ML | PINN Reading List

• ML | Physics-informed neural networks (PINN)
• ML | PINN | DeepXDE

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• 原来好多人都不知道PINN的鼻祖文章是这几篇啊

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• 自动微分（AutoDiff） - AI4Science的核心基础技术

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• 当AI学会算流体：用PINN求解圆柱绕流问题
• Ang EHW, Wang G, Ng BF. Physics-Informed Neural Networks for Low Reynolds Number Flows over Cylinder. Energies. 2023; 16(12):4558. https://doi.org/10.3390/en16124558
• 在 PaddleScience 的 2D 非定常圆柱绕流
  • R. Franke, W. Rodi, and B. Schönung, “Numerical calculation of laminar vortex-shedding flow past cylinders,” J. Wind Eng. Ind. Aerodyn., vol. 35, pp. 237–257, Jan. 1990, doi: 10.1016/0167-6105(90)90219-3
  • R. D. Henderson, “Details of the drag curve near the onset of vortex shedding,” Phys. Fluids, vol. 7, no. 9, pp. 2102–2104, 1995, doi: 10.1063/1.868459.
  • C. Y. Wen, C. L. Yeh, M. J. Wang, and C. Y. Lin, “On the drag of two-dimensional flow about a circular cylinder,” Phys. Fluids, vol. 16, no. 10, pp. 3828–3831, 2004, doi: 10.1063/1.1789071.
  • O. Posdziech and R. Grundmann, “A systematic approach to the numerical calculation of fundamental quantities of the two-dimensional flow over a circular cylinder,” J. Fluids Struct., vol. 23, no. 3, pp. 479–499, 2007, doi: 10.1016/j.jfluidstructs.2006.09.004.
  • J. Park and H. Choi, “Numerical solutions of flow past a circular cylinder at reynolds numbers up to 160,” KSME Int. J., vol. 12, no. 6, pp. 1200–1205, 1998, doi: 10.1007/BF02942594.

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• 谷歌学术引用2万+JCP顶刊：物理信息神经网络PINNs的本硕博入门第1课
• AI4PDEs全新综述：经典数值方法与机器学习求解 PDE 范式之争
• Nature综述：物理信息机器学习未来发展指南
• 【2026年最新PINNs投稿指南】物理信息神经网络期刊分区
• 物理信息神经网络PINNs训练为什么总是翻车？梯度病态的诊断与修复
• 一句提示词，Gemini 3.1 高效高精度实现物理信息神经网络PINNs求解高频PDEs——与Claude Sonnet 4.6的正面交锋（一）（附代码和提示词）
• Vibe Coding&Vibe Researching科研系列（一）：神经切线核NTK的自适应权重物理信息神经网络PINN求解高频波动方程 | NTK 自适应权重算法
• Physics-Informed Vibe Coding（2）||NTK自适应加权+多尺度物理信息神经网络求解高频PDEs | MultiScale-PINN
• Physics-Informed Vibe Coding（3）||变量尺度变换物理信息神经网络求解Navier-Stokes 方程 | Variable-Scaling PINN
• Physics-Informed Vibe Coding（4）：物理信息神经网络训练经常失败？试试梯度自适应加权PINN | Gradient-Weighted PINN
• Physics-Informed Vibe Coding（5）||仅需100秒，Scale-PINN仿真Navier-Stokes方程Re=7500 | Scale-PINN
  • Scale-PINN: Learning Efficient Physics-Informed Neural Networks Through Sequential Correction
• Science Advances||Physics-informed DeepONet 如何让算子学习摆脱数据依赖 | DeepONet
• Nature Machine Intelligence||PeRCNN物理硬编码与离散学习，让卷积核成为差分算子 | PeRCNN
• Nature Communications：物理约束符号回归与弱形式，破解不完备高噪实验数据的知识发现
• AI4PDE顶刊解读(一)【CMAME开源代码】条件自适应增广拉格朗日PINN求解PDEs的正、反问题
• AI4PDE顶刊解读(二)【PNAS综述】||物理引导深度学习，从数据中学习动力系统

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• Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs
  • https://github.com/vc-bonn/Spline_PINN

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• Nature Communications：物理信息深度学习的方程发现、模型发现、知识发现 | 当方程本身就是未知数：PINN-SR 如何从稀缺数据中"发现"物理定律

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参考文献

• Wang, S., Wang, H., & Perdikaris, P. (2021). On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 384, 113938. https://doi.org/10.1016/j.cma.2021.113938
• Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707. https://doi.org/10.1016/j.jcp.2018.10.045
• Jacot, A., Gabriel, F., & Hongler, C. (2018). Neural tangent kernel: Convergence and generalization in neural networks. In Advances in Neural Information Processing Systems (NeurIPS 2018), 31, 8571–8580.
• Tancik, M., Srinivasan, P. P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J. T., & Ng, R. (2020). Fourier features let networks learn high frequency functions in low dimensional domains. In Advances in Neural Information Processing Systems (NeurIPS 2020), 33, 7537–7547.
• Wang, S., Yu, X., & Perdikaris, P. (2022). When and why PINNs fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449, 110768. https://doi.org/10.1016/j.jcp.2021.110768
• Xiong, X., Lu, K., Zhang, Z., Zeng, Z., Zhou, S., Hu, R., & Deng, Z. (2025). High-frequency flow field super-resolution via physics-informed hierarchical adaptive Fourier feature networks. Physics of Fluids, 37(9). AIP Publishing.
• Xiong, X., Lu, K., Zhang, Z., Zeng, Z., Zhou, S., Deng, Z., & Hu, R. (2025). J-PIKAN: A physics-informed KAN network based on Jacobi orthogonal polynomials for solving fluid dynamics. Communications in Nonlinear Science and Numerical Simulation, 109414. Elsevier.
• Xiong, X., Zhang, Z., Hu, R., Gao, C., & Deng, Z. (2025). Separated-variable spectral neural networks: A physics-informed learning approach for high-frequency PDEs. arXiv preprint arXiv:2508.00628.