Expression Rates of Neural Operators for Linear Elliptic PDEs in Polytopes

TL;DR

This paper analyzes the approximation capabilities of neural operators, specifically deepONets, for solving linear elliptic PDEs in polytopes, demonstrating algebraic and exponential convergence rates based on data regularity.

Contribution

It provides theoretical bounds on the approximation rates of neural operators for elliptic PDEs in polygonal and polyhedral domains, highlighting the impact of data regularity.

Findings

Algebraic convergence rates for finite regularity data.

Exponential convergence rates for analytic data.

Neural operators effectively approximate elliptic PDE solution maps.

Abstract

We study the approximation rates of a class of deep neural network approximations of operators which arise as data-to-solution maps $\mathcal{S}$ of linear elliptic partial differential equations (PDEs), and act between pairs $X,Y$ of suitable infinite-dimensional spaces. We prove expression rate bounds for approximate neural operators $\mathcal{G}$ with the structure $\mathcal{G} = \mathcal{R} \circ \mathcal{A} \circ \mathcal{E}$, with linear encoders $\mathcal{E}$ and decoders $\mathcal{R}$. We focus in particular on deepONets emulating the coefficient-to-solution maps for elliptic PDEs set in polygons and in some polyhedra. Exploiting the regularity of the solution sets of elliptic PDEs in polytopes, we show algebraic rates of convergence for problems with data with finite regularity, and exponential rates for analytic data.