Research

Research Interests: Numerical Analysis and Machine Learning

My research areas cover a wide range of classical and modern topics in applied mathematics. I am interested in the numerical solution of nonlinear PDEs and underlying theoretical analysis. My primary focus is an interdisciplinary approach for both mathematical and numerical analysis of fluid equations, which provides deeper insights into the efficient and robust numerical approximation for PDEs. Recently, I am also interested in the mathematical analysis of machine learning with a particular emphasis on solving PDEs to establish theoretical backgrounds for effective computational methods using deep neural networks.

• Numerical methods for nonlinear PDEs: finite element methods and underlying PDE analysis

• Scientific machine learning: physics-informed machine learning, mathematical theory of neural network approximation

• Uncertainty quantification: numerical approximation of probabilistic PDEs, high-dimensional integration

Publications & Preprints

1. VS-PINN: A Fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behavior with Sang Hyeon Park,  arXiv:2406.06287 [math.NA], submitted (2024).

2. Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEs  with Youngjoon Hong, Jae Yong Lee, arXiv:2404.17868 [math.NA], submitted (2024).

3. Some Liouville-type theorems for the stationary 3D magneto-micropolar fluids with Jae-Myoung Kim, arXiv:2204.05759 [math.AP], Acta Math. Sci.  (accepted for publication) (2024).

4. Finite Element Operator Network for Solving Parametric PDEs with Jae Yong Lee, Youngjoon Hong, arXiv:2308.04690 [math.NA], submitted (2023).

5. A novel approach for wafer defect pattern classification based on topological data analysis with Dowan Koo,  Expert Systems with Applications, 120765, (2023). DOI: https://doi.org/10.1016/j.eswa.2023.120765

6. Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs with Youngjoon Hong, Seok-Bae Yun, arXiv:2211.08900 [math.NA], submitted (2022).

7. Quasi-Monte Carlo finite element approximation of the Navier–Stokes equations with random initial data with Guanglian Li, Yi Yu,  arXiv:2210.15572 [math.NA], submitted (2022)

8. Upper and lower bounds of convergence rates for strong solutions of the generalized Newtonian fluids with non-standard growth conditions with Jae-Myoung Kim,  Z. Angew. Math. Phys 73 (251),  (2022). DOI: https://doi.org/10.1007/s00033-022-01871-0.

9. Temporal decay of strong solutions for generalized Newtonian fluids with variable power-law index, J. Math. Phys. Volume 63(4), (2022). DOI: https://doi.org/10.1063/5.0074567.

10. Existence of global weak solutions for unsteady motions of incompressible chemically reacting generalized Newtonian fluids, J. Math. Anal. Appl. Volume 513(1) (2022). DOI: https://doi.org/10.1016/j.jmaa.2022.126206.

11. Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index with Endre Suli,  Mathematics of Computation, Volume 88(317): 1061-1090 (2019). DOI: https://doi.org/10.1090/mcom/3379.

12. Finite element approximation of an incompressible chemically reacting non-Newtonian fluid with Petra Pustejovska and Endre Suli,  ESAIM: M2AN, Volume 52 (2018) 509–541. DOI: https://doi.org/10.1051/m2an/2017043.

Works in Progress

1. Fully discrete finite element approximation for unsteady motions of incompressible non-Newtonian fluids with concentration-dependent power-law index with Endre Suli.

2. Multi-index Monte Carlo finite element approximation for the general class of parabolic equations with Josef Dick.

3. Error estimate for the fully-discrete finite element approximation of the generalized Stokes system with variable nonlinearity with Alex Kaltenbach.

4. On the existence of strong solutions for unsteady motion of incompressible chemically reacting generalized Newtonian fluid with Kyungkeun Kang and Kyueon Choi.

5. Finite element operator learning methods for incompressible Navier-Stokes equations with Yongjoon Hong and Jaeyong Lee.

6. Quasi-Monte Carlo finite element approximation of the singularly perturbed convection-diffusion equation with random velocity fields with Guanglian Li and Shubin Fu.