수학과

 

 

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1. 연사: 신동욱 (아주대학교)

2. 일시: 9월 2일(금) 오후 5~6시

3. 장소: 팔달관 621호

4. 제목: Numerical methods and neural networks for PDEs

5. 초록: Various natural and physical phenomena can be described by partial differential equations (PDEs).  In order to solve these equations, a number of numerical methods have been introduced and developed. In this talk, we consider various numerical methods for PDEs. The finite difference and finite element methods are popular and work very well for the  boundary value problems. These methods are simple and easy to implement, but the mass conservation property does not hold. On the other hands, nonstandard methods such as mixed finite element, Discontinuous Galerkin (DG), hybrid DG and hybrid difference methods are locally conservative and important for certain class of problems, e.g., Darcy flows in porous media and convection diffusion equations.  These methods allow to achieve high-order accuracy with high-order approximations. However, convergence rates are influenced by smoothness of the solutions. An adaptive algorithm aims to overcome slow convergence for non-smooth solutions. Here, we introduce two types of reliable, efficient, and fully computable a posteriori error estimators and show their numerical results.

Recently, deep learning models have been received large attention and applied to a wide range of fields. Most deep learning models are data-driven approaches. In these approaches, it is difficult to predict correctly outside the data domain. Physics informed neural networks (PINNs) overcome this issue by using the knowledge of any physical laws that govern a given data-set. Here, we consider a simple example of PINNs.

 

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