Neural Operator: Learning Maps Between Function Spaces

08/19/2021
by   Nikola Kovachki, et al.
17

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 tailored to learn operators mapping between infinite dimensional function spaces. We formulate the approximation of operators by composition of a class of linear integral operators and nonlinear activation functions, so that the composed operator can approximate complex nonlinear operators. We prove a universal approximation theorem for our construction. Furthermore, we introduce four classes of operator parameterizations: graph-based operators, low-rank operators, multipole graph-based operators, and Fourier operators and describe efficient algorithms for computing with each one. The proposed neural operators are resolution-invariant: they share the same network parameters between different discretizations of the underlying function spaces and can be used for zero-shot super-resolutions. Numerically, the proposed models show superior performance compared to existing machine learning based methodologies on Burgers' equation, Darcy flow, and the Navier-Stokes equation, while being several order of magnitude faster compared to conventional PDE solvers.

READ FULL TEXT

page 3

page 34

page 36

page 40

research
10/18/2020

Fourier Neural Operator for Parametric Partial Differential Equations

The classical development of neural networks has primarily focused on le...
research
06/06/2023

Globally injective and bijective neural operators

Recently there has been great interest in operator learning, where netwo...
research
09/12/2022

Bounding the Rademacher Complexity of Fourier neural operators

A Fourier neural operator (FNO) is one of the physics-inspired machine l...
research
08/28/2023

Scattering with Neural Operators

Recent advances in machine learning establish the ability of certain neu...
research
02/01/2023

Graph Neural Operators for Classification of Spatial Transcriptomics Data

The inception of spatial transcriptomics has allowed improved comprehens...
research
08/18/2023

A hybrid Decoder-DeepONet operator regression framework for unaligned observation data

Deep neural operators (DNOs) have been utilized to approximate nonlinear...
research
06/28/2023

The curse of dimensionality in operator learning

Neural operator architectures employ neural networks to approximate oper...

Please sign up or login with your details

Forgot password? Click here to reset