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Low-dimensional embedding, manifold learning, clustering, classification, and anomaly detection are among the most important problems in machine learning. The existing methods usually consider the case when each instance has a fixed,…

Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional…

Extreme Learning Machines (ELMs) have become a popular tool in the field of Artificial Intelligence due to their very high training speed and generalization capabilities. Another advantage is that they have a single hyper-parameter that…

To overcome these obstacles and improve computational accuracy and efficiency, this paper presents the Randomized Radial Basis Function Neural Network (RRNN), an innovative approach explicitly crafted for solving multiscale elliptic…

In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the…

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…

In this paper, neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. Neural network work approximation methods have advantages over classical approaches in that they…

This paper develops a randomized approach for incrementally building deep neural networks, where a supervisory mechanism is proposed to constrain the random assignment of the weights and biases, and all the hidden layers have direct links…

We show that discrete synaptic weights can be efficiently used for learning in large scale neural systems, and lead to unanticipated computational performance. We focus on the representative case of learning random patterns with binary…

Deep learning methods, which exploit auto-differentiation to compute derivatives without dispersion or dissipation errors, have recently emerged as a compelling alternative to classical mesh-based numerical schemes for solving hyperbolic…

Physics-informed machine learning frameworks such as Physics-Informed Neural Networks (PINNs) and Physics-Informed Extreme Learning Machines (PI-ELMs) have shown great promise for solving partial differential equations (PDEs) but struggle…

Conventional extreme learning machines solve a Moore-Penrose generalized inverse of hidden layer activated matrix and analytically determine the output weights to achieve generalized performance, by assuming the same loss from different…

Physics-informed neural networks solve partial differential equations by training neural networks. Since this method approximates infinite-dimensional PDE solutions with finite collocation points, minimizing discretization errors by…

The single-layer feedforward neural network with random weights is a recurring motif in the neural networks literature. The advantage of these networks is their simplified training, which reduces to solving a ridge-regression problem. A…

To address the sensitivity of parameters and limited precision for physics-informed extreme learning machines (PIELM) with common activation functions, such as sigmoid, tangent, and Gaussian, in solving high-order partial differential…

The boundary control problem is a non-convex optimization and control problem in many scientific domains, including fluid mechanics, structural engineering, and heat transfer optimization. The aim is to find the optimal values for the…

Elliptic Partial Differential Equations (PDEs) play a central role in computing the equilibrium conditions of physical problems (heat, gravitation, electrostatics, etc.). Efficient solutions to elliptic PDEs are also relevant to computer…

Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and engineering. In recent years, a great number of computational approaches have been developed, most of them relying on a combination…

Extreme learning machine (ELM) is a network model that arbitrarily initializes the first hidden layer and can be computed speedily. In order to improve the classification performance of ELM, a $\ell_2$ and $\ell_{0.5}$ regularization ELM…

We propose a simplified, biologically inspired predictive local learning rule that eliminates the need for global backpropagation in conventional neural networks and membrane integration in event-based training. Weight updates are triggered…