WeakIdent

Script for predicting differential equations using WeakIdent.

This repository provides implementation details of WeakIdent using Matlab and using Python.

Script for identifying differential equations using WeakIdent.

This repo provides implementation details of WeakIdent using Python.

Code author: Mengyi Tang Rajchel

For Paper, "🔗WeakIdent: Weak formulation for Identifying Differential Equation using Narrow-fit and Trimming" by Mengyi Tang, Wenjing Liao, Rachel Kuske and Sung Ha Kang.

😊 If you found WeakIdent useful in your research, please consider cite us:

@misc{tang2022weakident,
      title={WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming}, 
      author={Mengyi Tang and Wenjing Liao and Rachel Kuske and Sung Ha Kang},
      year={2022},
      eprint={2211.03134},
      archivePrefix={arXiv},
      primaryClass={math.NA}
}

What does WeakIdent do?

WeakIdent is a general and robust framework to recover differential equations using a weak formulation, for both ordinary and partial differential equations (ODEs and PDEs). Noisy time series data are taken in with spacing as input and output a governing equation for this data.

The structure of this repository.

├── README.md
├── WeakIdent-Matlab
└── WeakIdent-Python
    ├── README.md
    ├── configs
    ├── environment.yml
    ├── main.py
    ├── model.py
    ├── output
    └── utils

WeakIdent - Matlab

run main.m to use WeakIdent on partial differential equations and and ode systems.

weakIdentV4: script for predicting PDE equations using WeakIdent datasetV2: folder containing datasets used in the paper.

WeakIdent - Python

Environment set-up

Required packages sys, yaml, argparse, time, typing, pandas, tabular, numpy, numpy_index, Scipy

Set-up

[Option 1] If you do not have conda installed, you can use pip install to install the packages listed above.

[Option 2] (1) run conda env create -f environment.yml to create the environment. (2) run conda activate test_env1 to activate the environment.

Datasets

Sample datasets from various type of equations including true coefficients can be found in folder dataset-Python. For each dataset, there exists a configuration file in configs that specifies the input argument to run WeakIdent. The following table provides the name of equations for each dataset:

config file index	Equation name
1	Transport Equation
2	Reaction Diffusion Equation
3	Anisotropic Porous Medium (PM) Equation
4	Heat Equation
5	Korteweg-de Vires (KdV) Equation
6	Kuramoto-Sivashinsky (KS) Equation
7	Nonlinear Schrodinger (NLS)
8	2D Linear System
9	Nonlinear System (Van der Pol)
10	Nonlinear System (Duffing)
11	Noninear System (Lotka-Volterra)
12	Nonlinear System (Lorenz)
13	Noninear System 2D (Lotka-Volterra)

We refer details of each dataset to the experimental result section in our paper WeakIdent: Weak formulation for Identifying Differential Equation using Narrow-fit and Trimming

Remark

The dataset for reaction diffusion type equation and Nonlinear Lotka-Volterro equation are sligher larger (100-200 M). They are not provided in dataset-Python. Instead, I provided auto-simulation on these two dataset when running WeakIdent on these two equations.

To run WeakIdent on reaction diffusion type equation run python main.py --config configs/config_2.yaml. The auto simulation takes 1-3 mintues.
To run WeakIdent on Nonlinear System 2D (Lotka-Volterra), please run python main.py --config configs/config_13.yaml. The auto simulation takes 1 second.
To run other examples, WeakIdent directly takes provided datasets as input. Running time various between 1 - 30 seconds.

Run WeakIdent on provided datasets

There are 13 datasets provided in dataset-Python. To run WeakIdent on each indivial dataset, run python main.py --config configs/config_$n$ file name$.yaml to identify differential equation using a pre-simulated dataset specified in configs/config_$n$.yaml.

An example of running WeakIdent on Transport Equation with diffusion

Run python main.py --config configs/config_1.yaml to see the output:

Start loading arguments and dataset transportDiff.npy for Transport Equation
Start building feature matrix W:
[===========================================] 100.0% 
Start building scale matrix S:
[===========================================] 100.0% 
The number of rows in the highly dynamic region is  845

 Start finding support: 
[=========] 100.0% 
WeakIdent finished support trimming and narrow-fit for variable no.1 . A support is found this variable.

 ------------- coefficient vector overview ------noise-signal-ratio : 0.5  -------
+----+----------------+------------+------------+
|    | feature        |   true u_t |   pred u_t |
+====+================+============+============+
|  0 | 1              |       0    |  0         |
+----+----------------+------------+------------+
|  1 | u              |       0    |  0         |
+----+----------------+------------+------------+
|  2 | u_{x}          |      -1    | -1.00707   |
+----+----------------+------------+------------+
|  3 | u_{xx}         |       0.05 |  0.0526434 |
+----+----------------+------------+------------+
                    ......
+----+----------------+------------+------------+
| 42 | (u^6)_{xxxxxx} |       0    |  0         |
+----+----------------+------------+------------+

 ------------- equation overview ------noise-signal-ratio : 0.5  -------------------
+----+---------------------------------+------------------------------------+
|    | True equation                   | Predicted equation                 |
+====+=================================+====================================+
|  0 | u_t = - 1.0 u_{x} + 0.05 u_{xx} | u_t = - 1.007 u_{x} + 0.053 u_{xx} |
+----+---------------------------------+------------------------------------+

 ------------------------------ CPU time: 0.7 seconds ------------------------------

 Identification error for Transport Equation from WeakIdent: 
+----+-----------+----------------+-------------+---------+---------+
|    |     $e_2$ |   $e_{\infty}$ |   $e_{res}$ |   $tpr$ |   $ppv$ |
+====+===========+================+=============+=========+=========+
|  0 | 0.0075372 |      0.0528683 |    0.417006 |       1 |       1 |
+----+-----------+----------------+-------------+---------+---------+

More sample output for each dataset

We provide sample output for each equation(dataset) in output.

Credit/note

Build feature matrix through convolution (using fft), this part of the code is modified from get_lib_columns() (originally Matlab version) from WeakSindyPde.

udemirezen / WeakIdent