533 citations · 623 across the 21 of their papers we have counts for
21 papers
SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learning
Guangyuan Wang, Mads Toftrup, Sebastian Loeschcke +2
Physics-informed neural networks (PINNs) often face ill-conditioned objectives that limit high-accuracy training. Dense quasi-Newton methods improve local conditioning but require…
Fourier Feature Pyramids for Physics-Informed Neural Networks
Brandon Zhao, Yixuan Wang, Jonathan T. Barron +3
We present an improved neural field architecture for solving partial differential equations (PDEs). Current physics-informed neural networks (PINNs) provide a flexible framework fo…
Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches
Yixuan Wang
This thesis develops numerical and theoretical approaches for understanding and analyzing singularity formation in Partial Differential Equations (PDEs). The singularity formation…
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…
KANO: Kolmogorov-Arnold Neural Operator
Jin Lee, Ziming Liu, Xinling Yu +4
We introduce Kolmogorov--Arnold Neural Operator (KANO), a dual-domain neural operator jointly parameterized by both spectral and spatial bases with intrinsic symbolic interpretabil…
Initialization Schemes for Kolmogorov-Arnold Networks: An Empirical Study
Spyros Rigas, Dhruv Verma, Georgios Alexandridis +1
Kolmogorov-Arnold Networks (KANs) are a recently introduced neural architecture that replace fixed nonlinearities with trainable activation functions, offering enhanced flexibility…
Scale-Consistent Learning for Partial Differential Equations
Zongyi Li, Samuel Lanthaler, Catherine Deng +4
Machine learning (ML) models have emerged as a promising approach for solving partial differential equations (PDEs) in science and engineering. Previous ML models typically cannot…
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…
Finite time blowup for Keller-Segel equation with logistic damping in three dimensions
Jiaqi Liu, Yixuan Wang, Tao Zhou
The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with…
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…
KAN 2.0: Kolmogorov-Arnold Networks Meet Science
Ziming Liu, Pingchuan Ma, Yixuan Wang +2
A major challenge of AI + Science lies in their inherent incompatibility: today's AI is primarily based on connectionism, while science depends on symbolism. To bridge the two worl…
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…
Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier-Stokes
Thomas Y. Hou, Yixuan Wang
We study the singularity formation of a quasi-exact 1D model proposed by Hou-Li in \cite{hou2008dynamic}. This model is based on an approximation of the axisymmetric Navier-Stokes…
Exponentially Convergent Multiscale Finite Element Method
Yifan Chen, Thomas Y. Hou, Yixuan Wang
We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale s…
FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation
Adarsh Ganeshram, Haydn Maust, Valentin Duruisseaux +6
The physics-informed neural operator (PINO) is a machine learning paradigm that has demonstrated promising results for learning solutions to partial differential equations (PDEs).…
Second Order Ensemble Langevin Method for Sampling and Inverse Problems
Ziming Liu, Andrew M. Stuart, Yixuan Wang
We propose a sampling method based on an ensemble approximation of second order Langevin dynamics. The log target density is appended with a quadratic term in an auxiliary momentum…
Exponentially convergent multiscale methods for high frequency heterogeneous Helmholtz equations
Yifan Chen, Thomas Y. Hou, Yixuan Wang
In this paper, we present a multiscale framework for solving the Helmholtz equation in heterogeneous media without scale separation and in the high frequency regime where the waven…
Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions
Yifan Chen, Thomas Y. Hou, Yixuan Wang
In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main result…
Schrödinger PCA: On the Duality between Principal Component Analysis and Schrödinger Equation
Ziming Liu, Sitian Qian, Yixuan Wang +2
Principal component analysis (PCA) has achieved great success in unsupervised learning by identifying covariance correlations among features. If the data collection fails to captur…
Approximation to Singular Quadratic Collision Model in Fokker-Planck-Landau Equation
Ruo Li, Yanli Wang, Yixuan Wang
We propose a Hermite-Galerkin spectral method to numerically solve the spatially homogeneous Fokker-Planck-Landau equation with singular quadratic collision model. To compute the c…