533 citations · 537 across the 6 of their papers we have counts for
6 papers
Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes Equation
Thomas Hou, Yixuan Wang, Changhe Yang
The nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equations is one of the central open problems in mathematical fluid dynamics. In this pape…
High precision PINNs in unbounded domains: application to singularity formulation in PDEs
Yixuan Wang, Ziming Liu, Zongyi Li +2
We investigate the high-precision training of Physics-Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs…
Axisymmetric type II blowup solutions to the three-dimensional Keller-Segel system
Thomas Y. Hou, Van Tien Nguyen, Peicong Song
We construct axisymmetric solutions to the three-dimensional parabolic-elliptic Keller-Segel system that blow up in finite time. In particular, the singularity is of type II, which…
On the expressiveness and spectral bias of KANs
Yixuan Wang, Jonathan W. Siegel, Ziming Liu +1
Kolmogorov-Arnold Networks (KAN) \cite{liu2024kan} were very recently proposed as a potential alternative to the prevalent architectural backbone of many deep learning models, the…
On the stability of blowup solutions to the complex Ginzburg-Landau equation in R^d
Jiajie Chen, Thomas Y. Hou, Van Tien Nguyen +1
Building upon the idea in \cite{HNWarXiv24}, we establish stability of the type-I blowup with log correction for the complex Ginzburg-Landau equation. In the amplitude-phase repres…
KAN: Kolmogorov-Arnold Networks
Ziming Liu, Yixuan Wang, Sachin Vaidya +5
Inspired by the Kolmogorov-Arnold representation theorem, we propose Kolmogorov-Arnold Networks (KANs) as promising alternatives to Multi-Layer Perceptrons (MLPs). While MLPs have…