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Learning the solution of partial differential equations (PDEs) with a neural network is an attractive alternative to traditional solvers due to its elegance, greater flexibility and the ease of incorporating observed data. However, training…

Machine learning methods are commonly used to solve inverse problems, wherein an unknown signal must be estimated from few indirect measurements generated via a known acquisition procedure. In particular, neural networks perform well…

Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the…

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 studies deep neural networks for solving extremely large linear systems arising from highdimensional problems. Because of the curse of dimensionality, it is expensive to store both the solution and right-hand side vector in such…

Many DNN-enabled vision applications constantly operate under severe energy constraints such as unmanned aerial vehicles, Augmented Reality headsets, and smartphones. Designing DNNs that can meet a stringent energy budget is becoming…

In recent years, deep learning methods, exemplified by Physics-Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high…

A novel boundary element method (BEM) removes the classical dependence on explicit fundamental solutions and extends quasi-optimal BEM discretisations to strongly elliptic operators with variable coefficients. The approach constructs a…

We present an alternative to the pseudo-inverse method for determining the hidden to output weight values for Extreme Learning Machines performing classification tasks. The method is based on linear discriminant analysis and provides Bayes…

Deep learning has become a popular tool across many scientific fields, including the study of differential equations, particularly partial differential equations. This work introduces the basic principles of deep learning and the Deep…

The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolutionized the area of artificial intelligence, machine…

In this paper, we introduce a novel concept for learning of the parameters in a neural network. Our idea is grounded on modeling a learning problem that addresses a trade-off between (i) satisfying local objectives at each node and (ii)…

In the contemporary of deep learning, where models often grapple with the challenge of simultaneously achieving robustness against adversarial attacks and strong generalization capabilities, this study introduces an innovative Local Feature…

Training neural networks is a challenging non-convex optimization problem, and backpropagation or gradient descent can get stuck in spurious local optima. We propose a novel algorithm based on tensor decomposition for guaranteed training of…

We present a method combining affinity prediction with region agglomeration, which improves significantly upon the state of the art of neuron segmentation from electron microscopy (EM) in accuracy and scalability. Our method consists of a…

This paper proposes the Nerual Energy Descent (NED) via neural network evolution equations for a wide class of deep learning problems. We show that deep learning can be reformulated as the evolution of network parameters in an evolution…

Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace.…

Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…

We propose a quantum algorithm for `extremal learning', which is the process of finding the input to a hidden function that extremizes the function output, without having direct access to the hidden function, given only partial input-output…

In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the…