MS03: Recent advances in scientific computation for general mathematical models


Session A: Oct.1, 5:00pm-6:20pm, Classroom Building 102
Session B: Oct.2, 10:30am-11:50am, Classroom Building 102

5:00pm-5:20pm (Oct 1)
CLB 102

Shuhao Cao,
University of Missouri-Kansas City

Neural Finite Element Method


Finite Element Methods (FEM) are arguably one of the most widely used numerical methods to approve PDEs in science and engineering disciplines. One of the key elements involves using discrete basis functions on a mesh. Recently, Neural Networks are used to approximate the solutions in a mesh free way.  This method's popularity is largely facilitated by user-friendly interfaces of the auto-differentiation in various deep learning packages such as PyTorch and JAX. However, using NN changes a well-conditioned problem into a nonconvex ill-conditioned problem. The worse part is that, to achieve the same accuracy, NN and optimization-based PDE solver costs hundreds of thousand more Floating Point Operations (FLOPs) than the old faithful FEM. In this talk, inspired by the popular Transformer that powered multiple scientific breakthrough such as AlphaFold 2 and all new Natural Language models such as GPT-3, we shall introduce a new Finite Element Method algorithm natively implemented in PyTorch to accelerate the solution procedure in nonlinear PDEs.


5:20pm-5:40pm (Oct 1)
CLB 102

Weitao Chen,
University of California, Riverside

A chemical-mechanical coupled model for simulating cell morphogenesis and tissue development

  Chemical signals and mechanical properties, as well as the interplay between them, play a critical role in regulating cell growth and tissue development. Most existing mathematical models to study tissue growth focus on either chemical signals or mechanical forces only. In this study, we have developed a multiscale chemical-mechanical coupled model by integrating chemical signaling pathways, cell mechanical properties and cell-cell interaction. The model includes a discrete particle based mechanical submodel and a continuum PDE chemical submodel, integrated by an adaptive mesh generator. It has been applied to simulate the development of Drosophila wing disc tissue to investigate the general principles involved in growth regulation. Our results show that the spatial distribution of the morphogen is critical in determining tissue size and shape. A larger tissue size with a faster growth rate and more symmetric shape can be achieved if the gradient spreads in a larger domain. Together with the absorbing boundary conditions, the feedback regulation that downregulates receptors on the cell membrane allows the further spread of the morphogen away from its source region, resulting in prolonged tissue growth at a more spatially homogeneous growth rate.


5:40pm-6:00pm (Oct 1)
CLB 102

Juntao Huang,
Texas Tech University
Structure-preserving machine learning moment closures for the radiative transfer equation
  In this talk, we present our work on structure-preserving machine learning (ML) moment closure models for the radiative transfer equation. Most of the existing ML closure models are not able to guarantee the stability, which directly causes blow up in the long-time simulations. In our work, with carefully designed neural network architectures, the ML closure model can guarantee the stability (or hyperbolicity). Moreover, other mathematical properties, such as physical characteristic speeds, are also discussed. Extensive benchmark tests show the good accuracy, long-time stability, and good generalizability of our ML closure model.


6:00pm-6:20pm (Oct 1)
CLB 102

Tulin Kaman,
University of Arkansas
Modeling and simulation of turbulent mixing due to hydrodynamic instabilities
  Turbulence modeling is one of the most challenging scientific problem because it requires capturing the chaotic and capricious eddies of flows. Turbulence models are classified according to the governing equations and numerical methods. In this talk, we present a brief summary of numerical approaches that are widely used for simulating compressible turbulence mixing such as (i) Direct Numerical Simulation (DNS), the full Navier-Stokes Equations are resolved without any models for turbulence, (ii) Large Eddy Simulation (LES), the flow field is resolved down to a certain length scale and scales smaller than that are modeled rather than resolved, and (iii) Reynolds-Averaged Navier-Stokes (RANS), the time-averaged equations are solved for mean values of all quantities. We present an increasingly accurate and robust algorithm based on dynamic subgrid scale models and front tracking for the simulation of turbulent mixing.


10:30am-10:50am (Oct 2)
CLB 102

Tracy Stepien,
University of Florida
An approximate Bayesian computation approach for biological model selection and validation
  Mathematical models of cell migration in the context of wound healing, embryonic development, and cancer growth have been developed using a wide variety of frameworks, including reaction-diffusion equations, continuum mechanics, and agent-based models. However, studying model uncertainty or model selection in these settings is less common. We develop a method for studying the appropriateness of model equation components that combines approximate Bayesian computation (ABC) and sensitivity analysis (SA). We provide two case studies in cell migration where we apply this method to sparse experimental data sets of retina development in the eye and tumor-immune dynamics in the brain. We identify model components that can be removed via model reduction using ABC+SA and potential cancer treatment pathways.


10:50am-11:10am (Oct 2)
CLB 102

Chayu Yang,
University of Nebraska-Lincoln
Comparing R0 for a class of PDE epidemic models with ODE models
  We present a general numerical framework to compute the basic reproduction number R0 for a reaction-diffusion epidemic model and compare the value with the associated autonomous ODE model.


11:10am-11:30am (Oct 2)
CLB 102

Yanzhi Zhang,
Missouri University of Science and Technology
Numerical methods for heterogeneous problems with variable-order fractional Laplacian
  In this talk, I will introduce the recently developed meshfree methods based on the radial basis function to solve problems with the variable-order fractional Laplacian. The proposed methods take advantage of the analytical Laplacian of the radial basis functions so as to accommodate the discretization of the classical and variable-order fractional Laplacian in a single framework and avoid the large computational cost for numerical evaluation of the fractional derivatives. Moreover, our methods are simple and easy to handle complex geometry and local refinements, and their computer program implementation remains the same for any dimension d. The effects of variable-order fractional Laplacian will also be discussed.


11:30am-11:50am (Oct 2)
CLB 102

Jia Zhao,
Utah State University
Solving and learning phase field models using the modified physics informed neural networks
  In this talk, we introduce some recent results on solving and learning phase field models using deep neural networks. In the first part, we focus on using the deep neural network to design an automatic numerical solver for the Allen-Cahn and Cahn-Hilliard equations by proposing an adaptive physics informed neural network (PINN). In particular, we propose to embrace the adaptive idea in both space and time and introduce various sampling strategies, such that we are able to improve the efficiency and accuracy of the PINN on solving phase field equations. In the second part, we introduce a new deep learning framework for discovering the phase field models from existing image data. The new framework embraces the approximation power of physics informed neural networks (PINN), and the computational efficiency of the pseudo-spectral methods, which we named pseudo-spectral PINN or SPINN. We will illustrate its approximation power by some interesting examples.