Morning Session:11:35 -12:35 pm

Recent Advances in Statistics and Applied Math

1. Sarah H Murphy: shelfert@uncc.edu

Title: Stacking PINNs for Dynamical Systems 

(zoom link: https://charlotte-edu.zoom.us/j/92756582647?pwd=TlJVZVRLcEEzSDl5WTg4cEx4ejBRQT09)

2. Phuong (Mai) Nguyen: pnguye45@uncc.edu (in-person)

Title: Numerical differentiation by the polynomial-exponential basis

3. Elaine Gorom-Alexander: egorom@uncc.edu (in-person)

Title: A Study of Bi-Material Nonlocal Dynamic Properties

1 Title: Stacking PINNs for Dynamical Systems

Abstract: Physics-informed neural networks have demonstrated vast potential for effectively solving systems of equations for modeling physical systems. However, for some dynamical systems, PINNs can be difficult or impossible to train. This work is focused on modeling the behavior of chaotic dynamical systems using physics-informed neural networks. We consider a novel multifidelity framework for stacking physics-informed neural networks that will allow for modeling systems such as the Lorenz system. We iteratively train a multifidelity PINN for a number of steps, where the low-fidelity model at each step takes the output of the previous step as input. This method allows for fast computations of the dynamics of complex systems of equations, without the need for computationally expensive codes. The proposed method is particularly useful for dynamical systems where standard PINNs fail to train, such as the pendulum equation, the wave equation, and the Lorenz system. 

2. Title: Numerical differentiation by the polynomial-exponential basis

Abstract: In this talk, we studied the problem of how to compute derivatives of data that include noise. This task is challenging because even small amounts of noise can lead to significant errors in computation. To address this issue, we proposed an approach that involves approximating the data by removing high-frequency terms in its Fourier expansion with respect to a particular basis for the L2 normed space, called the polynomial-exponential basis. This truncation technique helps to regularize the problem, meanwhile, the use of the polynomial-exponential basis preserves the accuracy of the computation. We provided numerical examples in one and two dimensions to demonstrate the effectiveness of this approach.

3 Title: A Study of Bi-Material Nonlocal Dynamic Properties

Abstract: In this talk, we couple peridynamics to peridynamics at an interface with two materials. When two different materials meet, the resulting system is more susceptible to fracture near their interface. While peridynamics can be more computationally expensive than other models, it brings high accuracy and naturally allows the simulation of crack propagation in its model due to its use of integro-differentials and time derivatives instead of the spatial derivatives typical of classical models.

We prove conservation of momentum and conservation of energy for this peridynamics-to-peridynamics system. Utilizing the conservation properties, we calculate the reflection coefficients of a plane wave near the interface between the two materials. We use this coefficient of reflection to calculate the error between the peridynamics framework and classical models. Using the coefficient of reflection near the interface, we then design a novel numerical scheme that reduces numerical artifacts and preserves conservation laws of the system. A comprehensive stabilityanalysis and error estimate is conducted for the numerical scheme.