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Implicit Neural Representations (INRs) are powerful to parameterize continuous signals in computer vision. However, almost all INRs methods are limited to low-level tasks, e.g., image/video compression, super-resolution, and image…

Our goal is to provide a review of deep learning methods which provide insight into structured high-dimensional data. Rather than using shallow additive architectures common to most statistical models, deep learning uses layers of…

An object-oriented approach to implementing artificial neural networks is introduced in this article. The networks obtained in this way are highly connected in that they admit edges between nodes in any layers of the network, and dynamic,…

This paper aims to accelerate the test-time computation of deep convolutional neural networks (CNNs). Unlike existing methods that are designed for approximating linear filters or linear responses, our method takes the nonlinear units into…

A novel artificial neural network method is proposed for solving Cauchy inverse problems. It allows multiple hidden layers with arbitrary width and depth, which theoretically yields better approximations to the inverse problems. In this…

Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical…

We introduce a new class of non-linear models for functional data based on neural networks. Deep learning has been very successful in non-linear modeling, but there has been little work done in the functional data setting. We propose two…

Using neural networks to solve variational problems, and other scientific machine learning tasks, has been limited by a lack of consistency and an inability to exactly integrate expressions involving neural network architectures. We address…

Physics-informed neural networks (PINNs) are a new tool for solving boundary value problems by defining loss functions of neural networks based on governing equations, boundary conditions, and initial conditions. Recent investigations have…

We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…

Deep learning (DL) is increasingly used to solve ill-posed inverse problems in imaging, such as reconstruction from noisy or incomplete data, as DL offers advantages over explicit image feature extractions in defining the needed prior.…

This manuscript presents a new extended linear system for integral equation based techniques for solving boundary value problems on locally perturbed geometries. The new extended linear system is similar to a previously presented technique…

This paper proposes a new method based on neural networks for computing the high-dimensional committor functions that satisfy Fokker-Planck equations. Instead of working with partial differential equations, the new method works with an…

The paper proposes a novel hybrid method for solving equilibrium problems and fixed point problems. By constructing specially cutting-halfspaces, in this algorithm, only an optimization program is solved at each iteration without the…

The solution of linear inverse problems arising, for example, in signal and image processing is a challenging problem since the ill-conditioning amplifies, in the solution, the noise present in the data. Recently introduced algorithms based…

Despite the highest classification accuracy in wide varieties of application areas, artificial neural network has one disadvantage. The way this Network comes to a decision is not easily comprehensible. The lack of explanation ability…

As artificial intelligence increasingly drives critical decisions, the ability to genuinely explain how neural networks make predictions is essential for trust. Yet, most current explanation methods offer post-hoc rationalizations rather…

Identifying governing equations for a dynamical system is a topic of critical interest across an array of disciplines, from mathematics to engineering to biology. Machine learning -- specifically deep learning -- techniques have shown their…

Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when…

Many types of physics-informed neural network models have been proposed in recent years as approaches for learning solutions to differential equations. When a particular task requires solving a differential equation at multiple…