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

Understanding deep learning is increasingly emergent as it penetrates more and more into
industry and science. In recent years, a research line from Fourier analysis sheds light on …

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

We propose a new method to deal with the essential boundary conditions encountered in
the deep learning-based numerical solvers for partial differential equations. The trial …

Along with fruitful applications of Deep Neural Networks (DNNs) to realistic problems,
recently, some empirical studies of DNNs reported a universal phenomenon of Frequency …

In this paper, an adaptive multi-scale neural network with Resnet blocks (adaptive-MS-
Resnet) architecture is constructed for solving the Poisson equation, Helmholtz equation …

Neural operators have emerged as a powerful tool for learning the mapping between infinite-
dimensional parameter and solution spaces of partial differential equations (PDEs). In this …

In this paper we present a Fourier feature based deep domain decomposition method (F-
D3M) for partial differential equations (PDEs). Currently, deep neural network based …

We study the problem of learning a single neuron $\mathbf {x}\mapsto\sigma (\mathbf {w}^
T\mathbf {x}) $ with gradient descent (GD). All the existing positive results are limited to the …

Physics-informed neural networks (PINNs) have become a promising research direction in
the field of solving partial differential equations (PDEs). Dealing with singular perturbation …