| Name | folax JSON |
| Version |
0.0.2
JSON |
| download |
| home_page | None |
| Summary | A framework for solving and optimizing PDEs by integrating machine learning with numerical methods in computational mechanics. |
| upload_time | 2025-08-20 08:26:52 |
| maintainer | None |
| docs_url | None |
| author | Reza Najian Asl, email = <reza.najian-asl@tum.de> |
| requires_python | >=3.10 |
| license | BSD 4-Clause License
Copyright (c) 2024, Reza Najian Asl, Shahed Rezaei
All rights reserved.
Redistribution and use in source and binary forms, with or without
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This product includes software developed by Reza Najian Asl, Shahed Rezaei.
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| keywords |
numerical-simulation-optimization
pde-solver
physics-informed-neural-networks (pinns)
scientific machine learning (sciml)
|
| VCS |
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| bugtrack_url |
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| requirements |
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<p align=center><img height="54.125%" width="54.125%" src="https://github.com/RezaNajian/eFOL/assets/62375973/0e1ca4e0-0658-4f5d-aad9-1ae7c9f67574"></p>
[![License][license-image]][license]
[](https://github.com/RezaNajian/folax/actions/workflows/CI.yml)
[](https://pypi.org/project/folax/)
[license-image]: https://img.shields.io/badge/license-BSD-green.svg?style=flat
[license]: https://github.com/RezaNajian/FOL/LICENSE
# Folax: Solution and Optimization of parameterized PDEs
**F**inite **O**perator **L**earning (FOL) with [**JAX**](https://github.com/jax-ml/jax) constitutes a unified numerical framework that seamlessly integrates established numerical methods with advanced scientific machine learning techniques for solving and optimizing parametrized partial differential equations (PDEs). In constructing a physics-informed operator learning approach, FOL formulates a purely physics-based loss function derived from the Method of Weighted Residuals, allowing discrete residuals—computed using classical PDE solution techniques—to be directly incorporated into backpropagation during network training. This approach ensures that the learned operators rigorously satisfy the underlying governing equations while maintaining consistency with established numerical discretizations. Importantly, this loss formulation is agnostic to the network architecture and has been successfully applied to architectures such as Conditional Neural Fields, Fourier Neural Operators (FNO), and DeepONets.
FOL has been applied in the following scientific studies:
- A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries [[arXiv](https://arxiv.org/abs/2504.02459)].
- Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs [[arXiv](https://arxiv.org/abs/2407.04157)].
- Digitalizing metallic materials from image segmentation to multiscale solutions via physics informed operator learning [[npj Computational Materials](https://www.nature.com/articles/s41524-025-01718-y)].
- A Finite Operator Learning Technique for Mapping the Elastic Properties of Microstructures to Their Mechanical Deformations [[Numerical Methods in Eng.](https://onlinelibrary.wiley.com/doi/full/10.1002/nme.7637)].
- SPiFOL: A Spectral-based physics-informed finite operator learning for prediction of mechanical behavior of microstructures [[J. Mechanics and Physics of Solids](https://www.sciencedirect.com/science/article/pii/S0022509625001954)].
We built upon several widely adopted Python packages, including [JAX](https://github.com/jax-ml/jax) for high-performance array computations on CPUs and GPUs, [PETSc](https://petsc.org/release/) for the efficient solution of large-scale linear systems, [Metis](https://github.com/KarypisLab/METIS) for mesh partitioning (integration forthcoming), [Flax](https://github.com/google/flax?tab=readme-ov-file) for constructing modular and flexible neural networks, [Optax](https://github.com/google-deepmind/optax) for applying state-of-the-art gradient-based optimization algorithms, and [Orbax](https://github.com/google/orbax) for efficient checkpointing and serialization. This foundation ensures scalability, computational efficiency, and ease of use in large-scale training and simulation workflows.
## Installation
### CPU installation
To install folax using pip (recommended) for CPU usage you can type the following command
``pip install folax[cpu]``
### GPU installation
To install folax using pip (recommended) for GPU usage you can type the following command
``pip install folax[cuda]``
### Developer installation
If you would like to do development in folax, please first clone the repo and in the folax folder, run the following command
``pip install -e .[cuda,dev]``
## Contributing
If you would like to contribute to the project, please open a pull request with small changes. If you would like to see big changes in the source code, please open an issue or discussion so we can start a conversation.
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"description": "<p align=center><img height=\"54.125%\" width=\"54.125%\" src=\"https://github.com/RezaNajian/eFOL/assets/62375973/0e1ca4e0-0658-4f5d-aad9-1ae7c9f67574\"></p>\n\n[![License][license-image]][license] \n[](https://github.com/RezaNajian/folax/actions/workflows/CI.yml)\n[](https://pypi.org/project/folax/)\n\n\n[license-image]: https://img.shields.io/badge/license-BSD-green.svg?style=flat\n[license]: https://github.com/RezaNajian/FOL/LICENSE\n\n# Folax: Solution and Optimization of parameterized PDEs\n**F**inite **O**perator **L**earning (FOL) with [**JAX**](https://github.com/jax-ml/jax) constitutes a unified numerical framework that seamlessly integrates established numerical methods with advanced scientific machine learning techniques for solving and optimizing parametrized partial differential equations (PDEs). In constructing a physics-informed operator learning approach, FOL formulates a purely physics-based loss function derived from the Method of Weighted Residuals, allowing discrete residuals\u2014computed using classical PDE solution techniques\u2014to be directly incorporated into backpropagation during network training. This approach ensures that the learned operators rigorously satisfy the underlying governing equations while maintaining consistency with established numerical discretizations. Importantly, this loss formulation is agnostic to the network architecture and has been successfully applied to architectures such as Conditional Neural Fields, Fourier Neural Operators (FNO), and DeepONets. \n\nFOL has been applied in the following scientific studies:\n- A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries [[arXiv](https://arxiv.org/abs/2504.02459)].\n- Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs [[arXiv](https://arxiv.org/abs/2407.04157)].\n- Digitalizing metallic materials from image segmentation to multiscale solutions via physics informed operator learning [[npj Computational Materials](https://www.nature.com/articles/s41524-025-01718-y)].\n- A Finite Operator Learning Technique for Mapping the Elastic Properties of Microstructures to Their Mechanical Deformations [[Numerical Methods in Eng.](https://onlinelibrary.wiley.com/doi/full/10.1002/nme.7637)].\n- SPiFOL: A Spectral-based physics-informed finite operator learning for prediction of mechanical behavior of microstructures [[J. Mechanics and Physics of Solids](https://www.sciencedirect.com/science/article/pii/S0022509625001954)].\n\nWe built upon several widely adopted Python packages, including [JAX](https://github.com/jax-ml/jax) for high-performance array computations on CPUs and GPUs, [PETSc](https://petsc.org/release/) for the efficient solution of large-scale linear systems, [Metis](https://github.com/KarypisLab/METIS) for mesh partitioning (integration forthcoming), [Flax](https://github.com/google/flax?tab=readme-ov-file) for constructing modular and flexible neural networks, [Optax](https://github.com/google-deepmind/optax) for applying state-of-the-art gradient-based optimization algorithms, and [Orbax](https://github.com/google/orbax) for efficient checkpointing and serialization. This foundation ensures scalability, computational efficiency, and ease of use in large-scale training and simulation workflows.\n\n## Installation\n### CPU installation \nTo install folax using pip (recommended) for CPU usage you can type the following command\n\n``pip install folax[cpu]``\n\n### GPU installation\nTo install folax using pip (recommended) for GPU usage you can type the following command\n\n``pip install folax[cuda]``\n\n### Developer installation\nIf you would like to do development in folax, please first clone the repo and in the folax folder, run the following command\n\n``pip install -e .[cuda,dev]``\n\n## Contributing\nIf you would like to contribute to the project, please open a pull request with small changes. If you would like to see big changes in the source code, please open an issue or discussion so we can start a conversation.\n\n",
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