題目一：Adaptive sampling for PINNs & Deep Ritz
内容簡介：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.
報告人：周濤
報告人簡介：中國科學院數學與系統科學研究院研究員，主要研究方向為不确定性量化、偏微分方程數值方法以及時間并行算法等。在國際權威期刊SIAM Review、SINUM、JCP等發表論文80餘篇。國家高層次人才計劃入選者。2018年擔任國防科工局《核挑戰專題》不确定性量化方向首席科學家。2022年獲第三屆王選傑出青年學者獎。現擔任SIAM J Numer Anal.、SIAM J Sci Comput.、J Sci Comput.等十餘種國内外權威期刊編委，并擔任東亞工業與應用數學學會副主席及學會期刊EAJAM主編。
題目二：An energy-stable variable-step L1 scheme for time-fractional Navier-Stokes equations
内容簡介：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.
報告人：李東方
報告人簡介：華中科技大學數學與統計學院教授，博導，入選國家級青年人才計劃。主持國家自然科學基金2項，科技部課題3項，參與國家自然科學基金重點項目和863課題1項。主要從事微分方程數值解、機器學習和信号處理等領域的研究工作。尤其在微分方程保結構算法和分數階微分方程的高效數值算法和理論上取得一些有意義的進展。相關工作發表在《SIAM. J. Numer. Anal.》，《SIAM. J. Sci. Comput.》、《Math. Comput》、《J. Comp. Phys.》等多個國際著名計算學科SCI期刊上，其中10多篇為高被引論文。
時 間：2024年7月3日（周三）上午9：30開始
地 點：騰訊會議902-396-058
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