学　术

　　教师发展中心“学者论坛”活动特别邀请香港科技大学项阳教授、东方理工高等研究院/美国普渡大学沈捷教授来校作学术交流，具体安排如下，欢迎广大师生参加。

　　一、时　间：2023年6月27日（周二）14:30

　　二、地　点：清水河校区主楼A1-512

　　三、主持人：数学科学学院 魏朝祯研究员、顾亦奇研究员

　　四、主　题：微分方程数值解先进方法研讨会

　　五、讲座题目、内容和主讲人介绍

　　主题一：Deep Operator-Splitting Network for Solving PDEs

　　主讲人：香港科技大学 项阳 教授

　　内容简介：

　　Deep neural networks (DNNs) recently emerged as a promising tool for analyzing and solving complex differential equations arising in science and engineering applications. Alternative to traditional numerical schemes, learning based solvers utilize the representation power of DNNs to approximate the input-output relations in an automated manner. However, the lack of physics-in-the-loop often makes it difficult to construct a neural network solver that simultaneously achieves high accuracy, low computational burden, and interpretability. In this work, focusing on a class of evolutionary PDEs characterized by decomposable operators, we show that the classical “operator splitting” technique can be adapted to design neural network architectures. This gives rise to a learning-based PDE solver, which we name Deep Operator-Splitting Network (DOSnet). Such non-black-box network design is constructed from the physical rules and operators governing the underlying dynamics, and is more efficient and flexible than the classical numerical schemes and standard DNNs. To demonstrate the advantages of our new AI-enhanced PDE solver, we train and validate it on several types of operator-decomposable differential equations. We also apply DOSnet to nonlinear Schrodinger equations which have important applications in the signal processing for modern optical fiber transmission systems, and experimental results show that our model has better accuracy and lower computational complexity than numerical schemes and the baseline DNNs.

　　主讲人简介：

　　项阳，香港科技大学数学系教授，港深协同创新研究院机器学习和自动驾驶算法研究实验室负责人。现任东亚工业与应用数学学会主席。主要研究方向是计算数学，机器学习理论及应用，应用领域包括材料科学和人工智能等前沿方向。项阳教授在与缺陷相关的材料结构建模与计算和基于深度神经网络的高性能计算上取得了一系列具有重要意义的原创性成果，是2021年美国工业与应用数学协会（SIAM）材料科学中数学问题会议大会报告人，入选斯坦福大学发布的全球TOP 2%科学家。

　　主题二：Efficient positivity/bound/length preserving schemes for complex nonlinear systems

　　主讲人：东方理工高等研究院/美国普渡大学 沈捷 教授

　　内容简介：

　　Solutions of a large class of partial differential equations (PDEs) arising from sciences and engineering applications are required to be positive or within a specified bound or preserve the length, and also energy dissipative.

　　It is of critical importance that their numerical approximations preserve these structures at the discrete level, as violation of these structures may render the discrete problems ill-posed or inaccurate.

　　I will review the existing approaches for constructing positivity/bound preserving schemes, and then present several efficient and accurate approaches: (i) through reformulation as Wasserstein gradient flows; (ii) through a suitable functional transform; and (iii) through a Lagrange multiplier. These approaches have different advantages and limitations, are all relatively easy to implement, and can be combined with most spatial discretizations.

　　主讲人简介：

　　沈捷，东方理工高等研究院/美国普渡大学教授，国际著名计算数学家。沈捷教授的研究兴趣是数值分析和科学计算，以及在计算流体力学和材料科学中的应用。他在数值分析和科学计算的许多领域做出了重大贡献，包括Navier-Stokes方程高效数值格式的设计和分析，以及在一大类偏微分方程中的谱-伽辽金方法研究。2017年当选美国数学会会士，2020年当选美国工业与应用数学学会会士。沈捷教授已在SIAM Review，SIAM J. Numer. Anal.，SIAM J. Sci. Comput.，Numer. Math.，Math. Comp.等国际著名期刊上发表学术论文200余篇，文章引用次数已超过22000次。

　　六、主办单位：教师发展中心

　　　　承办单位：数学科学学院