Physics-informed-vibe-coding

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我们的课程【PINN & 神经算子 & AI 智能体全面加速科研】，自 2025 年暑期第 1 期开课以来，累计国内外硕博研究生和高校教师报名学员超 170 人，学员来自剑桥大学、慕尼黑工业大学、诺丁汉大学、新加坡国立大学、香港科技大学、北京大学、中国科学院、浙江大学、上海交通大学、哈尔滨工业大学、华中科技大学、西安交通大学、西北工业大学、中国科学技术大学、南京大学、深圳大学、山东大学、新疆大学、湘疆大学、上海科技大学、石家庄学院等国内外高校及科研单位，获一众好评！我们课程主打的就是帮助大家学明白，不仅会用，还要理解底层数学原理，以及代码算法高效准确实现，学会顶刊如何发表。2026 年秋季课程是我们第 2 期课程（第一期 40 小时录播 + 第二期 40 小时直播，共 80 小时视频课程），自 9 月 6 日起，每周六、周日晚上 7 点半–9 点半直播，欢迎大家扫码进群或者加我微信交流学习/课程咨询报名/论文辅导报名。

课程支持平台

• 微信公众号：AI4CFD
• 微信公众号：数据驱动与力学
• 微信公众号：AI智能体科研站
• 小红书：AI4CFD
• B站：AI4PDE

Philosophy

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。

Contents

#	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

Dependencies

jax, jaxlib (CUDA), optax, matplotlib, numpy, scipy

How to Run

Each case directory contains a single self-contained .py file:

cd NTK-PINN-jax/case2_wave1d/
python wave1d_ntk_pinn.py

cd MultiScalePINN_jax/case1_heat1d/
python heat1d_multiscale_pinn.py

cd VSPINN_jax/case1_ns2d/
python ns2d_vspinn_pinn.py

cd GradientWeighted_PINN_jax/case2_klein_gordon/
python klein_gordon_gw_pinn.py

cd ScalePINN-jax/case1_ldc_re7500/
python ldc_re7500_scalepinn.py

cd RFM_jax/case1_stokes_2d/
python stokes_2d_rfm.py

All results (data .txt, figures .png, checkpoints .pkl) are saved automatically.

License

MIT

Citation

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}
}