nPIML

---------- Noise-aware physics-informed machine learning (nPIML) ----------

Here, we provide the data that support the findings of the paper "Noise-aware Physics-informed Machine Learning for Robust PDE Discovery".

Citation

DOI: https://doi.org/10.1088/2632-2153/acb1f0
Pongpisit Thanasutives et al 2023 Mach. Learn.: Sci. Technol. 4 015009

Bibtex:

@article{Thanasutives_2023,
	doi = {10.1088/2632-2153/acb1f0},
	url = {https://dx.doi.org/10.1088/2632-2153/acb1f0},
	year = {2023},
	month = {feb},
	publisher = {IOP Publishing},
	volume = {4},
	number = {1},
	pages = {015009},
	author = {Pongpisit Thanasutives and Takashi Morita and Masayuki Numao and Ken-ichi Fukui},
	title = {Noise-aware physics-informed machine learning for robust PDE discovery},
	journal = {Machine Learning: Science and Technology}
}

Open Research Codebase

Please access OneDrive (password: nPIML) or GoogleDrive. Then you can extract the zip file containing the code you are looking for. The extracted directories are supposed to be at ~/Desktop/. The related primary directories in research.zip include but are not limited to pysindy, parametric-discovery, l0bnb_algos, abess, WSINDy_PDE_JCP, SciencePlots (for visualization) and PDE-FIND*. Note that all the data are included already in this codebase.

Concerning the main text >>> nPIML framework consists of the three main steps.
[Step 1]

• KdV: ls Multi-task-Physics-informed-neural-networks/inverse_small_KdV/Hyperparameter\ study-approx_l0-reproduced-for-pub*.ipynb
• KS: ls kuramoto-sivashinsky-solver/ks_selector*.py
• QHO: ls kuramoto-sivashinsky-solver/qho.py
• NLS: ls kuramoto-sivashinsky-solver/nls.py

[Step 2] Mostly in Multi-task-Physics-informed-neural-networks/

• KdV: ls inverse_small_KdV/Find\ lambda_str*.ipynb
• KS: ls inverse_small_KS2/Find\ lambda_str*.ipynb
• QHO: ls inverse_qho/Find\ lambda_str*ipynb
• NLS: ls inverse_NLS/Find\ lambda_str*.ipynb

Visit https://github.com/MathBioCU/WSINDy_PDE or try running research/WSINDy_PDE_JCP/wsindy_pde_script_nPIML.m (MATLAB required) to enable the extension to the convolutional weak formulation (CWF).

[Step 3] Mostly in kuramoto-sivashinsky-solver/

• KdV (a recommended demo): ls kdv_pinn_2000_pub.py kdv_pinn_2000_pub_20220517.py
• KS: ls deephpm_KS_chaotic_learned_coeffs_*_new.py deephpm_KS_chaotic_learned_coeffs_more_noise.py
• QHO: ls qho_pinn_learned_coeffs_20220613.py
• NLS: nls_pinn_learned_coeffs_20220614.py

Code regarding all the three steps of Burgers' PDE example resides in Multi-task-Physics-informed-neural-networks/inverse_burgers/.
Step 1: ls abess-approx_l0-gammaSwish-clean-DEMO-kappa-20220429\ \(pub\)*.ipynb abess-approx_l0-gammaSwish-clean-DEMO-noise-tolerence*.ipynb
Step 2: ls Find\ lambda_str*.ipynb
Step 3: ls Final\ PINN-wiener-V2-less-samples-20220627-newpub*.ipynb Final\ PINN-Visualize\ noise\ \(freezed\ success\)-pub-reproduced20221204.ipynb. The last notebook is for the denoising visualization. You may start exploring the notebooks for Burgers' PDE discovery by installing torch==1.11.0.

Concerning the appendix section >>> Code for the appendix section is all in research/PDE-FIND/ (a clone of the PDE-FIND repository).

1. Nonlinear Diffusion: ls Nonlinear-Diffusion-visual-nPIML.ipynb
2. Navier Stokes: ls Navier-Stokes\ with\ noise\ original\ data*.ipynb
3. 2D Reaction Diffusion: ls Reaction-Diffusion-2D-big-nPIML-pub*.ipynb
4. 3D Reaction Diffusion: ls Reaction-Diffusion-3D-nPIML.ipynb

Please install research/l0bnb_algos/v2/l0bnb and research/abess/python for running the best-subset regression solvers. The notebooks from 1. to 4. import best_subset.py (in research/parametric-discovery/), which has helper fuctions (e.g., backward_refinement) written for utilization with the best-subset solvers. To use the weak formulation (WF) algorithm, install research/pysindy. Since these repositories are open sources, you may update them by git pull.
Specification of the conda environment (namely "py3.7") used only for producing the results in the paper's appendix section is given in env_spec.
Please be aware that an act of using external libraries is under their licenses provided in the associated directories.

Notes

• In this work, hidden weights are initialized usually by either the uniform Xavier or default function PyTorch provides for building a linear layer.
• Any dubiousness, questions or requests may be forwarded to the corresponding author. Feel free to create an issue on this repository if you have trouble downloading the source codes.
• Erratum: "the preselector networks" in Figure 15 should be "the projection networks."