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The neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility in representing the solution of the partial differential equation. In training a neural…
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…
Neural operators aim to learn mappings between infinite-dimensional function spaces, but their performance often degrades on complex or irregular geometries due to the lack of geometry-aware representations. We propose the Finite Element…
Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…
In this paper, we present a finite element method (FEM) framework enhanced by an operator-adapted wavelet decomposition algorithm designed for the efficient analysis of multiscale electromagnetic problems. Usual adaptive FEM approaches,…
We extend the finite element interpolated neural network (FEINN) framework from partial differential equations (PDEs) with weak solutions in $H^1$ to PDEs with weak solutions in $H(\textbf{curl})$ or $H(\textbf{div})$. To this end, we…
Designing an optimal deep neural network for a given task is important and challenging in many machine learning applications. To address this issue, we introduce a self-adaptive algorithm: the adaptive network enhancement (ANE) method,…
We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach extends each of the aforementioned methods and unifies them within a single…
Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as…
This paper studies numerical solutions for parameterized partial differential equations (P-PDEs) with deep learning (DL). P-PDEs arise in many important application areas and the computational cost using traditional numerical schemes can be…
We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in $d\geq 1$ dimensions. Our main approach consists of…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error,…
Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete…
While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains $\Omega \subset \mathbb{R} ^d, $ $d=1,2,3$ in association with…
This paper introduces a novel two-stream deep model based on graph convolutional network (GCN) architecture and feed-forward neural networks (FFNN) for learning the solution of nonlinear partial differential equations (PDEs). The model aims…
One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural…
Physics-informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not…
We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…
We propose a new algorithm for Adaptive Finite Element Methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the…