This document presents comprehensive historical accounts on the developments of finite
element methods (FEM) since 1941, with a specific emphasis on developments related to …

In this paper, we introduce a new approach based on distance fields to exactly impose
boundary conditions in physics-informed deep neural networks. The challenges in satisfying …

Physics-informed neural network architectures have emerged as a powerful tool for
develo** flexible PDE solvers that easily assimilate data. When applied to problems in …

We introduce a class of Sparse, Physics-based, and partially Interpretable Neural Networks
(SPINN) for solving ordinary and partial differential equations (PDEs). By reinterpreting a …

Spectral methods are an important part of scientific computing's arsenal for solving partial
differential equations (PDEs). However, their applicability and effectiveness depend crucially …

In recent years, remarkable achievements have been made in artificial intelligence tasks
and applications based on deep neural networks (DNNs), especially in the fields of vision …

This paper presents a general Convolution Hierarchical Deep-learning Neural Networks (C-
HiDeNN) computational framework for solving partial differential equations. This is the first …

We propose to enhance the training of physics-informed neural networks. To this aim, we
introduce nonlinear additive and multiplicative preconditioning strategies for the widely used …

While deep learning and data-driven modeling approaches based on deep neural networks
(DNNs) have recently attracted increasing attention for solving partial differential equations …

We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet
boundary conditions, we estimate the error when the boundary values are imposed through …