The numerical solution of the mixed boundary value problem of the high wavenumber Helmholtz equation is a challenging issue. The high wavenumber and the oddity of the solution to the mixed boundary value problem bring great difficulties to the rapid and accurate numerical solution. The rapid development of AI technology today offers some new ideas for solving such problems. In this report, we introduce a class of learning-based numerical solutions for differential equations (LbNM). Using the existing solutions as data, learn the solution operators, and then quickly obtain high-precision numerical solutions. In particular, we can utilize the detailed analysis results of partial differential equations regarding the junction points of solutions under different boundary conditions to propose using a class of solutions with certain oddities as learning data, which greatly accelerates the accuracy and speed of the algorithm.

Speaker Profile: Cheng Jin, a doctoral supervisor and professor at the School of Mathematical Sciences, Fudan University, and the president of the Shanghai Society for Industrial and Applied Mathematics. Fellow of the Institute of Physics of the United Kingdom, Fellow of the Chinese Society for Industrial and Applied Mathematics, Executive Committee Member of the Eurasian Anti-Problem Alliance, etc. He used to be the vice president of the Chinese Mathematical Society and a member of the expert review group of the Department of Mathematics and Physics of the National Natural Science Foundation of China. member of the NSF review Panel in the United States, editorial board member of several internationally renowned journals, etc. More than 130 papers have been published in domestic and international academic journals. In 2019, I won the First Prize of Shanghai Natural Science Award. In 2020, I won the Second Prize of Shanghai Natural Science Award. In 2022, I won the First Prize of Shanghai Teaching Achievement Award. Significant progress has been made in the theoretical analysis of inverse problems of partial differential equations and efficient inversion algorithms for general inverse problems. In terms of application, effective cooperation has been carried out with domestic and foreign enterprises such as Nippon Steel and Huawei, achieving outstanding results and receiving high praise from the industry.