Despite great progress in simulating multiphysics problems using the numerical
discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate …

Neural networks are increasingly used to construct numerical solution methods for partial
differential equations. In this expository review, we introduce and contrast three important …

Deep learning has been shown to be an effective tool in solving partial differential equations
(PDEs) through physics-informed neural networks (PINNs). PINNs embed the PDE residual …

Physics-informed neural networks (PINNs) are demonstrating remarkable promise in
integrating physical models with gappy and noisy observational data, but they still struggle …

Recent advances of data-driven machine learning have revolutionized fields like computer
vision, reinforcement learning, and many scientific and engineering domains. In many real …

Physics-informed neural networks (PINNs) have been popularized as a deep learning
framework that can seamlessly synthesize observational data and partial differential …

Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm
for solving problems relating to differential equations. Compared to classical numerical …

We study the training process of Deep Neural Networks (DNNs) from the Fourier analysis
perspective. We demonstrate a very universal Frequency Principle (F-Principle)---DNNs …

We propose a new type of neural networks, Kronecker neural networks (KNNs), that form a
general framework for neural networks with adaptive activation functions. KNNs employ the …

In this paper, we propose multi-scale deep neural networks (MscaleDNNs) using the idea of
radial scaling in frequency domain and activation functions with compact support. The radial …