**2024 – 2025 Academic Year**

**When: ** October 4^{th}, 2024 from 2:30 p.m. – 3:30 p.m.

**Speaker: **Dongwei Chen (Colorado State University)

**Location:** Virtual via Zoom

**Abstract:** In this talk, I am going to present my latest work on the approximation in reproducing
kernel Hilbert spaces. We generalize the least square method to probabilistic approximation
in reproducing kernel Hilbert spaces and show the existence and uniqueness of the
optimizer. Furthermore, we generalize the celebrated representer theorem in this setting,
and especially when the probability measure is finitely supported or the Hilbert space
is finite-dimensional, we show that the approximation problem turns out to be a measure
quantization problem. Some discussions and examples are also given when the space
is infinite-dimensional, and the measure is infinitely supported. This is a joint
work with Kai-Hsiang Wang from Northwestern University.

**Previous Years**

**Organized by:** McKenzie Black, Thomas Hamori, Chunyan Li

** **Note: Due to the COVID-19 pandemic, we are currently leaving the format of the seminar
up to each individual speaker. To make the seminar as accessible as possible, we will
host a zoom for each in-person presentation live so that anyone who can't or would
prefer not to attend in person can still participate.

- February 25th
- 1:00 pm

**Abstract: **Partial differential equations are often used to model various physical phenomena,
such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics
and so on. Due to their important applications in scientific research and engineering,
many numerical methods have been developed in the past decades for efficient and accurate
solutions of these equations. Inspired by the rapidly growing impact of deep learning
techniques, we propose in this paper a novel neural network method, “GF-Net”, for
learning the Green’s functions of the classic linear reaction-diffusion equations
in the unsupervised fashion. The proposed method overcomes the challenges for finding
the Green’s functions of the equations on arbitrary domains by utilizing the physics-informed
neural network approach and domain decomposition. Consequently, it particularly leads
to a fast algorithm for solving the target equations subject to various sources and
Dirichlet boundary conditions without network retraining. We also numerically demonstrate
the effectiveness of the proposed method by extensive experiments in the square, annular
and L-shape domains.

** **

**Chunyan Li**, University of South Carolina

- February 25
^{th} - 1:00 pm

**Abstract:** In this talk, we will introduce a nonlinear dimensionality reduction method with
neural networks, called VAE. Two parameterized conditional distributions are learned
as the encoder and decoder by minimizing the so called variational lower bound objective
in VAE. We will go through the derivation and reparameterization trick used in this
whole process. Applications will be shown in the end.

**Zongyi Li**, California Institute of Technology

- February 11
^{th} - 1:00 pm

**Abstract:** The classical development of neural networks has primarily focused on learning mappings
between finite dimensional Euclidean spaces or finite sets. We propose a generalization
of neural networks tailored to learn operators mapping between infinite dimensional
function spaces. We formulate the approximation of operators by composition of a class
of linear integral operators and nonlinear activation functions, so that the composed
operator can approximate complex nonlinear operators. We prove a universal approximation
theorem for our construction. Furthermore, we introduce four classes of operator parameterizations:
graph-based operators, low-rank operators, multipole graph-based operators, and Fourier
operators and describe efficient algorithms for computing with each one. The proposed
neural operators are resolution-invariant: they share the same network parameters
between different discretizations of the underlying function spaces and can be used
for zero-shot super-resolutions. Numerically, the proposed models show superior performance
compared to existing machine learning based methodologies on Burgers' equation, Darcy
flow, and the Navier-Stokes equation, while being several order of magnitude faster
compared to conventional PDE solvers.

**Yunkai Teng**, University of South Carolina

- October 29
^{th} - 12:00 pm

**Abstract:** Level Set Learning and Function Approximations on Sparse Data through Pseudo-reversible
Neural Network

**Chunyan Li**, University of South Carolina

- October 15
^{th} - 12:00 pm

**Abstract:** PCA, one of the popular dimensionality reduction methods, is an orthogonal linear
transformation that transforms the data to a new coordinate system. In this talk,
we will learn how to derive this new basis and characterize the structure of all principal
components via SVD of covariance matrix of data. The variant of PCA, Dual PCA and
Kernel PCA are mentioned as well.

**McKenzie Black**, University of South Carolina

- October 1
^{st} - 12:00 pm

**Abstract:** In this talk, we will introduce the Pressure-less Euler Alignment system and update
the system with nonlinear velocity. We explore local well posedness of the system
while discussing varying method to get there. Focusing on the nonlinear velocity,
we introduce a similar system to determine how the magnitude of nonlinearity effects
unconditional flocking and all subsets to follow.

**Thomas Hamori**, University of South Carolina

- September 24
^{th} - 12:00 pm

**Abstract: ** Conservation laws are foundational in fluid dynamics. I will derive conservation
laws for traffic flow from conservation of mass for macroscopic traffic flow models.
A brief discussion will follow regarding the classical theory for macroscopic traffic
flow, and I will present joint work with my advisor Dr. Changhui Tan on a class of
nonlocal traffic models. In these models, the nonlocality is used to combat the nonlinearity
of the PDE. I will show that the nonlocality broadens the class of initial conditions
with global smooth solutions for these models.