Neural Operator: Learning Maps Between Function Spaces

TL;DR

This paper introduces neural operators, a new class of neural networks designed to learn mappings between infinite dimensional function spaces, enabling efficient approximation of PDE solution operators with discretization-invariance and superior performance.

Contribution

The paper proposes neural operators, proves their universal approximation capability, introduces four parameterization classes, and demonstrates their effectiveness in solving PDEs faster than traditional methods.

Findings

Neural operators can approximate any nonlinear continuous operator.

They are discretization-invariant, sharing parameters across different discretizations.

Neural operators outperform existing ML methods in PDE solution tasks, being significantly faster.

Abstract

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, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.