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This\nrepository consists of the code used to generate the datasets, to upload and\ndownload the datasets from the data repository, as well as to train and evaluate\ndifferent machine learning models as baselines. PDEBench features a much wider\nrange of PDEs than existing benchmarks and includes realistic and difficult\nproblems (both forward and inverse), larger ready-to-use datasets comprising\nvarious initial and boundary conditions, and PDE parameters. Moreover, PDEBench\nwas created to make the source code extensible and we invite active\nparticipation from the SciML community to improve and extend the benchmark.\n\n![Visualizations of some PDE problems covered by the benchmark.](https://github.com/pdebench/PDEBench/blob/main/pdebench_examples.PNG)\n\nCreated and maintained by Makoto Takamoto\n`\u003cmakoto.takamoto@neclab.eu, takamtmk@gmail.com\u003e`, Timothy Praditia\n`\u003ctimothy.praditia@iws.uni-stuttgart.de\u003e`, Raphael Leiteritz, Dan MacKinlay,\nFrancesco Alesiani, Dirk Pflüger, and Mathias Niepert.\n\n---\n\n## Datasets and Pretrained Models\n\nWe also provide datasets and pretrained machine learning models.\n\nPDEBench Datasets:\nhttps://darus.uni-stuttgart.de/dataset.xhtml?persistentId=doi:10.18419/darus-2986\n\nPDEBench Pre-Trained Models:\nhttps://darus.uni-stuttgart.de/dataset.xhtml?persistentId=doi:10.18419/darus-2987\n\nDOIs\n\n[![DOI:10.18419/darus-2986](https://img.shields.io/badge/DOI-doi%3A10.18419%2Fdarus--2986-red)](https://doi.org/10.18419/darus-2986)\n[![DOI:10.18419/darus-2987](https://img.shields.io/badge/DOI-doi%3A10.18419%2Fdarus--2987-red)](https://doi.org/10.18419/darus-2987)\n\n## Installation\n\n### Using pip\n\nLocally:\n\n```bash\npip install --upgrade pip wheel\npip install .\n```\n\nFrom PyPI:\n\n```bash\npip install pdebench\n```\n\nTo include dependencies for data generation:\n\n```bash\npip install \"pdebench[datagen310]\"\npip install \".[datagen310]\" # locally\n```\n\nor\n\n```bash\npip install \"pdebench[datagen39]\"\npip install \".[datagen39]\" # locally\n```\n\n### GPU Support\n\nFor GPU support there are additional platform-specific instructions:\n\nFor PyTorch, the latest version we support is v1.13.1\n[see previous-versions/#linux - CUDA 11.7](https://pytorch.org/get-started/previous-versions/#linux-and-windows-2).\n\nFor JAX, which is approximately 6 times faster for simulations than PyTorch in\nour tests,\n[see jax#pip-installation-gpu-cuda-installed-via-pip](https://github.com/google/jax#pip-installation-gpu-cuda-installed-via-pip-easier)\n\n## Installation using conda:\n\nIf you like you can also install dependencies using anaconda, we suggest to use\n[mambaforge](https://github.com/conda-forge/miniforge#mambaforge) as a\ndistribution. Otherwise you may have to **enable the conda-forge** channel for\nthe following commands.\n\nStarting from a fresh environment:\n\n```\nconda create -n myenv python=3.9\nconda activate myenv\n```\n\nInstall dependencies for model training:\n\n```\nconda install deepxde hydra-core h5py -c conda-forge\n```\n\nAccording to your hardware availability, either install PyTorch with CUDA\nsupport:\n\n- [see previous-versions/#linux - CUDA 11.7](https://pytorch.org/get-started/previous-versions/#linux-and-windows-2).\n\n```\nconda install pytorch==1.13.1 torchvision==0.14.1 torchaudio==0.13.1 pytorch-cuda=11.7 -c pytorch -c nvidia\n```\n\n- [or CPU only binaries](https://pytorch.org/get-started/previous-versions/#linux-and-windows-2).\n\n```\nconda install pytorch==1.13.1 torchvision==0.14.1 torchaudio==0.13.1 cpuonly -c pytorch\n```\n\nOptional dependencies for data generation:\n\n```\nconda install clawpack jax jaxlib python-dotenv\n```\n\n## Configuring DeepXDE\n\nIn our tests we used PyTorch as backend for DeepXDE. Please\n[follow the documentation](https://deepxde.readthedocs.io/en/latest/user/installation.html#working-with-different-backends)\nto enable this.\n\n## Data Generation\n\nThe data generation codes are contained in [data_gen](./pdebench/data_gen):\n\n- `gen_diff_react.py` to generate the 2D diffusion-reaction data.\n- `gen_diff_sorp.py` to generate the 1D diffusion-sorption data.\n- `gen_radial_dam_break.py` to generate the 2D shallow-water data.\n- `gen_ns_incomp.py` to generate the 2D incompressible inhomogeneous\n  Navier-Stokes data.\n- `plot.py` to plot the generated data.\n- `uploader.py` to upload the generated data to the data repository.\n- `.env` is the environment data to store Dataverse URL and API token to upload\n  the generated data. Note that the filename should be strictly `.env` (i.e.\n  remove the `example` from the filename)\n- `configs` directory contains the yaml files storing the configuration for the\n  simulation. Arguments for the simulation are problem-specific and detailed\n  explanation can be found in the simulation scripts.\n- `src` directory contains the simulation scripts for different problems:\n  `sim_diff_react-py` for 2D diffusion-reaction, `sim_diff_sorp.py` for 1D\n  diffusion-sorption, and `swe` for the shallow-water equation.\n\n### Data Generation for 1D Advection/Burgers/Reaction-Diffusion/2D DarcyFlow/Compressible Navier-Stokes Equations\n\nThe data generation codes are contained in\n[data_gen_NLE](./pdebench/data_gen/data_gen_NLE/):\n\n- `utils.py` util file for data generation, mainly boundary conditions and\n  initial conditions.\n- `AdvectionEq` directory with the source codes to generate 1D Advection\n  equation training samples\n- `BurgersEq` directory with the source codes to generate 1D Burgers equation\n  training samples\n- `CompressibleFluid` directory with the source codes to generate compressible\n  Navier-Stokes equations training samples\n\n  - `ReactionDiffusionEq` directory with the source codes to generate 1D\n    Reaction-Diffusion equation training samples (**Note:\n    [DarcyFlow data can be generated by run_DarcyFlow2D.sh](pdebench/data_gen/data_gen_NLE/README.md)\n    in this folder.**)\n\n- `save` directory saving the generated training samples\n\nA typical example to generate training samples (1D Advection Equation): (in\n`data_gen/data_gen_NLE/AdvectionEq/`)\n\n```bash\npython3 advection_multi_solution_Hydra.py +multi=beta1e0.yaml\n```\n\nwhich is assumed to be performed in each directory.\n\nExamples for generating other PDEs are provided in `run_trainset.sh` in each\nPDE's directories. The config files for Hydra are stored in `config` directory\nin each PDE's directory.\n\n#### Data Transformaion and Merge into HDF5 format\n\n1D Advection/Burgers/Reaction-Diffusion/2D DarcyFlow/Compressible Navier-Stokes\nEquations save data as a numpy array. So, to read those data via our\ndataloaders, the data transformation/merge should be performed. This can be done\nusing `data_gen_NLE/Data_Merge.py` whose config file is located at:\n`data_gen/data_gen_NLE/config/config.yaml`. After properly setting the\nparameters in the config file (type: name of PDEs, dim: number of\nspatial-dimension, bd: boundary condition), the corresponding HDF5 file could be\nobtained as:\n\n```bash\npython3 Data_Merge.py\n```\n\n## Configuration\n\nYou can set the default values for data locations for this project by putting\nconfig vars like this in the `.env` file:\n\n```\nWORKING_DIR=~/Data/Working\nARCHIVE_DATA_DIR=~/Data/Archive\n```\n\nThere is an example in `example.env`.\n\n## Data Download\n\nThe download scripts are provided in [data_download](./pdebench/data_download).\nThere are two options to download data.\n\n1. Using `download_direct.py` (**recommended**)\n   - Retrieves data shards directly using URLs. Sample command for each PDE is\n     given in the README file in the [data_download](./pdebench/data_download)\n     directory.\n2. Using `download_easydataverse.py` (might be slow and you could encounter\n   errors/issues; hence, not recommended!)\n   - Use the config files from the `config` directory that contains the yaml\n     files storing the configuration. Any files in the dataset matching\n     `args.filename` will be downloaded into `args.data_folder`.\n\n## Baseline Models\n\nIn this work, we provide three different ML models to be trained and evaluated\nagainst the benchmark datasets, namely\n[FNO](https://arxiv.org/pdf/2010.08895.pdf),\n[U-Net](https://www.sciencedirect.com/science/article/abs/pii/S0010482519301520?via%3Dihub),\nand [PINN](https://www.sciencedirect.com/science/article/pii/S0021999118307125).\nThe codes for the baseline model implementations are contained in\n[models](./pdebench/models):\n\n- `train_models_forward.py` is the main script to train and evaluate the model.\n  It will call on model-specific script based on the input argument.\n- `train_models_inverse.py` is the main script to train and evaluate the model\n  for inverse problems. It will call on model-specific script based on the input\n  argument.\n- `metrics.py` is the script to evaluate the trained models based on various\n  evaluation metrics described in our paper. Additionally, it also plots the\n  prediction and target data.\n- `analyse_result_forward.py` is the script to convert the saved pickle file\n  from the metrics calculation script into pandas dataframe format and save it\n  as a CSV file. Additionally it also plots a bar chart to compare the results\n  between different models.\n- `analyse_result_inverse.py` is the script to convert the saved pickle file\n  from the metrics calculation script into pandas dataframe format and save it\n  as a CSV file. This script is used for the inverse problems. Additionally it\n  also plots a bar chart to compare the results between different models.\n- `fno` contains the scripts of FNO implementation. These are partly adapted\n  from the\n  [FNO repository](https://github.com/zongyi-li/fourier_neural_operator).\n- `unet` contains the scripts of U-Net implementation. These are partly adapted\n  from the\n  [U-Net repository](https://github.com/mateuszbuda/brain-segmentation-pytorch).\n- `pinn` contains the scripts of PINN implementation. These utilize the\n  [DeepXDE library](https://github.com/lululxvi/deepxde).\n- `inverse` contains the model for inverse model based on gradient.\n- `config` contains the yaml files for the model training input. The default\n  templates for different equations are provided in the\n  [args](./pdebench/models/config/args) directory. User just needs to copy and\n  paste them to the args keyword in the\n  [config.yaml](./pdebench/models/config/config.yaml) file.\n\nAn example to run the forward model training can be found in\n[run_forward_1D.sh](./pdebench/models/run_forward_1D.sh), and an example to run\nthe inverse model training can be found in\n[run_inverse.sh](./pdebench/models/run_inverse.sh).\n\n### Short explanations on the config args\n\n- model_name: string, containing the baseline model name, either 'FNO', 'Unet',\n  or 'PINN'.\n- if_training: bool, set True for training, or False for evaluation.\n- continue_training: bool, set True to continue training from a checkpoint.\n- num_workers: int, number of workers for the PyTorch dataloader.\n- batch_size: int, training batch size.\n- initial_step: int, number of time steps used as input for FNO and U-Net.\n- t_train: int, number of the last time step used for training (for\n  extrapolation testing, set this to be \u003c Nt).\n- model_update: int, number of epochs to save model.\n- filename: str, has to match the dataset filename.\n- single_file: bool, set False for 2D diffusion-reaction, 1D diffusion-sorption,\n  and the radial dam break scenarios, and set True otherwise.\n- reduced_resolution: int, factor to downsample spatial resolution.\n- reduced_resolution_t: int, factor to downsample temporal resolution.\n- reduced_batch: int, factor to downsample sample size used for training.\n- epochs: int, total epochs used for training.\n- learning_rate: float, learning rate of the optimizer.\n- scheduler_step: int, number of epochs to update the learning rate scheduler.\n- scheduler_gamma: float, decay rate of the learning rate.\n\n#### U-Net specific args:\n\n- in_channels: int, number of input channels\n- out_channels: int, number of output channels\n- ar_mode: bool, set True for fully autoregressive or pushforward training.\n- pushforward: bool, set True for pushforward training, False otherwise (ar_mode\n  also has to be set True).\n- unroll_step: int, number of time steps to backpropagate in the pushforward\n  training.\n\n#### FNO specific args:\n\n- num_channels: int, number of channels (variables).\n- modes: int, number of Fourier modes to multiply.\n- width: int, number of channels for the Fourier layer.\n\n#### INVERSE specific args:\n\n- base_path: string, location of the data directory\n- training_type: string, type of training, autoregressive, single\n- mcmc_num_samples: int, number of generated samples\n- mcmc_warmup_steps: 10\n- mcmc_num_chains: 1\n- num_samples_max: 1000\n- in_channels_hid: 64\n- inverse_model_type: string, type of inverse inference model, ProbRasterLatent,\n  InitialConditionInterp\n- inverse_epochs: int, number of epochs for the gradient based method\n- inverse_learning_rate: float, learning rate for the gradient based method\n- inverse_verbose_flag: bool, some printing\n\n#### Plotting specific args:\n\n- plot: bool, set True to activate plotting.\n- channel_plot: int, determines which channel/variable to plot.\n- x_min: float, left spatial domain.\n- x_max: float, right spatial domain.\n- y_min: float, lower spatial domain.\n- y_max: float, upper spatial domain.\n- t_min: float, start of temporal domain.\n- t_max: float, end of temporal domain.\n\n## Datasets and pretrained models\n\nWe provide the benchmark datasets we used in the paper through our\n[DaRUS data repository](https://darus.uni-stuttgart.de/dataset.xhtml?persistentId=doi:10.18419/darus-2986).\nThe data generation configuration can be found in the paper. Additionally, the\npretrained models are also available to be downloaded from\n[PDEBench Pretrained Models](https://darus.uni-stuttgart.de/dataset.xhtml?persistentId=doi:10.18419/darus-2987)\nDaRus repository. To use the pretrained models, users can specify the argument\n`continue_training: True` in the\n[config file](./pdebench/models/config/config.yaml).\n\n---\n\n## Directory Tour\n\nBelow is an illustration of the directory structure of PDEBench.\n\n```\n📂 pdebench\n|_📁 models\n  |_📁 pinn    # Model: Physics-Informed Neural Network\n    |_📄 train.py\n    |_📄 utils.py\n    |_📄 pde_definitions.py\n  |_📁 fno     # Model: Fourier Neural Operator\n    |_📄 train.py\n    |_📄 utils.py\n    |_📄 fno.py\n  |_📁 unet    # Model: U-Net\n    |_📄 train.py\n    |_📄 utils.py\n    |_📄 unet.py\n  |_📁 inverse # Model: Gradient-Based Inverse Method\n    |_📄 train.py\n    |_📄 utils.py\n    |_📄 inverse.py\n  |_📁 config  # Config: All config files reside here\n  |_📄 train_models_inverse.py\n  |_📄 run_forward_1D.sh\n  |_📄 analyse_result_inverse.py\n  |_📄 train_models_forward.py\n  |_📄 run_inverse.sh\n  |_📄 metrics.py\n  |_📄 analyse_result_forward.py\n|_📁 data_download  # Data: Scripts to download data from DaRUS\n  |_📁 config\n  |_📄 download_direct.py\n  |_📄 download_easydataverse.py\n  |_📄 visualize_pdes.py\n  |_📄 README.md\n  |_📄 download_metadata.csv\n|_📁 data_gen   # Data: Scripts to generate data\n  |_📁 configs\n  |_📁 data_gen_NLE\n  |_📁 src\n  |_📁 notebooks\n  |_📄 gen_diff_sorp.py\n  |_📄 plot.py\n  |_📄 example.env\n  |_📄 gen_ns_incomp.py\n  |_📄 gen_diff_react.py\n  |_📄 uploader.py\n  |_📄 gen_radial_dam_break.py\n|_📄 __init__.py\n```\n\n---\n\n## Publications \u0026 Citations\n\nPlease cite the following papers if you use PDEBench datasets and/or source code\nin your research.\n\n\u003cdetails\u003e\n\u003csummary\u003e\n    \u003ca href=\"https://arxiv.org/abs/2210.07182\"\u003ePDEBench: An Extensive Benchmark for Scientific Machine Learning - NeurIPS'2022 \u003c/a\u003e\n\u003c/summary\u003e\n\u003cbr/\u003e\n\n```\n@inproceedings{PDEBench2022,\nauthor = {Takamoto, Makoto and Praditia, Timothy and Leiteritz, Raphael and MacKinlay, Dan and Alesiani, Francesco and Pflüger, Dirk and Niepert, Mathias},\ntitle = {{PDEBench: An Extensive Benchmark for Scientific Machine Learning}},\nyear = {2022},\nbooktitle = {36th Conference on Neural Information Processing Systems (NeurIPS 2022) Track on Datasets and Benchmarks},\nurl = {https://arxiv.org/abs/2210.07182}\n}\n```\n\n\u003c/details\u003e\n\n\u003cdetails\u003e\n\u003csummary\u003e\n    \u003ca href=\"https://doi.org/10.18419/darus-2986\"\u003ePDEBench Datasets - NeurIPS'2022 \u003c/a\u003e\n\u003c/summary\u003e\n\u003cbr/\u003e\n\n```\n@data{darus-2986_2022,\nauthor = {Takamoto, Makoto and Praditia, Timothy and Leiteritz, Raphael and MacKinlay, Dan and Alesiani, Francesco and Pflüger, Dirk and Niepert, Mathias},\npublisher = {DaRUS},\ntitle = {{PDEBench Datasets}},\nyear = {2022},\ndoi = {10.18419/darus-2986},\nurl = {https://doi.org/10.18419/darus-2986}\n}\n```\n\n\u003c/details\u003e\n\n\u003cdetails\u003e\n\u003csummary\u003e\n    \u003ca href=\"https://arxiv.org/abs/2304.14118\"\u003eLearning Neural PDE Solvers with Parameter-Guided Channel Attention - ICML'2023 \u003c/a\u003e\n\u003c/summary\u003e\n\u003cbr/\u003e\n\n```\n@article{cape-takamoto:2023,\n     author   = {Makoto Takamoto and\n                 Francesco Alesiani and\n                 Mathias Niepert},\n title        = {Learning Neural {PDE} Solvers with Parameter-Guided Channel Attention},\n journal      = {CoRR},\n volume       = {abs/2304.14118},\n year         = {2023},\n url          = {https://doi.org/10.48550/arXiv.2304.14118},\n doi          = {10.48550/arXiv.2304.14118},\n eprinttype    = {arXiv},\n eprint       = {2304.14118},\n }\n```\n\n\u003c/details\u003e\n\n\u003cdetails\u003e\n\u003csummary\u003e\n    \u003ca href=\"https://openreview.net/forum?id=I4WlXAA9Gd\"\u003e Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations - ICLR-W'2024 \u0026 ICML'2024 \u003c/a\u003e\n\u003c/summary\u003e\n\u003cbr/\u003e\n\n```\n@inproceedings{vcnef-vectorized-conditional-neural-fields-hagnberger:2024,\nauthor = {Hagnberger, Jan and Kalimuthu, Marimuthu and Musekamp, Daniel and Niepert, Mathias},\ntitle = {{Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations}},\nyear = {2024},\nbooktitle = {Proceedings of the 41st International Conference on Machine Learning (ICML 2024)}\n}\n```\n\n\u003c/details\u003e\n\n\u003cdetails\u003e\n\u003csummary\u003e\n    \u003ca href=\"https://arxiv.org/abs/2408.01536\"\u003e Active Learning for Neural PDE Solvers - NeurIPS-W'2024 \u003c/a\u003e\n\u003c/summary\u003e\n\u003cbr/\u003e\n\n```\n@article{active-learn-neuralpde-benchmark-musekamp:2024,\n author       = {Daniel Musekamp and\n                 Marimuthu Kalimuthu and\n                 David Holzm{\\\"{u}}ller and\n                 Makoto Takamoto and\n                 Mathias Niepert},\n title        = {Active Learning for Neural {PDE} Solvers},\n journal      = {CoRR},\n volume       = {abs/2408.01536},\n year         = {2024},\n url          = {https://doi.org/10.48550/arXiv.2408.01536},\n doi          = {10.48550/ARXIV.2408.01536},\n eprinttype    = {arXiv},\n eprint       = {2408.01536},\n}\n```\n\n\u003c/details\u003e\n\n---\n\n## Code contributors\n\n- [Makato Takamoto](https://github.com/mtakamoto-D)\n  ([NEC laboratories Europe](https://www.neclab.eu/))\n- [Timothy Praditia](https://github.com/timothypraditia)\n  ([Stuttgart Center for Simulation Science | University of Stuttgart](https://www.simtech.uni-stuttgart.de/))\n- [Raphael Leiteritz](https://github.com/leiterrl)\n  ([Stuttgart Center for Simulation Science | University of Stuttgart](https://www.simtech.uni-stuttgart.de/))\n- [Francesco Alesiani](https://github.com/falesiani)\n  ([NEC laboratories Europe](https://www.neclab.eu/))\n- [Dan MacKinlay](https://danmackinlay.name/)\n  ([CSIRO’s Data61](https://data61.csiro.au/))\n- [Marimuthu Kalimuthu](https://github.com/kmario23)\n  ([Stuttgart Center for Simulation Science | University of Stuttgart](https://www.simtech.uni-stuttgart.de/))\n- [John Kim](https://github.com/johnmjkim)\n  ([ANU TechLauncher](https://comp.anu.edu.au/TechLauncher/)/[CSIRO’s Data61](https://data61.csiro.au/))\n- [Gefei Shan](https://github.com/davecatmeow)\n  ([ANU TechLauncher](https://comp.anu.edu.au/TechLauncher/)/[CSIRO’s Data61](https://data61.csiro.au/))\n- [Yizhou Yang](https://github.com/verdantwynnd)\n  ([ANU TechLauncher](https://comp.anu.edu.au/TechLauncher/)/[CSIRO’s Data61](https://data61.csiro.au/))\n- [Ran Zhang](https://github.com/maphyca)\n  ([ANU TechLauncher](https://comp.anu.edu.au/TechLauncher/)/[CSIRO’s Data61](https://data61.csiro.au/))\n- [Simon Brown](https://github.com/SimonSyBrown)\n  ([ANU TechLauncher](https://comp.anu.edu.au/TechLauncher/)/[CSIRO’s Data61](https://data61.csiro.au/))\n\n## License\n\nMIT licensed, except where otherwise stated. See `LICENSE.txt` file.\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fpdebench%2FPDEBench","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fpdebench%2FPDEBench","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fpdebench%2FPDEBench/lists"}