Near-Optimal Learning of Gaussian Sobolev Operators

2026-07-08

AI Summary

This work addresses the challenges of high sample complexity and slow convergence rates in learning Sobolev operators with limited regularity. The authors propose a fully data-driven Hermite-PCA approximation algorithm that integrates principal component analysis, weighted least squares, and Hermite function expansions. This approach achieves, for the first time, near-optimal sample complexity for learning Gaussian Sobolev operators and exhibits spectral convergence—where higher regularity leads to faster convergence. A comprehensive error analysis is provided through systematic decomposition and spectral approximation theory, accounting for all sources of error. Numerical experiments confirm the theoretical convergence rates and demonstrate the algorithm’s efficiency in practical operator learning tasks.

Abstract

A key question in operator learning is how to design surrogate operators with provable approximation guarantees in reasonable computational time. Whereas smooth operators can be approximated efficiently, i.e., with at least algebraic convergence in the amount of training data, learning finitely regular operators is known to be less efficient. The reason is an intrinsic curse of sample complexity, which allows only subalgebraic sample complexity rates. This fact makes it all the more important to develop algorithms which provably achieve these rates. In this work, we present a fully data-driven algorithm, termed Hermite-PCA approximation, for learning Gaussian Sobolev operators with near-optimal sample complexity. It employs principal component analysis and weighted least-squares methods and is therefore computationally efficient. Moreover, it is spectral, in the sense that it achieves faster (and near-optimal) convergence the higher the Sobolev regularity. We provide a full error analysis of this algorithm, taking into account all sources of error, along with numerical experiments that verify our theoretical results and empirically confirm the efficacy of Hermite-PCA approximation for learning Sobolev operators.

Problem

• operator learning
• Sobolev operators
• sample complexity
• approximation guarantees
• Gaussian Sobolev

Innovation

• Hermite-PCA approximation
• Gaussian Sobolev operators
• near-optimal sample complexity
• spectral convergence
• operator learning