Qihong Yang

Qihong Yang, Qiaolin He

We propose a novel neural network architecture, termed Multi-Scale Separable Fourier Neural Networks (MS-SFNN), for the accurate and efficient solution of linear and nonlinear high-frequency partial differential equations (PDEs). MS-SFNN exploits a separable representation: given a $d$-dimensional input, it employs $d$ independent subnetworks -- each acting on a single coordinate -- and constructs basis functions via element-wise multiplication of their outputs. The PDE solution is approximated as a linear combination of these basis functions, with coefficients determined by least squares. Critically, all network weights and biases are randomly initialized once, from a uniform distribution with unit variance, and remain fixed thereafter. To enhance expressivity, a tunable scaling factor is introduced in each subnetwork to modulate the frequency content of the resulting basis functions. Fourier features are explicitly embedded through cosine activations, endowing the method with strong spectral approximation capabilities. To mitigate the memory bottleneck associated with dense collocation in high-frequency or three-dimensional problems, we replace automatic differentiation with analytically derived basis function derivatives and develop a memory-efficient batched QR decomposition algorithm for solving large-scale least-squares systems. Numerical experiments demonstrate that MS-SFNN achieves unprecedented accuracy across a range of challenging PDEs, significantly outperforming state-of-the-art methods such as Physics-Informed Neural Networks (PINN) and Separated-Variable Spectral Neural Networks (SV-SNN).

Qihong Yang, Yangtao Deng, Qiaolin He et al.

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear combination of basis functions integrated into the network, with coefficients determined by a direct least-squares solve, thereby bypassing traditional gradient-based training. The key methodological contribution include: (1) an efficient tensor-product scheme that generates multi-dimensional basis functions from combinations of two sets of subnetwork outputs, significantly reducing model complexity and parameter count while maintaining expressivity; (2) a block time-marching strategy to improve computational efficiency in long-time simulations; and (3) a linear reformulation strategy for handling nonlinear PDEs by treating known nonlinear terms as sources. TPNet achieves superior accuracy and shorter training times than conventional neural network solvers. This performance gain stems from its structured design and deterministic least-squares fitting, which contrast with the iterative, often computationally intensive optimization required by mainstream methods like Physics-Informed Neural Networks (PINNs).

Yu Yang, Helin Gong, Qihong Yang et al.

In practical engineering experiments, the data obtained through detectors are inevitably noisy. For the already proposed data-enabled physics-informed neural network (DEPINN) \citep{DEPINN}, we investigate the performance of DEPINN in calculating the neutron diffusion eigenvalue problem from several perspectives when the prior data contain different scales of noise. Further, in order to reduce the effect of noise and improve the utilization of the noisy prior data, we propose innovative interval loss functions and give some rigorous mathematical proofs. The robustness of DEPINN is examined on two typical benchmark problems through a large number of numerical results, and the effectiveness of the proposed interval loss function is demonstrated by comparison. This paper confirms the feasibility of the improved DEPINN for practical engineering applications in nuclear reactor physics.

Qihong Yang, Yangtao Deng, Yu Yang et al.

In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential operator is realized by automatic differentiation. The eigenfunction of the eigenvalue problem is learned by the neural network and the iterative algorithms are implemented by optimizing the specially defined loss function. The largest positive eigenvalue, smallest eigenvalue and interior eigenvalues with the given prior knowledge can be solved efficiently. We examine the applicability and accuracy of our methods in the numerical experiments in one dimension, two dimensions and higher dimensions. Numerical results show that accurate eigenvalue and eigenfunction approximations can be obtained by our methods.

Yu Yang, Qihong Yang, Yangtao Deng et al.

In this work, we propose an end-to-end adaptive sampling neural network (MMPDE-Net) based on the moving mesh method, which can adaptively generate new sampling points by solving the moving mesh PDE. This model focuses on improving the quality of sampling points generation. Moreover, we develop an iterative algorithm based on MMPDE-Net, which makes the sampling points more precise and controllable. Since MMPDE-Net is a framework independent of the deep learning solver, we combine it with physics-informed neural networks (PINN) to propose moving sampling PINN (MS-PINN) and demonstrate its effectiveness by error analysis under some assumptions. Finally, we demonstrate the performance improvement of MS-PINN compared to PINN through numerical experiments of four typical examples, which numerically verify the effectiveness of our method.

Zhi Bian, Shaojun Zhou, Hao Fu et al.

For better user satisfaction and business effectiveness, more and more attention has been paid to the sequence-based recommendation system, which is used to infer the evolution of users' dynamic preferences, and recent studies have noticed that the evolution of users' preferences can be better understood from the implicit and explicit feedback sequences. However, most of the existing recommendation techniques do not consider the noise contained in implicit feedback, which will lead to the biased representation of user interest and a suboptimal recommendation performance. Meanwhile, the existing methods utilize item sequence for capturing the evolution of user interest. The performance of these methods is limited by the length of the sequence, and can not effectively model the long-term interest in a long period of time. Based on this observation, we propose a novel CTR model named denoising user-aware memory network (DUMN). Specifically, the framework: (i) proposes a feature purification module based on orthogonal mapping, which use the representation of explicit feedback to purify the representation of implicit feedback, and effectively denoise the implicit feedback; (ii) designs a user memory network to model the long-term interests in a fine-grained way by improving the memory network, which is ignored by the existing methods; and (iii) develops a preference-aware interactive representation component to fuse the long-term and short-term interests of users based on gating to understand the evolution of unbiased preferences of users. Extensive experiments on two real e-commerce user behavior datasets show that DUMN has a significant improvement over the state-of-the-art baselines. The code of DUMN model has been uploaded as an additional material.

Hao Fu, Shaojun Zhou, Qihong Yang et al.

The pre-training models such as BERT have achieved great results in various natural language processing problems. However, a large number of parameters need significant amounts of memory and the consumption of inference time, which makes it difficult to deploy them on edge devices. In this work, we propose a knowledge distillation method LRC-BERT based on contrastive learning to fit the output of the intermediate layer from the angular distance aspect, which is not considered by the existing distillation methods. Furthermore, we introduce a gradient perturbation-based training architecture in the training phase to increase the robustness of LRC-BERT, which is the first attempt in knowledge distillation. Additionally, in order to better capture the distribution characteristics of the intermediate layer, we design a two-stage training method for the total distillation loss. Finally, by verifying 8 datasets on the General Language Understanding Evaluation (GLUE) benchmark, the performance of the proposed LRC-BERT exceeds the existing state-of-the-art methods, which proves the effectiveness of our method.