题目: Deep learning based iteration scheme approximation for solving PDEs
报告人:李颖 上海大学计算机学院 副教授
时间:2023年8月12 日星期六下午 15:00
地点:上海大学宝山校区东区土木楼 312
专家简介
李颖，博士，副教授，上海大学 计算机学院。2014 年毕业于西安交通大学数学与统计学院，博士期间曾在美国布朗大学和中科院深圳先进技术研究院访问学习。主要从事科学与工程计算、机器学习、迁移学习、图像处理方面的研究和应用。相应成果发表于《Jounal of computational Physics》、《Applied Mathematical Modelling》、《Computers & mathematics with applications》、《Journal of Alloys and Compounds》《Scripta Materialia》等国际顶级期刊中。主持国家自然科学基金青年基金，上海市科技英才扬帆计划，横向项目等。研究应用领域主要在工程数值模拟、机器学习等领域。
报告摘要
Solving the high dimensional partial differential equations (PDEs) with the classical nummerical methods is a challenge task. As possessing the power of progressing high dimensional data, deep learning is naturally considered to solve PDEs. This talk focus on deep learning framework based iteration scheme approximation. First, we adopt the implicit multistep method and Runge-Kutta method for timme iteration scheme. Then, such iteration scheme is approximated by a neural network. Due to integrating the physical information of governing equation into time iteration schemes and introducing time-dependent input, our method achieves the continuous time prediction without a mass of interior points. Here, the activation function with adaptive variable adjusts itself during the iteration process. Finally, we present numerical experiments results for some benchmark PDEs, including Burgers.Allen-Cahn. Schrodinger. carburizing and Black-Scholes equations, and verify that the proposed approach is superior to the state-of-the-art techniques on accuracy and flexibility. Moreover, the Frequency Principle is also illustrated by the changes of prediction at different iterations in this paper.
