# Convergence Rates for Learning Pseudo-Differential Operators

arXiv: https://arxiv.org/abs/2601.04473 · HTML: https://mlsanity.com/paper/2601.04473/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jan 2026.

## Built on by (most cited first, 3 of 3)

- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)
- [From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators](https://mlsanity.com/paper/2607.00320/) (2026)
- [Near-Optimal Learning of Gaussian Sobolev Operators](https://mlsanity.com/paper/2607.11921/) (2026)

## Abstract (arXiv)

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.