首个采用 Vibe Coding 理念进行科学机器学习（PINNs · PIELM · 随机特征方法 · 神经算子）研究的开源仓库
The first open-source repository for Scientific Machine Learning research — PINNs, PIELM, Random Feature Methods, Neural Operators, and beyond — via the Vibe Coding paradigm
Xiong Xiong (熊雄) · Northwestern Polytechnical University (NWPU)
AI4PDE · Physics-Informed Deep Learning · Data-Driven Discovery
Vibe Coding & Vibe Researching — Humans discover and formulate the important scientific problems; AI agents write code, execute experiments, and automatically optimize the computation pipeline; humans verify, validate, and accept the final results. Not a single line of code is written by hand.
We believe the future of computational research lies in a clear division of labor between human intellect and AI capability. Humans bring domain expertise to identify meaningful problems, design solution strategies, and make critical judgments on result quality. AI agents handle the entire implementation cycle — from writing code and running experiments to tuning hyperparameters and generating visualizations. This closed-loop workflow enables researchers to focus on scientific insight rather than engineering overhead. Every algorithm in this repository is implemented in JAX with GPU acceleration, and each case is fully reproducible with saved data, figures, and model checkpoints.
Vibe Coding & Vibe Researching — 人类负责发现并提出重要科学问题，AI 智能体负责编程、执行实验与自动优化计算过程，最终由人类核对、验证并验收结果。全程不手写一行代码，（尝试）完成完整的复杂科研项目。
本项目提供一系列完整、自包含的 JAX-GPU 实现，覆盖科学机器学习（Scientific Machine Learning）领域的前沿算法，包括 PINNs、物理信息极限学习机（PIELM）、随机特征方法、神经算子等，并配套详细的中文学术教程。人类凭借领域知识发现有价值的科学问题、设计求解策略、把控结果质量；AI 智能体承担从代码编写、实验运行到超参调优、可视化生成的全链路工作。这一闭环协作模式使研究者得以专注于科学洞察，而非工程细节。
| # | Algorithm | Directory | Tutorial | Status |
|---|---|---|---|---|
| 1 | NTK-PINN — Neural Tangent Kernel adaptive weighting | NTK-PINN-jax/ |
NTK-PINN 教程 | Done |
| 2 | MultiScale-PINN — Multi-scale Fourier feature networks for PDEs | MultiScalePINN_jax/ |
待发布 | Done |
| 3 | VS-PINN — Variable-Scaling PINN for Navier-Stokes | VSPINN_jax/ |
VS-PINN 教程 | Done |
| 4 | GW-PINN — Gradient-Weighted adaptive loss balancing | GradientWeighted_PINN_jax/ |
GW-PINN 教程 | Done |
| 5 | Scale-PINN — Evolutionary regularization for high-Re flows | ScalePINN-jax/ |
待发布 | Done |
| 6 | Maxwell-PINN (No BO) — Pure PINN for 2D EM scattering (Helmholtz + ABC) | MaxwellPINN_jax/ |
待发布 | Done |
| 7 | TINN — Time-Induced Neural Networks with Levenberg-Marquardt for 1D Burgers | TINN_jax/ |
待发布 | Done |
| 8 | RFM — Random Feature Method for PDEs (non-iterative least-squares solver) | RFM_jax/ |
待发布 | Done |
jax, jaxlib (CUDA), optax, matplotlib, numpy, scipy
Each case directory contains a single self-contained .py file:
cd NTK-PINN-jax/case2_wave1d/
python wave1d_ntk_pinn.pycd MultiScalePINN_jax/case1_heat1d/
python heat1d_multiscale_pinn.pycd VSPINN_jax/case1_ns2d/
python ns2d_vspinn_pinn.pycd GradientWeighted_PINN_jax/case2_klein_gordon/
python klein_gordon_gw_pinn.pycd ScalePINN-jax/case1_ldc_re7500/
python ldc_re7500_scalepinn.pycd RFM_jax/case1_stokes_2d/
python stokes_2d_rfm.pyAll results (data .txt, figures .png, checkpoints .pkl) are saved automatically.
MIT
If you find this repository useful, please consider citing it:
@software{xiong2026vibecoding,
author = {Xiong, Xiong},
title = {Physics-Informed Vibe Coding: Scientific Machine Learning in JAX via Human-AI Collaboration},
year = {2026},
publisher = {GitHub},
url = {https://github.com/xgxgnpu/Physics-informed-vibe-coding},
note = {PINNs, PIELM, Random Feature Methods, Neural Operators, and beyond}
}You may also cite the following related works:
@article{WANG2022110768,
title={When and why PINNs fail to train: A neural tangent kernel perspective},
author={Wang, Sifan and Yu, Xinling and Perdikaris, Paris},
journal={Journal of Computational Physics},
volume={449},
pages={110768},
year={2022},
doi={https://doi.org/10.1016/j.jcp.2021.110768},
publisher={Elsevier}
}
@article{WANG2021113938,
title={On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks},
author={Wang, Sifan and Wang, Hanwen and Perdikaris, Paris},
journal={Computer Methods in Applied Mechanics and Engineering},
volume={384},
pages={113938},
year={2021},
doi={https://doi.org/10.1016/j.cma.2021.113938},
publisher={Elsevier}
}
@article{xiong2025high,
title={High-frequency flow field super-resolution via physics-informed hierarchical adaptive Fourier feature networks},
author={Xiong, Xiong and Lu, Kang and Zhang, Zhuo and Zeng, Zheng and Zhou, Sheng and Hu, Rongchun and Deng, Zichen},
journal={Physics of Fluids},
volume={37},
number={9},
year={2025},
publisher={AIP Publishing}
}
@article{xiong2025j,
title={J-PIKAN: A physics-informed KAN network based on Jacobi orthogonal polynomials for solving fluid dynamics},
author={Xiong, Xiong and Lu, Kang and Zhang, Zhuo and Zeng, Zheng and Zhou, Sheng and Deng, Zichen and Hu, Rongchun},
journal={Communications in Nonlinear Science and Numerical Simulation},
pages={109414},
year={2025},
publisher={Elsevier}
}
@article{xiong2025separated,
title={Separated-variable spectral neural networks: a physics-informed learning approach for high-frequency pdes},
author={Xiong, Xiong and Zhang, Zhuo and Hu, Rongchun and Gao, Chen and Deng, Zichen},
journal={arXiv preprint arXiv:2508.00628},
year={2025}
}
@article{chen2022rfm,
title={Bridging Traditional and Machine Learning-based Algorithms for Solving PDEs: The Random Feature Method},
author={Chen, Jingrun and Chi, Xurong and E, Weinan and Yang, Zhouwang},
journal={Journal of Machine Learning},
volume={1},
number={3},
pages={268--298},
year={2022},
doi={10.4208/jml.220726}
}
@article{dwivedi2020pielm,
title={Physics Informed Extreme Learning Machine (PIELM)--A rapid method for the numerical solution of partial differential equations},
author={Dwivedi, Vikas and Srinivasan, Balaji},
journal={Neurocomputing},
volume={391},
pages={96--118},
year={2020},
doi={10.1016/j.neucom.2019.12.099}
}
@article{zhang2025legend,
title={Legend-KINN: A Legendre Polynomial-Based Kolmogorov-Arnold-Informed Neural Network for Efficient PDE Solving},
author={Zhang, Zhuo and Xiong, Xiong and Zhang, Sen and Wang, Wei and Zhong, Yanxu and Yang, Canqun and Yang, Xi},
journal={Expert Systems with Applications},
pages={129839},
year={2025},
publisher={Elsevier}
}