n myenv python=3.9
conda activate myenv

Install dependencies for model training:

conda install deepxde hydra-core h5py -c conda-forge

According to your hardware availability, either install PyTorch with CUDA support:

• see previous-versions/#linux - CUDA 11.7.

conda install pytorch==1.13.1 torchvision==0.14.1 torchaudio==0.13.1 pytorch-cuda=11.7 -c pytorch -c nvidia

• or CPU only binaries.

conda install pytorch==1.13.1 torchvision==0.14.1 torchaudio==0.13.1 cpuonly -c pytorch

Optional dependencies for data generation:

conda install clawpack jax jaxlib python-dotenv

Configuring DeepXDE

In our tests we used PyTorch as backend for DeepXDE. Please follow the documentation to enable this.

Data Generation

The data generation codes are contained in data_gen:

• gen_diff_react.py to generate the 2D diffusion-reaction data.
• gen_diff_sorp.py to generate the 1D diffusion-sorption data.
• gen_radial_dam_break.py to generate the 2D shallow-water data.
• gen_ns_incomp.py to generate the 2D incompressible inhomogeneous Navier-Stokes data.
• plot.py to plot the generated data.
• uploader.py to upload the generated data to the data repository.
• .env is the environment data to store Dataverse URL and API token to upload the generated data. Note that the filename should be strictly .env (i.e. remove the example from the filename)
• configs directory contains the yaml files storing the configuration for the simulation. Arguments for the simulation are problem-specific and detailed explanation can be found in the simulation scripts.
• src directory contains the simulation scripts for different problems: sim_diff_react-py for 2D diffusion-reaction, sim_diff_sorp.py for 1D diffusion-sorption, and swe for the shallow-water equation.

Data Generation for 1D Advection/Burgers/Reaction-Diffusion/2D DarcyFlow/Compressible Navier-Stokes Equations

The data generation codes are contained in data_gen_NLE:

• utils.py util file for data generation, mainly boundary conditions and initial conditions.

• AdvectionEq directory with the source codes to generate 1D Advection equation training samples

• BurgersEq directory with the source codes to generate 1D Burgers equation training samples

• CompressibleFluid directory with the source codes to generate compressible Navier-Stokes equations training samples

  • ReactionDiffusionEq directory with the source codes to generate 1D Reaction-Diffusion equation training samples (Note: DarcyFlow data can be generated by run_DarcyFlow2D.sh in this folder.)

• save directory saving the generated training samples

A typical example to generate training samples (1D Advection Equation): (in data_gen/data_gen_NLE/AdvectionEq/)

python3 advection_multi_solution_Hydra.py +multi=beta1e0.yaml

which is assumed to be performed in each directory.

Examples for generating other PDEs are provided in run_trainset.sh in each PDE's directories. The config files for Hydra are stored in config directory in each PDE's directory.

Data Transformaion and Merge into HDF5 format

1D Advection/Burgers/Reaction-Diffusion/2D DarcyFlow/Compressible Navier-Stokes Equations save data as a numpy array. So, to read those data via our dataloaders, the data transformation/merge should be performed. This can be done using data_gen_NLE/Data_Merge.py whose config file is located at: data_gen/data_gen_NLE/config/config.yaml. After properly setting the parameters in the config file (type: name of PDEs, dim: number of spatial-dimension, bd: boundary condition), the corresponding HDF5 file could be obtained as:

python3 Data_Merge.py

Configuration

You can set the default values for data locations for this project by putting config vars like this in the .env file:

WORKING_DIR=~/Data/Working
ARCHIVE_DATA_DIR=~/Data/Archive

There is an example in example.env.

Data Download

The download scripts are provided in data_download. There are two options to download data.

1. Using download_direct.py (recommended)
   • Retrieves data shards directly using URLs. Sample command for each PDE is given in the README file in the data_download directory.
2. Using download_easydataverse.py (might be slow and you could encounter errors/issues; hence, not recommended!)
   • Use the config files from the config directory that contains the yaml files storing the configuration. Any files in the dataset matching args.filename will be downloaded into args.data_folder.

Baseline Models

In this work, we provide three different ML models to be trained and evaluated against the benchmark datasets, namely FNO, U-Net, and PINN. The codes for the baseline model implementations are contained in models:

• train_models_forward.py is the main script to train and evaluate the model. It will call on model-specific script based on the input argument.
• train_models_inverse.py is the main script to train and evaluate the model for inverse problems. It will call on model-specific script based on the input argument.
• metrics.py is the script to evaluate the trained models based on various evaluation metrics described in our paper. Additionally, it also plots the prediction and target data.
• analyse_result_forward.py is the script to convert the saved pickle file from the metrics calculation script into pandas dataframe format and save it as a CSV file. Additionally it also plots a bar chart to compare the results between different models.
• analyse_result_inverse.py is the script to convert the saved pickle file from the metrics calculation script into pandas dataframe format and save it as a CSV file. This script is used for the inverse problems. Additionally it also plots a bar chart to compare the results between different models.
• fno contains the scripts of FNO implementation. These are partly adapted from the FNO repository.
• unet contains the scripts of U-Net implementation. These are partly adapted from the U-Net repository.
• pinn contains the scripts of PINN implementation. These utilize the DeepXDE library.
• inverse contains the model for inverse model based on gradient.
• config contains the yaml files for the model training input. The default templates for different equations are provided in the args directory. User just needs to copy and paste them to the args keyword in the config.yaml file.

An example to run the forward model training can be found in run_forward_1D.sh, and an example to run the inverse model training can be found in run_inverse.sh.

Short explanations on the config args

• model_name: string, containing the baseline model name, either 'FNO', 'Unet', or 'PINN'.
• if_training: bool, set True for training, or False for evaluation.
• continue_training: bool, set True to continue training from a checkpoint.
• num_workers: int, number of workers for the PyTorch dataloader.
• batch_size: int, training batch size.
• initial_step: int, number of time steps used as input for FNO and U-Net.
• t_train: int, number of the last time step used for training (for extrapolation testing, set this to be < Nt).
• model_update: int, number of epochs to save model.
• filename: str, has to match the dataset filename.
• single_file: bool, set False for 2D diffusion-reaction, 1D diffusion-sorption, and the radial dam break scenarios, and set True otherwise.
• reduced_resolution: int, factor to downsample spatial resolution.
• reduced_resolution_t: int, factor to downsample temporal resolution.
• reduced_batch: int, factor to downsample sample size used for training.
• epochs: int, total epochs used for training.
• learning_rate: float, learning rate of the optimizer.
• scheduler_step: int, number of epochs to update the learning rate scheduler.
• scheduler_gamma: float, decay rate of the learning rate.

U-Net specific args:

• in_channels: int, number of input channels
• out_channels: int, number of output channels
• ar_mode: bool, set True for fully autoregressive or pushforward training.
• pushforward: bool, set True for pushforward training, False otherwise (ar_mode also has to be set True).
• unroll_step: int, number of time steps to backpropagate in the pushforward training.

FNO specific args:

• num_channels: int, number of channels (variables).
• modes: int, number of Fourier modes to multiply.
• width: int, number of channels for the Fourier layer.

INVERSE specific args:

• base_path: string, location of the data directory
• training_type: string, type of training, autoregressive, single
• mcmc_num_samples: int, number of generated samples
• mcmc_warmup_steps: 10
• mcmc_num_chains: 1
• num_samples_max: 1000
• in_channels_hid: 64
• inverse_model_type: string, type of inverse inference model, ProbRasterLatent, InitialConditionInterp
• inverse_epochs: int, number of epochs for the gradient based method
• inverse_learning_rate: float, learning rate for the gradient based method
• inverse_verbose_flag: bool, some printing

Plotting specific args:

• plot: bool, set True to activate plotting.
• channel_plot: int, determines which channel/variable to plot.
• x_min: float, left spatial domain.
• x_max: float, right spatial domain.
• y_min: float, lower spatial domain.
• y_max: float, upper spatial domain.
• t_min: float, start of temporal domain.
• t_max: float, end of temporal domain.

Datasets and pretrained models

We provide the benchmark datasets we used in the paper through our DaRUS data repository. The data generation configuration can be found in the paper. Additionally, the pretrained models are also available to be downloaded from PDEBench Pretrained Models DaRus repository. To use the pretrained models, users can specify the argument continue_training: True in the config file.

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Directory Tour

Below is an illustration of the directory structure of PDEBench.

📂 pdebench
|_📁 models
  |_📁 pinn    # Model: Physics-Informed Neural Network
    |_📄 train.py
    |_📄 utils.py
    |_📄 pde_definitions.py
  |_📁 fno     # Model: Fourier Neural Operator
    |_📄 train.py
    |_📄 utils.py
    |_📄 fno.py
  |_📁 unet    # Model: U-Net
    |_📄 train.py
    |_📄 utils.py
    |_📄 unet.py
  |_📁 inverse # Model: Gradient-Based Inverse Method
    |_📄 train.py
    |_📄 utils.py
    |_📄 inverse.py
  |_📁 config  # Config: All config files reside here
  |_📄 train_models_inverse.py
  |_📄 run_forward_1D.sh
  |_📄 analyse_result_inverse.py
  |_📄 train_models_forward.py
  |_📄 run_inverse.sh
  |_📄 metrics.py
  |_📄 analyse_result_forward.py
|_📁 data_download  # Data: Scripts to download data from DaRUS
  |_📁 config
  |_📄 download_direct.py
  |_📄 download_easydataverse.py
  |_📄 visualize_pdes.py
  |_📄 README.md
  |_📄 download_metadata.csv
|_📁 data_gen   # Data: Scripts to generate data
  |_📁 configs
  |_📁 data_gen_NLE
  |_📁 src
  |_📁 notebooks
  |_📄 gen_diff_sorp.py
  |_📄 plot.py
  |_📄 example.env
  |_📄 gen_ns_incomp.py
  |_📄 gen_diff_react.py
  |_📄 uploader.py
  |_📄 gen_radial_dam_break.py
|_📄 __init__.py

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Publications & Citations

Please cite the following papers if you use PDEBench datasets and/or source code in your research.

PDEBench: An Extensive Benchmark for Scientific Machine Learning - NeurIPS'2022

@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}
}

PDEBench Datasets - NeurIPS'2022

@data{darus-2986_2022,
author = {Takamoto, Makoto and Praditia, Timothy and Leiteritz, Raphael and MacKinlay, Dan and Alesiani, Francesco and Pflüger, Dirk and Niepert, Mathias},
publisher = {DaRUS},
title = {{PDEBench Datasets}},
year = {2022},
doi = {10.18419/darus-2986},
url = {https://doi.org/10.18419/darus-2986}
}

Learning Neural PDE Solvers with Parameter-Guided Channel Attention - ICML'2023

@article{cape-takamoto:2023,
     author   = {Makoto Takamoto and
                 Francesco Alesiani and
                 Mathias Niepert},
 title        = {Learning Neural {PDE} Solvers with Parameter-Guided Channel Attention},
 journal      = {CoRR},
 volume       = {abs/2304.14118},
 year         = {2023},
 url          = {https://doi.org/10.48550/arXiv.2304.14118},
 doi          = {10.48550/arXiv.2304.14118},
 eprinttype    = {arXiv},
 eprint       = {2304.14118},
 }

Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations - ICLR-W'2024 & ICML'2024

@inproceedings{vcnef-vectorized-conditional-neural-fields-hagnberger:2024,
author = {Hagnberger, Jan and Kalimuthu, Marimuthu and Musekamp, Daniel and Niepert, Mathias},
title = {{Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations}},
year = {2024},
booktitle = {Proceedings of the 41st International Conference on Machine Learning (ICML 2024)}
}

Active Learning for Neural PDE Solvers - NeurIPS-W'2024

@article{active-learn-neuralpde-benchmark-musekamp:2024,
 author       = {Daniel Musekamp and
                 Marimuthu Kalimuthu and
                 David Holzm{\"{u}}ller and
                 Makoto Takamoto and
                 Mathias Niepert},
 title        = {Active Learning for Neural {PDE} Solvers},
 journal      = {CoRR},
 volume       = {abs/2408.01536},
 year         = {2024},
 url          = {https://doi.org/10.48550/arXiv.2408.01536},
 doi          = {10.48550/ARXIV.2408.01536},
 eprinttype    = {arXiv},
 eprint       = {2408.01536},
}

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Code contributors

• Makato Takamoto (NEC laboratories Europe)
• Timothy Praditia (Stuttgart Center for Simulation Science | University of Stuttgart)
• Raphael Leiteritz (Stuttgart Center for Simulation Science | University of Stuttgart)
• Francesco Alesiani (NEC laboratories Europe)
• Dan MacKinlay (CSIRO’s Data61)
• Marimuthu Kalimuthu (Stuttgart Center for Simulation Science | University of Stuttgart)
• John Kim (ANU TechLauncher/CSIRO’s Data61)
• Gefei Shan (ANU TechLauncher/CSIRO’s Data61)
• Yizhou Yang (ANU TechLauncher/CSIRO’s Data61)
• Ran Zhang (ANU TechLauncher/CSIRO’s Data61)
• Simon Brown (ANU TechLauncher/CSIRO’s Data61)

License

MIT licensed, except where otherwise stated. See LICENSE.txt file.