Chang-Yeol Jung

Gung-Min Gie, Youngjoon Hong, Chang-Yeol Jung

We propose a new semi-analytic physics informed neural network (PINN) to solve singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that offers a promising perspective for finding numerical solutions to partial differential equations. The PINNs have shown impressive performance in solving various differential equations including time-dependent and multi-dimensional equations involved in a complex geometry of the domain. However, when considering stiff differential equations, neural networks in general fail to capture the sharp transition of solutions, due to the spectral bias. To resolve this issue, here we develop the semi-analytic PINN methods, enriched by using the so-called corrector functions obtained from the boundary layer analysis. Our new enriched PINNs accurately predict numerical solutions to the singular perturbation problems. Numerical experiments include various types of singularly perturbed linear and nonlinear differential equations.

Chang-Yeol Jung, Thien Binh Nguyen

A new adaptive weighted essentially non-oscillatory WENO-$θ$ scheme in the context of finite difference is proposed. Depending on the smoothness of the large stencil used in the reconstruction of the numerical flux, a parameter $θ$ is set adaptively to switch the scheme between a 5th-order upwind and 6th-order central discretization. A new indicator $τ^θ$ measuring the smoothness of the large stencil is chosen among two candidates which are devised based on the possible highest-order variations of the reconstruction polynomials in $L^2$ sense. In addition, a new set of smoothness indicators $\tildeβ_k$'s of the sub-stencils is introduced. These are constructed in a central sense with respect to the Taylor expansions around the point $x_{j}$. Numerical results show that the new scheme combines good properties of both 5th-order upwind schemes, e.g., WENO-JS ([Jiang and Shu, JCP 126 (1996)]), WENO-Z ([Borges et al., JCP 227 (2008)]), and 6th-order central schemes, e.g., WENO-NW6 ([Yamaleev and Carpenter, JCP 228 (2009)]), WENO-CU6 ([Hu el al., JCP 229 (2010)]). In particular, the new scheme captures discontinuities and resolves small-scaled structures much better than the 5th-order schemes; overcomes the loss of accuracy near some critical regions and is able to maintain symmetry which are drawbacks detected in the 6th-order ones.