Yingda Cheng

Why did you choose Lund University?

I chose to apply for the visiting professorship because I was drawn by the excellent research reputation of Lund University. I have been visiting Sweden since 2019, and in particular Uppsala University and KTH. The open and collaborative research environment in Sweden is very appealing to me. 

I knew of the excellent reputation of Lund University in mathematics and met Prof. Philipp Birken before on some committee meetings. The prospect of visiting Lund is very exciting because we will have potential to work on projects of mutual interests with Prof. Birken and Prof. Guo on numerical methods for PDEs and reduced order models. I have also learned about the COMPUTE research schools which focuses on scientific discovery using computing. This initiative fits well with my research and I am eager to make contributions to this effort.

Can you briefly describe your current research? 

My research is in the general are of applied mathematics, high-dimensional scientific computing, and scientific machine learning, including applications in fusion energy and quantum device simulations. The focus of my research is on algorithm design, analysis and applications for partial differential equation (PDE) based models and using/developing effectivetools from data science to compute complex physical behaviors. 

I worked extensively on numerical methods for PDEs, in particular Discontinuous Galerkin (DG) finite element methods. We designed new DG schemes for Hamilton-Jacobi (HJ) equations, equations with higher order derivatives, Maxwell's equations in nonlinear optics and kinetic equations. We provided rigorous analysis of the methods including stability, convergence and superconvergence. 

In recent years, my work has been focused on simulations for kinetic transport models. Kinetic models that describe the evolution of probability distribution functions have wide applications in many areas of mathematical physics, such as rarefied gas dynamics, plasma physics, nuclear engineering, semiconductor device design, astrophysics, traffic network, and swarming. Those models are hard to compute due to the high dimensionality and multiscale nature of the models, yet their understanding are fundamental for important applications such as nuclear fusion. We developed novel numerical techniques for computing such models using dimension reduction techniques (sparse grid and low rank methods) in conjunction with standard PDE solvers such as DG methods. Our methods have been applied to a class of high dimensional PDE problems and showed promise. 

Most recently, we have been using tools from scientific machine learning (ML) to build surrogate models for kinetic systems. We consider moment closure problems, which reduce kinetic equations to a set of moment equations by evolving only the first few moments of the probability density functions. The moment closure problem seeks to represent the highest order moment by low order moments. Traditional analytic techniques run into bottlenecks because they are based on an ansatz which may not hold true for all regimes. 

We proposed an approach to directly learn the gradient of the unclosed high order moment by neural networks, and we further propose to enforce structures in the learned models. Another approach is by the reduced basis method (RBM). We developed the first RBM for kinetic simulations. The novelty is that we treat the angular direction as our parameters to utilize the underlying low rank structure.

What will be your main research focus during your time at Lund University? 

My research focus will be on developing novel numerical methods for physical applications, and developing scientific machine learning tools for data-driven methods in reduced order models. My current research focus on fusion applications, and I am eager to make new contributions in methods that can effectively model and predict fusion plasmas.