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In exchange for large quantities of data and processing power, deep neural networks have yielded models that provide state of the art predication capabilities in many fields. However, a lack of strong guarantees on their behaviour have…
Verification of Neural Networks (NNs) that approximate the solution of Partial Differential Equations (PDEs) is a major milestone towards enhancing their trustworthiness and accelerating their deployment, especially for safety-critical…
In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure…
Artificial intelligence and machine learning paves the way to achieve greater technical feats. In this endeavor to hone these techniques, quantum machine learning is budding to serve as an important tool. Using the techniques of deep…
Convoulutional Neural Networks (CNNs) exhibit extraordinary performance on a variety of machine learning tasks. However, their mathematical properties and behavior are quite poorly understood. There is some work, in the form of a framework,…
Neural Networks (NNs) are the method of choice for building learning algorithms. Their popularity stems from their empirical success on several challenging learning problems. However, most scholars agree that a convincing theoretical…
In this review paper, we give a comprehensive overview of the large variety of approximation results for neural networks. Approximation rates for classical function spaces as well as benefits of deep neural networks over shallow ones for…
Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but…
The fields of neural computation and artificial neural networks have developed much in the last decades. Most of the works in these fields focus on implementing and/or learning discrete functions or behavior. However, technical, physical,…
We present a subspace method based on neural networks for solving the partial differential equation in weak form with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a…
Recurrent Neural Networks (RNNs) are a class of machine learning algorithms used for applications with time-series and sequential data. Recently, there has been a strong interest in executing RNNs on embedded devices. However, difficulties…
This chapter aims to provide next-level understanding of the problems of the world and the solutions available to those problems, which lie very well within the domain of neural computing, and at the same time are intelligent in their…
Training deep neural networks is a highly nontrivial task, involving carefully selecting appropriate training algorithms, scheduling step sizes and tuning other hyperparameters. Trying different combinations can be quite labor-intensive and…
Whilst the partial differential equations that govern the dynamics of our world have been studied in great depth for centuries, solving them for complex, high-dimensional conditions and domains still presents an incredibly large…
In this study, we explore the integration of Neural Networks, a powerful class of functions known for their exceptional approximation capabilities. Our primary emphasis is on the integration of multi-layer Neural Networks, a challenging…
Residual deep neural networks (ResNets) are mathematically described as interacting particle systems. In the case of infinitely many layers the ResNet leads to a system of coupled system of ordinary differential equations known as neural…
Neural Networks are function approximators that have achieved state-of-the-art accuracy in numerous machine learning tasks. In spite of their great success in terms of accuracy, their large training time makes it difficult to use them for…
Can neural networks learn to solve partial differential equations (PDEs)? We investigate this question for two (systems of) PDEs, namely, the Poisson equation and the steady Navier--Stokes equations. The contributions of this paper are…
As machine learning has been deployed ubiquitously across applications in modern data science, algorithmic fairness has become a great concern. Among them, imposing fairness constraints during learning, i.e. in-processing fair training, has…
The landscape of low-energy effective field theories stemming from string theory is too vast for a systematic exploration. However, the meadows of the string landscape may be fertile ground for the application of machine learning…