The superior multi-functional properties of polymer composites have made them an ideal
choice for aerospace, automobile, marine, civil, and many other technologically demanding …

Operator learning refers to the application of ideas from machine learning to approximate
(typically nonlinear) operators mapping between Banach spaces of functions. Such …

The classical development of neural networks has primarily focused on learning mappings
between finite dimensional Euclidean spaces or finite sets. We propose a generalization of …

Partial differential equations (PDEs) play a central role in the mathematical analysis and
modeling of complex dynamic processes across all corners of science and engineering …

Fourier neural operators (FNOs) have recently been proposed as an effective framework for
learning operators that map between infinite-dimensional spaces. We prove that FNOs are …

DeepONets have recently been proposed as a framework for learning nonlinear operators
mapping between infinite-dimensional Banach spaces. We analyze DeepONets and prove …

We prove rigorous bounds on the errors resulting from the approximation of the
incompressible Navier–Stokes equations with (extended) physics-informed neural networks …

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

We describe the new field of the mathematical analysis of deep learning. This field emerged
around a list of research questions that were not answered within the classical framework of …

We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation
of Sobolev-regular as well as analytic functions by neural networks with the hyperbolic …