Dimensionless learning, the next generation machine leanring algorithm for scientific problems

What is dimensionless learning

In short, dimensionless learning is a data-driven dimensional analysis method.

Compared with standard machine learning methods, dimensionless learning embeds a physical constraint, dimensional invariance in different variables, into machine learning algorithms, which means we restrict the large possible solution space into a smaller one. That is to say, we change the function set of machine leanring model and make it is feasible to be trained even for a small number of dataset (about 100-200 data points).

Compared with classical dimensional analysis which only can know the required number of dimensionless numbers for a complex problem, dimensionless learning provides a systematic way to identify unique and dominant dimensionless numbers.

Where to find the paper

Title: Data-driven discovery of dimensionless numbers and scaling laws from experimental measurements.

You can find the preprint paper at here.

Paper highlights

Automatically discover unique and dominant dimensionless numbers with clear physical meaning and scaling laws from complex systems based on experimental measurements;
Identify dimensionally homogeneous differential equations with minimal parameters by leveraging sparsity-promoting techniques (SINDy).

Requirements

matplotlib==3.1.3
derivative==0.3.1
pandas==1.3.4
pyyaml==6.0
scikit-learn==0.24.2
pysindy==1.3.0

1. Local version

You can install these packages with by creating an virtual environment via Anaconda:

conda create --name PyDimension --file requirements.txt

Activate the virtual environment:

conda activate PyDimension

2. Online version - Binder

Note that you can also use the online version code in tutorials folder by simply clicking the icon:

3. Online version - Colab

You can run the code on Colab by clicking the icon:

Tutorials and examples

Go to the folder called tutorials and run the jupyter notebook.

discover_pde_from_data.ipynb: discover dimensionless numbers and dimensionally homogeneous differential equations in spring-mass-damping systems.
keyhole_example.ipynb: discover dimensionless numbers and scaling laws based on experimental measurements.

We will update other examples and simplify the codes in this repository.

How to find the basis vectors

For the turbulent Rayleigh-Benard convection case, you can calculate the basis vectors using python or matlab.

Python:

from sympy import Matrix
import numpy as np
D = np.array([
    [1, 0, 1, 1, 0, 2, 2], 
    [0, 0, -3, -2,0, -1, -1], 
    [0, 0, 1, 0, 0, 0, 0], 
    [0, 1, -1, 0, -1, 0, 0]
])
D = Matrix(D)
basis_vectors = D.nullspace()

Matlab:

D = [1 0 1 1 0 2 2; 0 0 -3 -2 0 -1 -1; 0 0 1 0 0 0 0; 0 1 -1 0 -1 0 0]
basis_vectors = null(D, 'r')
# basis_vectors =
# 
#          0   -1.5000   -1.5000
#     1.0000         0         0
#          0         0         0
#          0   -0.5000   -0.5000
#     1.0000         0         0
#          0    1.0000         0
#          0         0    1.0000

To simplify the basis vectors, the second column can substract the third column, and the second column can times 2. Then, we obtain the final basis vectors:

$\boldsymbol{w_{b1}} = [0,1,0,0,1,0,0]^T,$

$\boldsymbol{w_{b2}} = [0,0,0,0,0,1,-1]^T,$

$\boldsymbol{w_{b3}} = [3,0,0,1,0,-2,0]^T.$

Code structure

.
├── LICENSE
├── README.md
├── __init__.py
├── configs
│   ├── __init__.py
│   └── config_oscillation.yml	            # configuration for spring-mass-damping example
├── dataset
│   ├── dataset_oscillation.csv	            # dataset for spring-mass-damping example
│   └── keyhole_data.csv		    # dataset for keyhole dynamics
├── models
├── requirements.txt			    # required python packages
├── tutorials
│   ├── __init__.py
│   ├── discover_pde_from_data.ipynb	    # example for spring-mass-damping
│   └── keyhole_example.ipynb		    # example for keyhole dynamics
└── utils
    ├── __init__.py
    ├── BIC.py			            # calculate BIC to select the best model
    ├── MSolver.py			    # calculate basis vectors
    ├── config_parser.py		    # configer to parse configuration
    ├── dimension_zoo.py		    # store dimension matrix and others matrix information
    ├── gen_pde_dataset.py		    # generate spring-mass-damping dataset based on the analytical solution
    └── tools.py			    # some tools

Citations

@article{xie2021data,
  title={Data-driven discovery of dimensionless numbers and scaling laws from experimental measurements},
  author={Xie, Xiaoyu and Liu, Wing Kam and Gan, Zhengtao},
  journal={arXiv preprint arXiv:2111.03583},
  year={2021}
}
@article{gan2021universal,
  title={Universal scaling laws of keyhole stability and porosity in 3D printing of metals},
  author={Gan, Zhengtao and Kafka, Orion L and Parab, Niranjan and Zhao, Cang and Fang, Lichao and Heinonen, Olle and Sun, Tao and Liu, Wing Kam},
  journal={Nature communications},
  volume={12},
  number={1},
  pages={1--8},
  year={2021},
  publisher={Nature Publishing Group}
}
@article{saha2021hierarchical,
  title={Hierarchical Deep Learning Neural Network (HiDeNN): An artificial intelligence (AI) framework for computational science and engineering},
  author={Saha, Sourav and Gan, Zhengtao and Cheng, Lin and Gao, Jiaying and Kafka, Orion L and Xie, Xiaoyu and Li, Hengyang and Tajdari, Mahsa and Kim, H Alicia and Liu, Wing Kam},
  journal={Computer Methods in Applied Mechanics and Engineering},
  volume={373},
  pages={113452},
  year={2021},
  publisher={Elsevier}
}

Contact

If you have any questions or want to contribute to thsi respository, please contact:

Xiaoyu Xie
Northwestern University, Mechanical Engineering
xiaoyuxie2020@u.northwestern.edu

ZheruiMartinezGuo/PyDimension