Jiahui Cheng

Research Interests: Computational Math and Statistical Analysis, Generative Modeling.

I have a broad interest in tackling challenging problems , Neural Operator Learning, PDE Identification, and Diffusion Model Distillation.

My previous interest lies high dimensional problem (e.g., Double Descent phenomenon), image-to-image translation (e.g., Pix2pix, Cycle GAN).

I am currently a fourth-year graduate student at Georgia Tech, and I am very fortunate to be advised by Wenjing Liao. Prior to that, I received my bachelor's degree from School of Gifted Young at University of Science and Technology of China (USTC).

Note: Click on [summary] or [highlight] to view summaries or highlight of the papers/projects!

Selected Publications

PDE Identification

• Ensemble Dynamics Guided Weak Formulation for Identifying Differential Equations.
  Jiahui Cheng, Sungha Kang, Haomin Zhou, Wenjing Liao.
  Preprint, 2023
  

Estimation Theory

• Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation. [arxiv] [Summary]
  Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nevertheless, the majority of these studies examine how well deep neural networks can model functions with uniform regularity. In this paper, we explore a different angle: how deep neural networks can adapt to different regularity in functions across different locations and scales and nonuniform data distributions. More precisely, we focus on a broad class of functions defined by nonlinear treebased approximation. This class encompasses a range of function types, such as functions with uniform regularity and discontinuous functions. We develop nonparametric approximation and estimation theories for this function class using deep ReLU networks. Our results show that deep neural networks are adaptive to different regularity of functions and nonuniform data distributions at different locations and scales. We apply our results to several function classes, and derive the corresponding approximation and generalization errors. The validity of our results is demonstrated through numerical experiments. Hao Liu, Jiahui Cheng, Wenjing Liao.
  Preprint, 2024
  

High Dimensional Statistics

• High Dimensional Binary Classification under Label Shift: Phase Transition and Regularization. [arxiv] [Summary]
  Label Shift has been widely believed to be harmful to the generalization performance of machine learning models. Researchers have proposed many approaches to mitigate the impact of the label shift, e.g., balancing the training data. However, these methods often consider the underparametrized regime, where the sample size is much larger than the data dimension. The research under the overparametrized regime is very limited. To bridge this gap, we propose a new asymptotic analysis of the Fisher Linear Discriminant classifier for binary classification with label shift. Specifically, we prove that there exists a phase transition phenomenon: Under certain overparametrized regime, the classifier trained using imbalanced data outperforms the counterpart with reduced balanced data. Moreover, we investigate the impact of regularization to the label shift: The aforementioned phase transition vanishes as the regularization becomes strong. Jiahui Cheng, Minshuo Chen, Hao Liu, Tuo Zhao, Wenjing Liao.
  Sampling Theory, Signal Processing, and Data Analysis, 2022
  

Inverse Problem

• Estimate the spectrum of affine dynamical systems from partial observations of a single trajectory data [arxiv] [Summary]
  In this paper, we study the nonlinear inverse problem of estimating the spectrum of a system matrix, that drives a finite-dimensional affine dynamical system, from partial observations of a single trajectory data. In the noiseless case, we prove an annihilating polynomial of the system matrix, whose roots are a subset of the spectrum, can be uniquely determined from data. We then study which eigenvalues of the system matrix can be recovered and derive various sufficient and necessary conditions to characterize the relationship between the recoverability of each eigenvalue and the observation locations. We propose various reconstruction algorithms, with theoretical guarantees, generalizing the classical Prony method, ESPIRIT, and matrix pencil method. We test the algorithms over a variety of examples with applications to graph signal processing, disease modeling and a real-human motion dataset. The numerical results validate our theoretical results and demonstrate the effectiveness of the proposed algorithms, even when the data did not follow an exact linear dynamical system. Jiahui Cheng, Sui Tang
  Inverse Problem, 2021
  

Solids and Fluids Simulation

• Continuum simulation based on Material Point Method [Summary]
  In this thesis, we study the hybrid Eulerian/ Lagrangian Material Point Method (MPM) and propose a new discretization for the weak form of MPM based on the weak form of force balance and alleviated the energy dissipation. We improve the stability and energy dissipation compared with MPM and Affine Particle in Cell method. We implemented a Java Graphical User Interface (GUI) for visualization of our algorithm for snow simulation. Jiahui Cheng, Weihua Tong
  Bachelor's thesis, 2020
  

Full Publications