Convergence Rates for Learning Pseudo-Differential Operators

2026-01-08

AI Summary

This study addresses the efficient learning of elliptic pseudodifferential operators with guaranteed convergence and numerical stability. Within a wavelet Galerkin framework, operator learning is formulated as an infinite-dimensional regression problem endowed with a multiscale sparse structure, leading to a data- and computation-efficient sparse estimator. The key innovation lies in the introduction of a task-oriented matrix compression scheme combined with a nested support strategy, which unifies operator learning, data-driven solvers, and wavelet methods. Theoretical analysis establishes the convergence rate of the proposed estimator and demonstrates that the learned operator yields an efficient and stable Galerkin solver whose numerical error is commensurate with the statistical estimation accuracy.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• pseudo-differential operators
• convergence rates
• operator learning
• elliptic operators
• scientific computing

Innovation

• pseudo-differential operators
• wavelet-Galerkin framework
• matrix compression
• multiscale sparsity
• convergence rates