In general, Dr. Meng is broadly interested in computational mathematics, inverse problems, machine learning, and data science.  His long-term research goal is to work as an applied mathematician to address interdisciplinary challenges spanning science, engineering, and technology.  

In particular, Dr. Meng is interested in wave propagation and inverse problems: such as inverse scattering problem, inverse source problem, eigenvalue problem, homogenization, periodic structure, reduced order modeling, and machine learning.

(1) Learning theory and algorithm for inverse problems

In recent decades, machine learning has demonstrated its enormous success in recognition, computer vision, and many problems in science, engineering, economics and technology.  Dr. Meng is interested in exploring powerful learning theory and algorithm to develop more efficient and robust imaging algorithms with better accuracy evaluated within a reduced order model setting. Current research topics include: learning data-driven basis, feature learning, kernel machine, and deep learning.

(2) sampling method and eigenvalue problem in inverse scattering

Different from classical iterative methods,  the so-called sampling (or qualitative) methods  require less a prior information (i.e., avoids info such as number of connectivity components and types of boundary conditions), and moreover, they provide solid mathematical justifications in addition to efficient numerical algorithms. The sampling methods utilize only the measurement data and therefore they are data-driven. Dr. Meng's research has been focused on development of sampling methods in complex media with various types of data such as (1) full multi-static data at a fixed frequency, (2) limited-aperture data, (3) sparse data (i.e. data measured at a few sparse points/directions), and (4) multi-frequency data. 

On the other hand, eigenvalues associated with certain wave phenomena contain information about the material properties of  the scattering medium and can be determined from measurements of the scattered wave field. Thus, they can be used as target signatures and provide an alternative way for carrying out non-destructive testing and imaging. They play a central role in the analysis of sampling methods. Dr. Meng's research has been focused on the spectral analysis of such eigenvalues and their corresponding inverse problems.

(3) Wave propagation in periodic media

In recent decades, periodic and complex media have been used with remarkable success to manipulate waves toward achieving super-focusing, sub-wavelength imaging, cloaking, and topological insulation. Dr. Meng's research has been focosed on higher-order homogenization and novel physical phenomena in periodic structures.