主持人：郑海标

报告人简介：
张镭，上海交通大学自然科学研究院特别研究员。北京大学学士(1999)，中国科学院计算数学研究所硕士(2002)，加州理工学院博士(2007)，Max-Planck研究所和牛津大学博士后。研究着眼于解决多尺度建模，分析和计算的根本性问题，内容包括偏微分方程，数值分析，以及在材料科学，地球物理科学，生命科学等领域中的广泛应用问题。 在数值均匀化和原子连续耦合方法这两个亟具代表性和挑战性且具有重要的科学与工程背景的领域中做出了一系列开创性的工作。 研究结果发表在Comm. Pure. Appl. Math., SIAM J. Numer. Anal., SIAM MMS (Multiscale Modeling and Simulation)等国际著名学术刊物上。1993年获得国际数学奥赛金牌。2007年获加州理工学院优秀博士论文奖(W.P. Carey Prize in applied mathematics)。 2010-2012年任牛津大学Wolfson学院研究员(Research Fellow)。

报告摘要：
Multiscale partial differential equations (MsPDEs) play a crucial role in applications involving heterogeneous and random media, enabling the prediction of complex phenomena such as reservoir modeling, atmospheric and ocean circulation, and high-frequency scattering. However, classical numerical methods encounter significant challenges when dealing with MsPDEs, especially when their computational cost scales inversely with the finest scale, denoted as $\varepsilon$, of the problem. To address this issue, multiscale solvers have been developed, incorporating microscopic information to achieve computational efficiency independent of $\varepsilon$.

In recent years, there has been a growing interest in utilizing neural network methods to solve multiscale PDEs, despite the spectral bias or frequency principle [1,2], which suggests that deep neural networks (DNNs) struggle to effectively capture high-frequency components of functions. To overcome this challenge, we have developed dedicated neural solvers [3,4] that mitigate the spectral bias and accurately solve multiscale PDEs with fixed parameters.

Moreover, neural operators have emerged as powerful tools for learning the mapping between the infinite-dimensional parameter and solution spaces of PDEs. However, when it comes to multiscale PDEs, the spectral bias towards low-frequency components poses a significant obstacle for existing neural operators. To tackle this challenge, we propose models based on hierarchical attention and convolution [5,6], inspired by the hierarchical matrix approach. Our model incorporates a scale-adaptive interaction range and self-attentions across multiple levels, allowing nested feature computation with controllable linear cost and the encoding/decoding of the multiscale solution space. Additionally, we introduce an empirical $H^1$ loss function to enhance the learning of high-frequency components. Through numerical experiments, we demonstrate that our approach outperforms state-of-the-art methods for representative multiscale problems.

In the end, we show a few applications of those methods in, e.g., ultrasonic imaging and climate science.