Juncai He

Huang Huang, Fei Yu, Jianqing Zhu et al.

This paper is devoted to the development of a localized Large Language Model (LLM) specifically for Arabic, a language imbued with unique cultural characteristics inadequately addressed by current mainstream models. Significant concerns emerge when addressing cultural sensitivity and local values. To address this, the paper proposes a comprehensive solution that includes further pre-training with Arabic texts, Supervised Fine-Tuning (SFT) utilizing native Arabic instructions, and GPT-4 responses in Arabic, alongside Reinforcement Learning with AI Feedback (RLAIF) employing a reward model attuned to local culture and values. The goal is to cultivate culturally cognizant and value-aligned Arabic LLMs capable of accommodating the diverse, application-specific needs of Arabic-speaking communities. Comprehensive evaluations reveal that the resulting model, dubbed `AceGPT', sets the state-of-the-art standard for open Arabic LLMs across various benchmarks. Codes, data, and models are in https://github.com/FreedomIntelligence/AceGPT.

Juncai He, Lin Li, Jinchao Xu et al.

In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the simplicial linear finite element method (FEM). We first consider the special case of FEM. By exploring the DNN representation of its nodal basis functions, we present a ReLU DNN representation of CPWL in FEM. We theoretically establish that at least $2$ hidden layers are needed in a ReLU DNN to represent any linear finite element functions in $Ω\subseteq \mathbb{R}^d$ when $d\ge2$. Consequently, for $d=2,3$ which are often encountered in scientific and engineering computing, the minimal number of two hidden layers are necessary and sufficient for any CPWL function to be represented by a ReLU DNN. Then we include a detailed account on how a general CPWL in $\mathbb R^d$ can be represented by a ReLU DNN with at most $\lceil\log_2(d+1)\rceil$ hidden layers and we also give an estimation of the number of neurons in DNN that are needed in such a representation. Furthermore, using the relationship between DNN and FEM, we theoretically argue that a special class of DNN models with low bit-width are still expected to have an adequate representation power in applications. Finally, as a proof of concept, we present some numerical results for using ReLU DNNs to solve a two point boundary problem to demonstrate the potential of applying DNN for numerical solution of partial differential equations.

Juncai He, Kaibo Hu, Jinchao Xu

We prove generalized Gaffney inequalities and the discrete compactness for finite element differential forms on $s$-regular domains, including general Lipschitz domains. In computational electromagnetism, special cases of these results have been established for edge elements with weakly imposed divergence-free conditions and used in the analysis of nonlinear and eigenvalue problems. In this paper, we generalize these results to discrete differential forms, not necessarily with strongly or weakly imposed constraints. The analysis relies on a new Hodge mapping and its approximation property. As an application, we show $L^{p}$ estimates for several finite element approximations of the scalar and vector Laplacian problems.

Jianqing Zhu, Juncai He, Lian Zhang et al.

By investigating iterative methods for a constrained linear model, we propose a new class of fully connected V-cycle MgNet for long-term time series forecasting, which is one of the most difficult tasks in forecasting. MgNet is a CNN model that was proposed for image classification based on the multigrid (MG) methods for solving discretized partial differential equations (PDEs). We replace the convolutional operations with fully connected operations in the existing MgNet and then apply them to forecasting problems. Motivated by the V-cycle structure in MG, we further propose the FV-MgNet, a V-cycle version of the fully connected MgNet, to extract features hierarchically. By evaluating the performance of FV-MgNet on popular data sets and comparing it with state-of-the-art models, we show that the FV-MgNet achieves better results with less memory usage and faster inference speed. In addition, we develop ablation experiments to demonstrate that the structure of FV-MgNet is the best choice among the many variants.

Juncai He, Richard Tsai, Rachel Ward

The low-dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low-dimensional manifolds embedded in a high-dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. This paper considers the case in which the training data is distributed in a linear subspace of $\mathbb R^d$. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network's depth and noise in the codimension of the data manifold. We also present additional side effects in training due to the presence of noise.

Juncai He, Xinliang Liu, Jinchao Xu

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 $O_i(u) = σ\left( \sum_j W_{ij} u + B_{ij}\right)$. Here, $ W_{ij}$ denotes the bounded linear operator connecting $j$-th input neuron to $i$-th output neuron, and the bias $ 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 other 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).

Juncai He

This paper is devoted to studying the optimal expressive power of ReLU deep neural networks (DNNs) and its application in approximation via the Kolmogorov Superposition Theorem. We first constructively prove that any continuous piecewise linear functions on $[0,1]$, comprising $O(N^2L)$ 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 the DNNs, achieved through investigating the shattering capacity of ReLU DNNs. Moreover, by invoking the Kolmogorov Superposition Theorem, we achieve an enhanced approximation rate for ReLU DNNs of arbitrary width and depth when dealing with continuous functions in high-dimensional spaces.

Liangchen Liu, Juncai He, Richard Tsai

In this paper, we study linear regression applied to data structured on a manifold. We assume that the data manifold is smooth and is embedded in a Euclidean space, and our objective is to reveal the impact of the data manifold's extrinsic geometry on the regression. Specifically, we analyze the impact of the manifold's curvatures (or higher order nonlinearity in the parameterization when the curvatures are locally zero) on the uniqueness of the regression solution. Our findings suggest that the corresponding linear regression does not have a unique solution when the embedded submanifold is flat in some dimensions. Otherwise, the manifold's curvature (or higher order nonlinearity in the embedding) may contribute significantly, particularly in the solution associated with the normal directions of the manifold. Our findings thus reveal the role of data manifold geometry in ensuring the stability of regression models for out-of-distribution inferences.

Jianqing Zhu, Juncai He, Qiumei Huang

This study used a multigrid-based convolutional neural network architecture known as MgNet in operator learning to solve numerical partial differential equations (PDEs). Given the property of smoothing iterations in multigrid methods where low-frequency errors decay slowly, we introduced a low-frequency correction structure for residuals to enhance the standard V-cycle MgNet. The enhanced MgNet model can capture the low-frequency features of solutions considerably better than the standard V-cycle MgNet. The numerical results obtained using some standard operator learning tasks are better than those obtained using many state-of-the-art methods, demonstrating the efficiency of our model.Moreover, numerically, our new model is more robust in case of low- and high-resolution data during training and testing, respectively.

Shuang Chen, Juncai He, Xue-Cheng Tai

We introduce an abstract neural flow framework for neural networks and neural operators. The framework contains two continuous-depth models, namely neural flows with composition and separation structures, and covers both finite-dimensional function approximation and infinite-dimensional operator approximation. We prove well-posedness and universal approximation properties for the corresponding neural flows, including, to the best of our knowledge, the first universal approximation result for flow-based models between infinite-dimensional spaces. We also obtain universal approximation results for convolutional neural flow models. Through suitable time discretizations, the composition structure recovers ResNet-type architectures, while the separation structure, via a splitting-based discretization, yields plain architectures. This gives a unified flow-based route to both residual and plain architectures for neural networks and neural operators with fully connected or convolutional linear layers.

Ziyi Yang, Zitong Tian, Yinjun Jia et al.

D-peptide binders targeting L-proteins have promising therapeutic potential. Despite rapid advances in machine learning-based target-conditioned peptide design, generating D-peptide binders remains largely unexplored. In this work, we show that by injecting axial features to $E(3)$-equivariant (polar) vector features,it is feasible to achieve cross-chirality generalization from homo-chiral (L--L) training data to hetero-chiral (D--L) design tasks. By implementing this method within a latent diffusion model, we achieved D-peptide binder design that not only outperforms existing tools in in silico benchmarks, but also demonstrates efficacy in wet-lab validation. To our knowledge, our approach represents the first wet-lab validated generative AI for the de novo design of D-peptide binders, offering new perspectives on handling chirality in protein design.

Guangsheng Bao, Yanbin Zhao, Juncai He et al.

Advanced large language models (LLMs) can generate text almost indistinguishable from human-written text, highlighting the importance of LLM-generated text detection. However, current zero-shot techniques face challenges as white-box methods are restricted to use weaker open-source LLMs, and black-box methods are limited by partial observation from stronger proprietary LLMs. It seems impossible to enable white-box methods to use proprietary models because API-level access to the models neither provides full predictive distributions nor inner embeddings. To traverse the divide, we propose **Glimpse**, a probability distribution estimation approach, predicting the full distributions from partial observations. Despite the simplicity of Glimpse, we successfully extend white-box methods like Entropy, Rank, Log-Rank, and Fast-DetectGPT to latest proprietary models. Experiments show that Glimpse with Fast-DetectGPT and GPT-3.5 achieves an average AUROC of about 0.95 in five latest source models, improving the score by 51% relative to the remaining space of the open source baseline. It demonstrates that the latest LLMs can effectively detect their own outputs, suggesting that advanced LLMs may be the best shield against themselves. We release our code and data at https://github.com/baoguangsheng/glimpse.

Juhao Liang, Zhenyang Cai, Jianqing Zhu et al.

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 `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.

Jianqing Zhu, Huang Huang, Zhihang Lin et al.

This paper addresses the critical need for democratizing large language models (LLM) in the Arab world, a region that has seen slower progress in developing models comparable to state-of-the-art offerings like GPT-4 or ChatGPT 3.5, due to a predominant focus on mainstream languages (e.g., English and Chinese). One practical objective for an Arabic LLM is to utilize an Arabic-specific vocabulary for the tokenizer that could speed up decoding. However, using a different vocabulary often leads to a degradation of learned knowledge since many words are initially out-of-vocabulary (OOV) when training starts. Inspired by the vocabulary learning during Second Language (Arabic) Acquisition for humans, the released AraLLaMA employs progressive vocabulary expansion, which is implemented by a modified BPE algorithm that progressively extends the Arabic subwords in its dynamic vocabulary during training, thereby balancing the OOV ratio at every stage. The ablation study demonstrated the effectiveness of Progressive Vocabulary Expansion. Moreover, AraLLaMA achieves decent performance comparable to the best Arabic LLMs across a variety of Arabic benchmarks. Models, training data, benchmarks, and codes will be all open-sourced.

Juncai He, Xinliang Liu, Zitong Tian

This paper studies the numerical approximation of divergence-free vector fields by linearized shallow neural networks, also referred to as random feature models or finite neuron spaces. Combining the stable potential lifting for divergence-free fields with the scalar Sobolev integral representation theory via ReLU$^k$ networks, we derive a core integral representation of divergence-free Sobolev vector fields through antisymmetric potentials parameterized by linearized ReLU$^k$ neural networks. This representation, together with a quasi-uniform distribution argument for the inner parameters, yields optimal approximation rates for such linearized ReLU$^k$ neural networks under an exact divergence-free constraint. Numerical experiments in two and three spatial dimensions, including $L^2$ projection and steady Stokes problems, confirm the theoretical rates and illustrate the effectiveness of exactly divergence-free conditions in computation.

Yahong Yang, Juncai He

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.

Juncai He, Jinchao Xu

In this study, we establish that deep neural networks employing ReLU and ReLU$^2$ activation functions can effectively represent Lagrange finite element functions of any order on various simplicial meshes in arbitrary dimensions. We introduce two novel formulations for globally expressing the basis functions of Lagrange elements, tailored for both specific and arbitrary meshes. These formulations are based on a geometric decomposition of the elements, incorporating several insightful and essential properties of high-dimensional simplicial meshes, barycentric coordinate functions, and global basis functions of linear elements. This representation theory facilitates a natural approximation result for such deep neural networks. Our findings present the first demonstration of how deep neural networks can systematically generate general continuous piecewise polynomial functions on both specific or arbitrary simplicial meshes.

Juncai He, Tong Mao, Jinchao Xu

In this paper, we investigate the expressivity and approximation properties of deep neural networks employing the ReLU$^k$ activation function for $k \geq 2$. Although deep ReLU networks can approximate polynomials effectively, deep ReLU$^k$ networks have the capability to represent higher-degree polynomials precisely. Our initial contribution is a comprehensive, constructive proof for polynomial representation using deep ReLU$^k$ networks. This allows us to establish an upper bound on both the size and count of network parameters. Consequently, we are able to demonstrate a suboptimal approximation rate for functions from Sobolev spaces as well as for analytic functions. Additionally, through an exploration of the representation power of deep ReLU$^k$ networks for shallow networks, we reveal that deep ReLU$^k$ networks can approximate functions from a range of variation spaces, extending beyond those generated solely by the ReLU$^k$ activation function. This finding demonstrates the adaptability of deep ReLU$^k$ networks in approximating functions within various variation spaces.

Yahong Yang, Juncai He

This paper establishes a comprehensive approximation result for deep fully-connected neural networks with commonly-used and general activation functions in Sobolev spaces $W^{n,\infty}$, with errors measured in the $W^{m,p}$-norm for $m < n$ and $1\le p \le \infty$. The derived rates surpass those of classical numerical approximation techniques, such as finite element and spectral methods, exhibiting a phenomenon we refer to as \emph{super-convergence}. Our analysis shows that deep networks with general activations can approximate weak solutions of partial differential equations (PDEs) with superior accuracy compared to traditional numerical methods at the approximation level. Furthermore, this work closes a significant gap in the error-estimation theory for neural-network-based approaches to PDEs, offering a unified theoretical foundation for their use in scientific computing.

Juncai He, Xinliang Liu, Jinchao Xu

In this work, we propose a novel framework to enhance the efficiency and accuracy of neural operators through self-composition, offering both theoretical guarantees and practical benefits. Inspired by iterative methods in solving numerical partial differential equations (PDEs), we design a specific neural operator by repeatedly applying a single neural operator block, we progressively deepen the model without explicitly adding new blocks, improving the model's capacity. To train these models efficiently, we introduce an adaptive train-and-unroll approach, where the depth of the neural operator is gradually increased during training. This approach reveals an accuracy scaling law with model depth and offers significant computational savings through our adaptive training strategy. Our architecture achieves state-of-the-art (SOTA) performance on standard benchmarks. We further demonstrate its efficacy on a challenging high-frequency ultrasound computed tomography (USCT) problem, where a multigrid-inspired backbone enables superior performance in resolving complex wave phenomena. The proposed framework provides a computationally tractable, accurate, and scalable solution for large-scale data-driven scientific machine learning applications.

Hao Wu, Yuan Gao, Ruijian Gou et al.

Reliable long-term forecasting of Earth system dynamics is fundamentally limited by instabilities in current artificial intelligence (AI) models during extended autoregressive simulations. These failures often originate from inherent spectral bias, leading to inadequate representation of critical high-frequency, small-scale processes and subsequent uncontrolled error amplification. Inspired by the nested grids in numerical models used to resolve small scales, we present TritonCast. At the core of its design is a dedicated latent dynamical core, which ensures the long-term stability of the macro-evolution at a coarse scale. An outer structure then fuses this stable trend with fine-grained local details. This design effectively mitigates the spectral bias caused by cross-scale interactions. In atmospheric science, it achieves state-of-the-art accuracy on the WeatherBench 2 benchmark while demonstrating exceptional long-term stability: executing year-long autoregressive global forecasts and completing multi-year climate simulations that span the entire available $2500$-day test period without drift. In oceanography, it extends skillful eddy forecast to $120$ days and exhibits unprecedented zero-shot cross-resolution generalization. Ablation studies reveal that this performance stems from the synergistic interplay of the architecture's core components. TritonCast thus offers a promising pathway towards a new generation of trustworthy, AI-driven simulations. This significant advance has the potential to accelerate discovery in climate and Earth system science, enabling more reliable long-term forecasting and deeper insights into complex geophysical dynamics.

Juncai He, Liangchen Liu, Yen-Hsi Richard Tsai

This paper investigates the impact of multiscale data on machine learning algorithms, particularly in the context of deep learning. A dataset is multiscale if its distribution shows large variations in scale across different directions. This paper reveals multiscale structures in the loss landscape, including its gradients and Hessians inherited from the data. Correspondingly, it introduces a novel gradient descent approach, drawing inspiration from multiscale algorithms used in scientific computing. This approach seeks to transcend empirical learning rate selection, offering a more systematic, data-informed strategy to enhance training efficiency, especially in the later stages.

Juncai He, Jinchao Xu, Lian Zhang et al.

We propose a constrained linear data-feature-mapping model as an interpretable mathematical model for image classification using a convolutional neural network (CNN). From this viewpoint, we establish detailed connections between the traditional iterative schemes for linear systems and the architectures of the basic blocks of ResNet- and MgNet-type models. Using these connections, we present some modified ResNet models that compared with the original models have fewer parameters and yet can produce more accurate results, thereby demonstrating the validity of this constrained learning data-feature-mapping assumption. Based on this assumption, we further propose a general data-feature iterative scheme to show the rationality of MgNet. We also provide a systematic numerical study on MgNet to show its success and advantages in image classification problems and demonstrate its advantages in comparison with established networks.

Juncai He, Lin Li, Jinchao Xu

This paper focuses on establishing $L^2$ approximation properties for deep ReLU convolutional neural networks (CNNs) in two-dimensional space. The analysis is based on a decomposition theorem for convolutional kernels with a large spatial size and multi-channels. Given the decomposition result, the property of the ReLU activation function, and a specific structure for channels, a universal approximation theorem of deep ReLU CNNs with classic structure is obtained by showing its connection with one-hidden-layer ReLU neural networks (NNs). Furthermore, approximation properties are obtained for one version of neural networks with ResNet, pre-act ResNet, and MgNet architecture based on connections between these networks.

Qipin Chen, Wenrui Hao, Juncai He

Weight initialization plays an important role in training neural networks and also affects tremendous deep learning applications. Various weight initialization strategies have already been developed for different activation functions with different neural networks. These initialization algorithms are based on minimizing the variance of the parameters between layers and might still fail when neural networks are deep, e.g., dying ReLU. To address this challenge, we study neural networks from a nonlinear computation point of view and propose a novel weight initialization strategy that is based on the linear product structure (LPS) of neural networks. The proposed strategy is derived from the polynomial approximation of activation functions by using theories of numerical algebraic geometry to guarantee to find all the local minima. We also provide a theoretical analysis that the LPS initialization has a lower probability of dying ReLU comparing to other existing initialization strategies. Finally, we test the LPS initialization algorithm on both fully connected neural networks and convolutional neural networks to show its feasibility, efficiency, and robustness on public datasets.

Juncai He, Lin Li, Jinchao Xu

We study ReLU deep neural networks (DNNs) by investigating their connections with the hierarchical basis method in finite element methods. First, we show that the approximation schemes of ReLU DNNs for $x^2$ and $xy$ are composition versions of the hierarchical basis approximation for these two functions. Based on this fact, we obtain a geometric interpretation and systematic proof for the approximation result of ReLU DNNs for polynomials, which plays an important role in a series of recent exponential approximation results of ReLU DNNs. Through our investigation of connections between ReLU DNNs and the hierarchical basis approximation for $x^2$ and $xy$, we show that ReLU DNNs with this special structure can be applied only to approximate quadratic functions. Furthermore, we obtain a concise representation to explicitly reproduce any linear finite element function on a two-dimensional uniform mesh by using ReLU DNNs with only two hidden layers.

Juncai He, Yuyan Chen, Lian Zhang et al.

In this paper, we propose a constrained linear data-feature mapping model as an interpretable mathematical model for image classification using convolutional neural network (CNN) such as the ResNet. From this viewpoint, we establish the detailed connections in a technical level between the traditional iterative schemes for constrained linear system and the architecture for the basic blocks of ResNet. Under these connections, we propose some natural modifications of ResNet type models which will have less parameters but still maintain almost the same accuracy as these corresponding original models. Some numerical experiments are shown to demonstrate the validity of this constrained learning data-feature mapping assumption.

Juncai He, Jinchao Xu

We develop a unified model, known as MgNet, that simultaneously recovers some convolutional neural networks (CNN) for image classification and multigrid (MG) methods for solving discretized partial differential equations (PDEs). This model is based on close connections that we have observed and uncovered between the CNN and MG methodologies. For example, pooling operation and feature extraction in CNN correspond directly to restriction operation and iterative smoothers in MG, respectively. As the solution space is often the dual of the data space in PDEs, the analogous concept of feature space and data space (which are dual to each other) is introduced in CNN. With such connections and new concept in the unified model, the function of various convolution operations and pooling used in CNN can be better understood. As a result, modified CNN models (with fewer weights and hyper parameters) are developed that exhibit competitive and sometimes better performance in comparison with existing CNN models when applied to both CIFAR-10 and CIFAR-100 data sets.

Juncai He, Xiaodong Jia, Jinchao Xu et al.

Compressed Sensing using $\ell_1$ regularization is among the most powerful and popular sparsification technique in many applications, but why has it not been used to obtain sparse deep learning model such as convolutional neural network (CNN)? This paper is aimed to provide an answer to this question and to show how to make it work. We first demonstrate that the commonly used stochastic gradient decent (SGD) and variants training algorithm is not an appropriate match with $\ell_1$ regularization and then replace it with a different training algorithm based on a regularized dual averaging (RDA) method. RDA was originally designed specifically for convex problem, but with new theoretical insight and algorithmic modifications (using proper initialization and adaptivity), we have made it an effective match with $\ell_1$ regularization to achieve a state-of-the-art sparsity for CNN compared to other weight pruning methods without compromising accuracy (achieving 95\% sparsity for ResNet18 on CIFAR-10, for example).