Statistical Learning Theory for Neural Operators

TL;DR

This paper develops statistical convergence results for learning nonlinear operators between infinite-dimensional Hilbert spaces, applying empirical process theory to neural operator architectures like FrameNet, and demonstrates algebraic convergence rates.

Contribution

It generalizes finite-dimensional nonparametric regression results to infinite-dimensional spaces and provides convergence guarantees for neural operators like FrameNet.

Findings

Proves convergence rates for least-squares estimators in infinite-dimensional settings.

Shows algebraic convergence rates for neural operators assuming holomorphicity.

Demonstrates applicability to learning solution operators of PDEs.

Abstract

We present statistical convergence results for the learning of (possibly) non-linear mappings in infinite-dimensional spaces. Specifically, given a map $G_0:\mathcal X\to\mathcal Y$ between two separable Hilbert spaces, we analyze the problem of recovering $G_0$ from $n\in\mathbb N$ noisy input-output pairs $(x_i, y_i)_{i=1}^n$ with $y_i = G_0 (x_i)+\varepsilon_i$; here the $x_i\in\mathcal X$ represent randomly drawn 'design' points, and the $\varepsilon_i$ are assumed to be either i.i.d. white noise processes or subgaussian random variables in $\mathcal{Y}$. We provide general convergence results for least-squares-type empirical risk minimizers over compact regression classes $\mathbf G\subseteq L^\infty(X,Y)$, in terms of their approximation properties and metric entropy bounds, which are derived using empirical process techniques. This generalizes classical results from finite-dimensional nonparametric regression to an infinite-dimensional setting. As a concrete application, we study an encoder-decoder based neural operator architecture termed FrameNet. Assuming $G_0$ to be holomorphic, we prove algebraic (in the sample size $n$) convergence rates in this setting, thereby overcoming the curse of dimensionality. To illustrate the wide applicability, as a prototypical example we discuss the learning of the non-linear solution operator to a parametric elliptic partial differential equation.