[NA] Hyea Hyun Kim: Partitioned neural network approximation to partial differential equations and its training performance enhancement utilizing domain decomposition algorithms

31 May 2024 13:00 till 14:00 - Location: Lipkenszaal LB 01.150 | Add to my calendar

With the success of deep learning technologies in many scientific and engineering applications, neural network approximation methods have emerged as an active research area in numerical partial differential equations. However, the new approximation methods still need further validations on their accuracy, stability, and efficiency so as to be used as alternatives to classical approximation methods. In this talk, we first introduce partitioned neural network approximation to partial differential equations, where neural network functions localized in each small subdomains are employed as a solution surrogate in order to reduce the approximation and optimization errors in the standard single large neural network approximation. The parameters in each local neural network function are then optimized to minimize the corresponding cost function to the model problem. To enhance the parameter training efficiency further, iterative algorithms for the partitioned neural network function can be developed by utilizing classical domain decomposition algorithms and their convergence theory. We finally present promising features in this new approach as a way of enhancing the neural network solution accuracy, stability, and efficiency with some supporting numerical results.