Related papers: Nonlocality and Nonlinearity Implies Universality …

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 neural networks to learn operators, termed neural operators,…

Neural operators can learn nonlinear mappings between function spaces and offer a new simulation paradigm for real-time prediction of complex dynamics for realistic diverse applications as well as for system identification in science and…

While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning…

The study of universal approximation of arbitrary functions $f: \mathcal{X} \to \mathcal{Y}$ by neural networks has a rich and thorough history dating back to Kolmogorov (1957). In the case of learning finite dimensional maps, many authors…

Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both…

Despite the recent popularity of attention-based neural architectures in core AI fields like natural language processing (NLP) and computer vision (CV), their potential in modeling complex physical systems remains under-explored. Learning…

Classical neural networks are known for their ability to approximate mappings between finite-dimensional spaces, but they fall short in capturing complex operator dynamics across infinite-dimensional function spaces. Neural operators, in…

We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator…

Neural operators have emerged as a powerful, data-driven paradigm for learning solution operators of partial differential equations (PDEs). State-of-the-art architectures, such as the Fourier Neural Operator (FNO), have achieved remarkable…

Operator learning refers to the application of ideas from machine learning to approximate (typically nonlinear) operators mapping between Banach spaces of functions. Such operators often arise from physical models expressed in terms of…

Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined…

Fourier neural operators (FNOs) are a recently introduced neural network architecture for learning solution operators of partial differential equations (PDEs), which have been shown to perform significantly better than comparable deep…

Fourier Neural Operators (FNOs) have emerged as promising surrogates for partial differential equation solvers. In this work, we extensively tested FNOs on a variety of systems with non-linear and non-stationary properties, using a wide…

Neural operators have emerged as powerful data-driven surrogates for learning solution operators of parametric partial differential equations (PDEs). However, widely used Fourier Neural Operators (FNOs) rely on global Fourier…

Operator learning is the approximation of operators between infinite dimensional Banach spaces using machine learning approaches. While most progress in this area has been driven by variants of deep neural networks such as the Deep Operator…

Time-periodic quantum systems exhibit a rich variety of far-from-equilibrium phenomena and serve as ideal platforms for quantum engineering and control. However, simulating their dynamics with conventional numerical methods remains…

Neural Operators (NOs) are a leading method for surrogate modeling of partial differential equations. Unlike traditional neural networks, which approximate individual functions, NOs learn the mappings between function spaces. While NOs have…

Modeling high-frequency information is a critical challenge in scientific machine learning. For instance, fully turbulent flow simulations of the Navier-Stokes equations at Reynolds numbers 3500 and above can generate high-frequency signals…

Neural operator (NO) architectures learn nonlinear maps between infinite-dimensional function spaces and are widely used to accelerate simulation and enable data-driven model discovery. While universality results ensure expressivity, they…

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically…