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overview paper

[61] (Alphabetic order) Zhi-Qin John Xu*, Yaoyu Zhang, Zhangchen Zhou, An overview of condensation phenomenon in deep learning. In pdf and in arXiv:2504.09484.

[56] Zhi-Qin John Xu#, Lulu Zhang#, Wei Cai*, On Understanding And Overcoming Spectral Biases of Deep Neural Network Learning Methods for Solving Pdes. Journal of Computational Physics, 2025, pdf and in arXiv 2024.

[27] Zhi-Qin John Xu*, Yaoyu Zhang, Tao Luo, Overview frequency principle/spectral bias in deep learning. Communications on Applied Mathematics and Computation 2024 (dedicated to the memory of Professor Zhong-Ci Shi), arxiv 2201.07395 (2022) pdf, and in arxiv.

科普

Transformer是推断还是记忆？初始化大小很重要 文献[49]

频率原则理解深度学习的媒体报道，见 《麻省理工科技评论》中文官网，DeepTech深科技，络绎科学

神经网络的简单偏好 (2017.11-2022.11 五年的理论研究小结)

Embedding Principle

调参，注意神经网络处于哪种相态 文献[10]

从频率角度理解为什么深度可以加速神经网络的训练 文献[11]

线性Frequency Principle动力学：定量理解深度学习的一种有效模型 文献[13,17]

F-Principle：初探深度学习在计算数学的应用 文献[4,9,12]

F-Principle：初探理解深度学习不能做什么 [4]

从傅里叶分析角度解读深度学习的泛化能力 [1,2,4,17]

多尺度神经网络解微分方程 文献[9,12]

code

code at github.

Toy example of Reasoning

1d example of F-Principle

Fourier Domain F-Principle: DNNs often fit target functions from low to high frequencies. Each frame is several training steps. Red: FFT of the target function; Blue: FFT of DNN output. Abscissa: frequency; Ordinate: amplitude.

Condensation

Weight condensation: input weights of neurons in a group are same.

Phase diagram of two-layer ReLU NN

Embedding principle

MscaleDNN structure