Research 

Current Research

My general interests in research lies in applied math and scientific computing. Currently, I am working on (1) intrinsic complexity of models and data sets;  (2) imaging with  transport/diffusion models rising from biomedical imaging;  (3) scientific computing of radiative transfer ; (4) PDE learning and neural networks.  Here is a selection of past projects I have worked on in the past years.

PDE learning and neural networks

The PDEs are popular models for various phenomena. Unveiling the underlying laws in PDE forms from the data has been an active research topic in the machine learning community. However, the actual setting of the problem fits in the framework of typical inverse problems.  There are many computational works based on regression that emphasize "interpretability" in terms of sparsity (or certain penalization from domain knowledge). At the same time, there is very little attention to the feasibility (well-posedness) of the proposed problem, that is, does the data include sufficient information for the learning task?  Neural networks are quite prevailing in various fields of research,  although the flexibility enables extraordinarily good approximation error for general approximation, it also complicates the training process. Normally, the oscillatory functions are complicated to approximate with the networks due to the implicit integral geometry of activation.  Related works: 

Intrinsic complexity of models and data sets

Big data is a popular topic nowadays,  despite the high dimension of the phase space, it has been widely believed that the nature of data approximately lives on a low dimensional space. The intrinsic complexity is studying the minimal dimensions or parameters to describe the data with sufficient accuracy. In the linear space, the problem is equivalent to the Kolmogorov n-width, which is to find the best linear space to approximate the data.  For special settings, such as a random field or sampled random vectors, the intrinsic complexity will depend on the covariance function's smoothness and the scales. For integral equations derived from physical systems such as the Lippmann-Schwinger equation or Peierls integral equation, the intrinsic complexity will depend on the separability of the integral kernels.  In the inverse problem, the intrinsic complexity has a strong relationship with the instability estimate. Related works:

Imaging with transport/diffusion models

The transport model has found its applications on various areas, including biomedical imaging, astronomy, nuclear engineering, remote sensing, seismology, oceanology and so on. Depending on the medium's property, the model could live in different scales. The most common type of inverse transport problem consists of reconstructing medium's parameters from the boundary measurements (Albedo operator or its variants, travel-times).  Under the diffusion regime, that is, the mean-free-path is much smaller than the medium's characteristic length, then the transport model is often reduced to the diffusion model.  Related works:

Computations of radiative transport equation

The computing of radiative transport equation (or even Boltzmann equation) is regarded as a difficult task since it involves high dimensionality and has quite different natures under different settings.  The average lemma has shown that the angular space's integration has better regularity which could be a way to reduce the complexity of the computation. In the simplest cases that the scattering phase function is isotropic or highly separable, it is possible to formulate an integral system with weakly singular kernels.  If Krylov subspace method is used to solve the linear system, then fast algorithms could be utilized to accelerate. If the scattering phase function involves too many modes, then the resulting kernels are highly oscillatory and the intrinsic numerical complexity will grow accordingly. Related works:

Implicit boundary integration

Numerically, the implicit boundary integral method provides a flexible quadrature rule depending on the surface that the integral lives on. Instead of on the surface, the discretization will live in a tube around the surface and be approximated through grid points. The method could be used to solve many quadrature-related problems, like integral equations on complicated surfaces. One of the applications is the macromolecular electronic potential. The macromolecular electronic potential is an important quantity in molecular biology and chemistry. The electronic potential could be used to determine the nature of the molecule's chemical bond. In biology, a DNA-binding protein mostly possesses a pocket of positive charges, which could help to bind the DNA from its phosphates with negative charges.  For interactions of two protein molecules, we have to look for a pocket of negative charges on one protein and a bump of positive charge on the other (or vice versa). Calculating the electronic potential map on the molecule could help to indicate a possible interaction site.   Related work:

Undergraduate Research Programs

I was the project co-leader (with Prof. Hongkai Zhao) for the summer DOmath project "Mathematical clairvoyant: computational inverse problems", the project amounts to letting the undergraduate students put their hands on the CT scan reconstruction from the perspectives of both theory and practical implementation.  See details of the project here.

I was a core team member (with Prof. Richard Moore,  et al.)  for the project "Actuating Platform Systems" proposed by Boeing company,  the workshop offers an opportunity for academia researchers to apply their knowledge to specific problems posed by a group of participating governmental and private companies.  See the news about the event here.  

I have taken place in the DRP twice when I was a graduate student. The DRP is to pair undergraduates with graduate mentors to undertake independent projects in mathematics, the project is based on reading through a book or an article, see the website of the program here.  

Future Research Projects

Besides the primary research topics, I keep expanding a variety of interests in other areas, including optimization, graph theory, random networks, dynamic systems, randomized algorithms, high dimensional probability,  mathematical problems in data science, etc. I have a GitHub profile hosting my research codes and I also used to keep a research blog on GitHub (stopped in  Jan. 2020 due to the pandemic).

In the following, I list a few questions (most are possibly open,  and some pages are not maintained since the Google site is not friendly to LaTeX) that I have been working on, thinking about, or am interested in. The underlying physics looks straightforward but the rigorous mathematical theories are always more difficult.  

If you find any of these resolved or any topic interesting, please feel free to discuss them with me at yimin.zhong@auburn.edu.