Neural networks (NNs) are the method of choice for building learning algorithms. They are
now being investigated for other numerical tasks such as solving high-dimensional partial …

Diffusion models, a powerful and universal generative artificial intelligence technology, have
achieved tremendous success and opened up new possibilities in diverse applications. In …

We formulate a general framework for hp-variational physics-informed neural networks (hp-
VPINNs) based on the nonlinear approximation of shallow and deep neural networks and …

We develop a general framework for data-driven approximation of input-output maps
between infinitedimensional spaces. The proposed approach is motivated by the recent …

Abstract The classical Universal Approximation Theorem holds for neural networks of
arbitrary width and bounded depth. Here we consider the natural 'dual'scenario for networks …

Physics-informed neural networks (PINNs)[31] use automatic differentiation to solve partial
differential equations (PDEs) by penalizing the PDE in the loss function at a random set of …

This paper studies the universal approximation property of deep neural networks for
representing probability distributions. Given a target distribution $\pi $ and a source …

Deep learning (DL) has had unprecedented success and is now entering scientific
computing with full force. However, current DL methods typically suffer from instability, even …

While deep learning is successful in a number of applications, it is not yet well understood
theoretically. A theoretical characterization of deep learning should answer questions about …

The termsurrogate modeling'in computational science and engineering refers to the
development of computationally efficient approximations for expensive simulations, such as …