Morning Session:11:35 -12:25 pm

Recent Advances in Statistics and Applied Math

1. Sarah H Murphy: shelfert@uncc.edu (in-person)

Title: Finite Basis Kolmogorov-Arnold Networks: Domain Decomposition for Data-Driven and Physics-Informed Problems

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

Title: A global approach for the inverse scattering problem using a Carleman contraction map

1 Title: Finite Basis Kolmogorov-Arnold Networks: Domain Decomposition for Data-Driven and Physics-Informed Problems

Abstract: This talk will introduce Kolmogorov-Arnold networks and the use of domain decomposition techniques for improving their training. Kolmogorov-Arnold networks (KANs) have attracted attention recently as an alternative to multilayer perceptrons (MLPs) for scientific machine learning. However, KANs can be expensive to train, even for relatively small networks. Inspired by finite basis physics-informed neural networks (FBPINNs), in this work, we develop a domain decomposition method for KANs that allows for several small KANs to be trained in parallel to give accurate solutions for multiscale problems. We show that finite basis KANs (FBKANs) can provide accurate results with noisy data and for physics-informed training.

2. Title: A global approach for the inverse scattering problem using a Carleman contraction map

Abstract: I will present the inverse scattering problem in a domain Omega in this talk. The input data, measured outside Omega,  involve the waves generated by the interaction of plane waves with various directions and unknown scatterers fully occluded inside Omega. The output of this problem is the spatially dielectric constant of these scatterers. Our approach to solving this problem consists of two primary stages. Initially, we eliminate the unknown dielectric constant from the governing equation, resulting in a system of partial differential equations. Subsequently, we develop the Carleman contraction mapping method to effectively tackle this system. It is noteworthy to highlight this method's robustness. It does not request a precise initial guess of the true solution, and its computational cost is not expensive.Some numerical examples are presented.