Xiaoqiang Wang

Ze Cheng, Zhuoyu Li, Xiaoqiang Wang et al.

PDE-Constrained Optimization (PDECO) problems can be accelerated significantly by employing gradient-based methods with surrogate models like neural operators compared to traditional numerical solvers. However, this approach faces two key challenges: (1) **Data inefficiency**: Lack of efficient data sampling and effective training for neural operators, particularly for optimization purpose. (2) **Instability**: High risk of optimization derailment due to inaccurate neural operator predictions and gradients. To address these challenges, we propose a novel framework: (1) **Optimization-oriented training**: we leverage data from full steps of traditional optimization algorithms and employ a specialized training method for neural operators. (2) **Enhanced derivative learning**: We introduce a *Virtual-Fourier* layer to enhance derivative learning within the neural operator, a crucial aspect for gradient-based optimization. (3) **Hybrid optimization**: We implement a hybrid approach that integrates neural operators with numerical solvers, providing robust regularization for the optimization process. Our extensive experimental results demonstrate the effectiveness of our model in accurately learning operators and their derivatives. Furthermore, our hybrid optimization approach exhibits robust convergence.

Gaoxiang Zhao, Li Zhou, Xiaoqiang Wang

Long-term time series forecasting focuses on leveraging historical data to predict future trends. The core challenge lies in effectively modeling dependencies both within sequences and channels. Convolutional Neural Networks and Linear models often excel in sequence modeling but frequently fall short in capturing complex channel dependencies. In contrast, Transformer-based models, with their attention mechanisms applied to both sequences and channels, have demonstrated strong predictive performance. Our research proposes a new approach for capturing sequence and channel dependencies: AverageTime, an exceptionally simple yet effective structure. By employing mixed channel embedding and averaging operations, AverageTime separately captures correlations for sequences and channels through channel mapping and result averaging. In addition, we integrate clustering methods to further accelerate the model's training process. Experiments on real-world datasets demonstrate that AverageTime surpasses state-of-the-art models in predictive performance while maintaining efficiency comparable to lightweight linear models. This provides a new and effective framework for modeling long time series.

Xiaoqiang Wang, Lejia Gu, Joseph Heung-wing Joseph Lee et al.

In this paper, we present a quantum singular value decomposition algorithm for third-order tensors inspired by the classical algorithm of tensor singular value decomposition (t-svd) and then extend it to order-$p$ tensors. It can be proved that the quantum version of the t-svd for a third-order tensor $\mathcal{A} \in \mathbb{R}^{N\times N \times N}$ achieves the complexity of $\mathcal{O}(N{\rm polylog}(N))$, an exponential speedup compared with its classical counterpart. As an application, we propose a quantum algorithm for recommendation systems which incorporates the contextual situation of users to the personalized recommendation. We provide recommendations varying with contexts by measuring the output quantum state corresponding to an approximation of this user's preferences. This algorithm runs in expected time $\mathcal{O}(N{\rm polylog}(N){\rm poly}(k)),$ if every frontal slice of the preference tensor has a good rank-$k$ approximation. At last, we provide a quantum algorithm for tensor completion based on a different truncation method which is tested to have a good performance in dynamic video completion.