Jianqing Zhu, Huang Huang, Zhihang Lin, Juhao Liang, Zhengyang Tang, Khalid Almubarak, Mosen Alharthi, Bang An, Juncai He, Xiangbo Wu, Fei Yu, Junying Chen, Ma Zhuoheng, Yuhao Du, He Zhang, Saied Alshahrani, Emad A. Alghamdi, Lian Zhang, Ruoyu Sun, Haizhou Li, Benyou Wang, Jinchao Xu. Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2025.

This article is devoted to studying the optimal expressive power of rectified linear unit (ReLU) deep neural networks (DNNs) and its application in approximation via the Kolmogorov superposition theorem (KST). We first constructively prove that any continuous piecewise linear (CPwL) functions on $[{0,1}]$ , comprising $\mathcal {O}(N^{2}L)$ segments, can be represented by ReLU DNNs with L hidden layers and N neurons per layer. Subsequently, we demonstrate that this construction is optimal regarding the parameter count of DNNs, achieved through investigating the shattering capacity of ReLU DNNs. Moreover, by invoking the KST, we achieve an enhanced approximation rate for ReLU DNNs of arbitrary width and depth when dealing with continuous functions in high-dimensional spaces.

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the $i$-th neuron in a nonlinear operator layer is defined by $\mathcal O_i(u) = \sigma\left( \sum_j \mathcal W_{ij} u + \mathcal B_{ij}\right)$. Here, $\mathcal W_{ij}$ denotes the bounded linear operator connecting $j$-th input neuron to $i$-th output neuron, and the bias $\mathcal B_{ij}$ takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question. This paper explores a comparison between deeper neural networks (DeNNs) with a flexible number of layers and wider neural networks (WeNNs) with limited hidden layers, focusing on their optimal generalization error in Sobolev losses. Analytical investigations reveal that the architecture of a neural network can be significantly influenced by various factors, including the number of sample points, parameters within the neural networks, and the regularity of the loss function. Specifically, a higher number of parameters tends to favor WeNNs, while an increased number of sample points and greater regularity in the loss function lean towards the adoption of DeNNs. We ultimately apply this theory to address partial differential equations using deep Ritz and physics-informed neural network (PINN) methods, guiding the design of neural networks.

Huang Huang, Fei Yu, Jianqing Zhu, Xuening Sun, Hao Cheng, Song Dingjie, Zhihong Chen, Mosen Alharthi, Bang An, Juncai He, Ziche Liu, Junying Chen, Jianquan Li, Benyou Wang, Lian Zhang, Ruoyu Sun, Xiang Wan, Haizhou Li, Jinchao Xu. Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers). 2024.

The alignment of large language models (LLMs) is critical for developing effective and safe language models. Traditional approaches focus on aligning models during the instruction tuning or reinforcement learning stages, referred to in this paper as `\textit{post alignment}'. We argue that alignment during the pre-training phase, which we term 'native alignment', warrants investigation. Native alignment aims to prevent unaligned content from the beginning, rather than relying on post-hoc processing. This approach leverages extensively aligned pre-training data to enhance the effectiveness and usability of pre-trained models. Our study specifically explores the application of native alignment in the context of Arabic LLMs. We conduct comprehensive experiments and ablation studies to evaluate the impact of native alignment on model performance and alignment stability. Additionally, we release open-source Arabic LLMs that demonstrate state-of-the-art performance on various benchmarks, providing significant benefits to the Arabic LLM community.