报告题目:Tensor decomposition-based neural operator with dynamic mode decomposition for parameterized time-dependent problems
报告人：孙祥 (中国海洋大学)
时间：2025年8月21日 9:00—9:35
地点：数学学院425报告厅
报告摘要：
Deep operator networks (DeepONets), as a powerful tool to approximate nonlinear mappings between different function spaces, have gained significant attention recently for applications in modeling parameterized partial differential equations. However, limited by the poor extrapolation ability of purely data-driven neural operators, these models tend to fail in predicting solutions with high accuracy outside the training time interval. To address this issue, a novel operator learning framework, TDMD-DeepONet, is proposed in this work, based on tensor train decomposition (TTD) and dynamic mode decomposition (DMD). We first demonstrate the mathematical agreement of the representation of TTD and DeepONet. Then the TTD is performed on a higher-order tensor consisting of given spatial-temporal snapshots collected under a set of parameter values to generate the parameter-, space- and time-dependent cores. DMD is then utilized to model the evolution of the time-dependent core, which is combined with the space-dependent cores to represent the trunk net. Similar to DeepONet, the branch net employs a neural network, with the parameters as inputs and outputs merged with the trunk net for prediction. Furthermore, the feature-enhanced TDMD-DeepONet (ETDMD-DeepONet) is proposed to improve the accuracy, in which an additional linear layer is incorporated into the branch network compared with TDMD-DeepONet. The input to the linear layer is obtained by projecting the initial conditions onto the trunk network. The proposed methods good performance is demonstrated through several classical examples, in which the results demonstrate that the new methods are more accurate in forecasting solutions than the standard DeepONet..
报告人简介： 孙祥，中国海洋大学副教授，硕士生导师。主要研究领域为模型降阶、不确定性量化以及机器学习。在Journal of Computational Physics, Journal of Scientific Computing以及Communications in Computational Physics等计算数学高水平期刊上发表学术论文20余篇。现主持国家自然科学基金青年项目、山东省自然科学基金青年项目、国家实验室科技创新项目、以及国家重点实验稳定支持项目等。
报告题目:AAROC: Reduced Over-Collocation Method with Adaptive Time Partitioning and Adaptive Enrichment for Parametric Time-Dependent Equations
报告人：纪丽洁 (上海大学)
时间：2025年8月21日 9:35—10:10
地点：数学学院425报告厅
报告摘要： Nonlinear and nonaffine terms in parametric partial differential equations can potentially lead to a computational cost of a reduced order model (ROM) that is comparable to the cost of the original full order model (FOM). To address this, the Reduced Residual Reduced Over-Collocation method (R2-ROC) is developed as a hyper-reduction method within the framework of the reduced basis method in the collocation setting. The vanilla R2-ROC method can face instability when applied to parametric fluid dynamic problems. To address this, an adaptive enrichment strategy has been proposed to stabilize the ROC method. However, this strategy can involve in an excessive number of reduced collocation points, thereby negatively impacting online efficiency. To ensure both efficiency and accuracy, we propose an adaptive time partitioning and adaptive enrichment strategy-based ROC method (AAROC). The adaptive time partitioning dynamically captures the low-rank structure, necessitating fewer reduced collocation points being sampled in each time segment. Numerical experiments on the parametric viscous Burgers’ equation and lid-driven cavity problems demonstrate the efficiency, enhanced stability, and accuracy of the proposed AAROC method.
报告人简介： 纪丽洁，上海大学数学系讲师。2021年博士毕业于上海交通大学。2019年至2020年在马萨诸塞大学达特茅斯分校访学一年。2021年至2023年，在上海交通大学博士后流动站从事博士后研究。主要研究方向为电荷输运问题的理论和数值分析、参数化偏微分方程的模型降阶算法、等离子体物理的数值算法以及黑盒优化问题等。在SIAM J. Appl. Math., SIAM J. Sci. Comput., J. Comput. Phys.，J. Sci. Comput.等期刊发表论文。获得2021年上海市超级博士后奖励计划资助，主持过中国博士后科学基金面上1项，现主持国家自然科学基金青年基金1项。
报告题目: Curvature-Dependent Elastic Bending Total Variation Model for Image Inpainting
报告人：南彩霞 (南京师范大学)
时间：2025年8月21日 10:30—11:05
地点：数学学院425报告厅
报告摘要： Image inpainting is pivotal within the realm of image processing, and many efforts have been dedicated to modeling, theory, and numerical analysis in this research area. In this paper, we propose a curvature-dependent elastic bending total variation model for the inpainting problem, in which the elastic bending energy in the phase-field framework introduces geometric information and the total variation term maintains the sharpness of the inpainting edge, referred to as elastic bending-TV model. The energy stability is theoretically proved based on the scalar auxiliary variable method. Additionally, an adaptive time-stepping algorithm is used to further improve the computational efficiency. Numerical experiments illustrate the effectiveness of the proposed model and verify the capability of our model in image inpainting.
报告人简介： 2022年12月毕业于湖南大学数学院，2023年2月-2025年2月在香港理工大学做博后，合作导师是乔中华教授，2025年3月入职南京师范大学数学科学学院。研究方向是非局部模型和图像处理。
报告题目: Modeling human behaviors on epidemic dynamics via an interacting particle system and Keller-Segel PDEs
报告人：熊云丰 (北京师范大学)
时间：2025年8月22日 9:30—10:30
地点：数学学院425报告厅
报告摘要： Human behaviors have non-negligible impacts on spread of contagious disease. For instance, large-scale gathering and high mobility of population could lead to accelerated disease transmission, while public behavioral changes in response to pandemics may effectively reduce contacts and suppress the peak of the outbreak. In order to understand how spatial characteristics like population mobility and clustering interplay with epidemic outbreaks, we formulate an interacting particle system via an agent-based biased random walk model on a two-dimensional lattice. The “popularity” and “awareness” variables are taken into consideration to capture human natural and preventive behavioral factors, which are assumed to guide and bias agent movement in a combined way. Numerical study of the particle system and its continuum limit, a class of Keller-Segel PDEs with logarithm sensitivity, reveals that the spatial heterogeneity, like social influence locality and spatial clustering induced by self-aggregation, potentially suppresses the contacts between agents and consequently flats the epidemic curve. Surprisedly, disease responses might not necessarily reduce the susceptibility of informed individuals and even aggravate disease outbreak if each individual responds independently upon their awareness. The disease control is achieved effectively only if there are coordinated public-health interventions and public compliance to these measures. Our model may help evaluate a variety of public-health policies.
报告人简介： 熊云丰目前为北京师范大学数学科学学院讲师，他2020年毕业于北京大学获理学博士学位，主要研究方向为高维问题的数值方法，包括谱方法和随机粒子方法在高维偏微分方程、随机系统中的应用。目前在国际权威期刊SIAM Journal on Numerical Analysis，SIAM Journal on Scientific Computing, Journal of Computational Physics，PLOS Computational Biology上发表论文十余篇。熊云丰博士获国家自然基金委青年科学基金项目、中国博士后科学基金会站前特别资助项目、面上项目资助，并参与一项国家重点研发计划青年科学家项目。
