题目一：Adaptive sampling for PINNs & Deep Ritz
内容简介：We present a deep adaptive sampling method for solving PDEs where deep neural networks are utilized to approximate the solutions. More precisely, we propose the failure informed adaptive sampling for PINNs and an adaptive important sampling scheme for deep Ritz. Both approaches can adaptively refine the training set with the goal of reducing the failure probability. Applications to both forward and inverse PDEs problems will be presented.
报告人：周涛
报告人简介：中国科学院数学与系统科学研究院研究员，主要研究方向为不确定性量化、偏微分方程数值方法以及时间并行算法等。在国际权威期刊SIAM Review、SINUM、JCP等发表论文80余篇。国家高层次人才计划入选者。2018年担任国防科工局《核挑战专题》不确定性量化方向首席科学家。2022年获第三届王选杰出青年学者奖。现担任SIAM J Numer Anal.、SIAM J Sci Comput.、J Sci Comput.等十余种国内外权威期刊编委，并担任东亚工业与应用数学学会副主席及学会期刊EAJAM主编。
题目二：An energy-stable variable-step L1 scheme for time-fractional Navier-Stokes equations
内容简介：We propose a structure-preserving scheme and its error analysis for time-fractional Navier-Stokes equations (TFNSEs) with periodic boundary conditions. The equations are first rewritten as an equivalent system by eliminating the pressure explicitly. Then, the spatial and temporal discretization are done by the Fourier spectral method and variable-step L1 scheme, respectively. It is proved that the fully-discrete scheme is energy-stable and divergence-free. The energy is an asymptotically compatible one since it recovers the classical energy when $\alpha\rightarrow 1$. Moreover, optimal error estimates are presented very technically by the obtained boundedness of the numerical solutions and some Sobolev inequalities. To our knowledge, they are the first results of the construction and analysis ofstructure-preserving schemes for TFNSEs. Several interesting numerical examples are given to confirm the theoretical results at last.
报告人：李东方
报告人简介：华中科技大学数学与统计学院教授，博导，入选国家级青年人才计划。主持国家自然科学基金2项，科技部课题3项，参与国家自然科学基金重点项目和863课题1项。主要从事微分方程数值解、机器学习和信号处理等领域的研究工作。尤其在微分方程保结构算法和分数阶微分方程的高效数值算法和理论上取得一些有意义的进展。相关工作发表在《SIAM. J. Numer. Anal.》，《SIAM. J. Sci. Comput.》、《Math. Comput》、《J. Comp. Phys.》等多个国际著名计算学科SCI期刊上，其中10多篇为高被引论文。
时 间：2024年7月3日（周三）上午9：30开始
地 点：腾讯会议902-396-058
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2024年6月25日