Convergence Rates for Learning Pseudo-Differential Operators

TL;DR

This paper develops convergence rates for learning elliptic pseudo-differential operators using a wavelet-Galerkin approach, introducing a sparse estimator with a novel matrix compression scheme that ensures efficient, stable numerical solutions.

Contribution

It introduces a new sparse estimator for pseudo-differential operators with proven convergence rates, combining wavelet methods, matrix compression, and multiscale sparsity in a novel way.

Findings

Convergence rates established for the proposed estimator.

The learned operator enables an efficient, stable Galerkin solver.

Numerical errors match statistical accuracy, validating the approach.

Abstract

This paper establishes convergence rates for learning elliptic pseudo-differential operators, a fundamental operator class in partial differential equations and mathematical physics. In a wavelet-Galerkin framework, we formulate learning over this class as a structured infinite-dimensional regression problem with multiscale sparsity. Building on this structure, we propose a sparse, data- and computation-efficient estimator, which leverages a novel matrix compression scheme tailored to the learning task and a nested-support strategy to balance approximation and estimation errors. In addition to obtaining convergence rates for the estimator, we show that the learned operator induces an efficient and stable Galerkin solver whose numerical error matches its statistical accuracy. Our results therefore contribute to bringing together operator learning, data-driven solvers, and wavelet methods in scientific computing.