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Deep Neural Networks (DNNs) are widely used for their ability to effectively approximate large classes of functions. This flexibility, however, makes the strict enforcement of constraints on DNNs an open problem. Here we present a framework…

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on…

Deep neural networks (DNNs) have been widely used to solve partial differential equations (PDEs) in recent years. In this work, a novel deep learning-based framework named Particle Weak-form based Neural Networks (ParticleWNN) is developed…

Deep neural networks (DNNs) have provided brilliant performance across various tasks. However, this success often comes at the cost of unnecessarily large model sizes, high computational demands, and substantial memory footprints.…

In this paper, we will introduce a novel deep model named Reconciled Polynomial Network (RPN) for deep function learning. RPN has a very general architecture and can be used to build models with various complexities, capacities, and levels…

Spectral methods employing non-standard polynomial bases, such as M\"untz polynomials, have proven effective for accurately solving problems with solutions exhibiting low regularity, notably including sub-diffusion equations. However, due…

Neural Networks (NNs) can be used to solve Ordinary and Partial Differential Equations (ODEs and PDEs) by redefining the question as an optimization problem. The objective function to be optimized is the sum of the squares of the PDE to be…

Artificial Intelligence and Machine learning have been widely used in various fields of mathematical computing, physical modeling, computational science, communication science, and stochastic analysis. Approaches based on Deep Artificial…

This book aims to provide an introduction to the topic of deep learning algorithms. We review essential components of deep learning algorithms in full mathematical detail including different artificial neural network (ANN) architectures…

To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve…

Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity…

Graph neural networks (GNNs) are powerful machine learning models for various graph learning tasks. Recently, the limitations of the expressive power of various GNN models have been revealed. For example, GNNs cannot distinguish some…

Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…

We investigate the concept of Best Approximation for Feedforward Neural Networks (FNN) and explore their convergence properties through the lens of Random Projection (RPNNs). RPNNs have predetermined and fixed, once and for all, internal…

The connection of Taylor maps and polynomial neural networks (PNN) to solve ordinary differential equations (ODEs) numerically is considered. Having the system of ODEs, it is possible to calculate weights of PNN that simulates the dynamics…

Deep neural networks (DNNs) lack the precise semantics and definitive probabilistic interpretation of probabilistic graphical models (PGMs). In this paper, we propose an innovative solution by constructing infinite tree-structured PGMs that…

Neural additive models (NAMs) enhance the transparency of deep neural networks by handling input features in separate additive sub-networks. However, they lack inherent mechanisms that provide calibrated uncertainties and enable selection…

In this work we propose a deep adaptive sampling (DAS) method for solving partial differential equations (PDEs), where deep neural networks are utilized to approximate the solutions of PDEs and deep generative models are employed to…

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…

Approximate nearest-neighbor search (ANNS) algorithms are a key part of the modern deep learning stack due to enabling efficient similarity search over high-dimensional vector space representations (i.e., embeddings) of data. Among various…