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Convergence Rates for Learning Pseudo-Differential Operators

Jiaheng Chen, Daniel Sanz-Alonso

TL;DR

This work tackles the problem of learning elliptic pseudo-differential operators from noisy data by formulating operator learning as estimation of a bi-infinite wavelet matrix within a wavelet–Galerkin framework. It introduces a learning-oriented matrix compression and a nested-support regression to exploit multiscale sparsity, and proves a nearly parametric convergence rate $N^{-1/(2+\rho)}$ (up to poly-log factors) for the operator estimator in the $H^{t}\to H^{-t'}$ norm, with the estimator achieving $\mathcal{O}(2^{Jn})$ nonzero coefficients and near-linear computational cost $\mathcal{O}(N2^{Jn})$. The learned operator yields an efficient, stable wavelet–Galerkin solver whose numerical error tracks the statistical accuracy of the estimator, providing a principled link between operator learning, data-driven PDE solvers, and wavelet methods. In the noiseless case, convergence improves to super-algebraic rates, and the framework accommodates Green’s functions of elliptic PDEs as a key example. Overall, the paper delivers a rigorous multiscale statistical theory for learning operator maps and demonstrates compatibility with downstream PDE solution workflows.

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.

Convergence Rates for Learning Pseudo-Differential Operators

TL;DR

This work tackles the problem of learning elliptic pseudo-differential operators from noisy data by formulating operator learning as estimation of a bi-infinite wavelet matrix within a wavelet–Galerkin framework. It introduces a learning-oriented matrix compression and a nested-support regression to exploit multiscale sparsity, and proves a nearly parametric convergence rate (up to poly-log factors) for the operator estimator in the norm, with the estimator achieving nonzero coefficients and near-linear computational cost . The learned operator yields an efficient, stable wavelet–Galerkin solver whose numerical error tracks the statistical accuracy of the estimator, providing a principled link between operator learning, data-driven PDE solvers, and wavelet methods. In the noiseless case, convergence improves to super-algebraic rates, and the framework accommodates Green’s functions of elliptic PDEs as a key example. Overall, the paper delivers a rigorous multiscale statistical theory for learning operator maps and demonstrates compatibility with downstream PDE solution workflows.

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.
Paper Structure (17 sections, 3 theorems, 69 equations, 1 figure, 1 table)

This paper contains 17 sections, 3 theorems, 69 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main Contributions and Outline
  3. Related Work
  4. Operator learning
  5. Wavelet--Garlekin methods
  6. Notation
  7. Problem Setup
  8. Pseudo-Differential Operators
  9. Operator Learning
  10. Wavelet–Matrix Formulation of Operator Learning
  11. Biorthogonal Wavelets
  12. Operator Learning in the Bi-Infinite Matrix Framework
  13. Construction of the Estimator
  14. First Stage: Truncation and Compression
  15. Second Stage: Estimation
  16. ...and 2 more sections

Key Result

Proposition 3.2

Under Assumption assumption:operator_data_noise and Assumption assumption:wavelets (i)-(ii), the matrices $\mathbf{A}, \mathbf{U}$, and $\mathbf{W}$ satisfy:

Figures (1)

  • Figure 1: Illustration of the $(J,t,t')$-compression of Definition \ref{["def:supp_Jtt'"]} in the $(j,j')$–plane. The index set is partitioned into regions $D_1$–$D_6$: blocks in $D_1\cup D_2$ are discarded; in $D_3$ and $D_4$, entries are retained only when $\mathrm{dist}(S_{j,k},S_{j',k'})\le\tau_{jj'}$ with $\tau_{jj'}\asymp 2^{-j'}$ and $\tau_{jj'}\asymp 2^{-j}$, respectively; in $D_5$, entries are retained only when $\mathrm{dist}(S_{j,k},S_{j',k'})\le\tau_{jj'}$ with $\tau_{jj'}\asymp 2^{(J(t+t'-r)-jt'-j't-(j+j')\widetilde{d})/(2\widetilde{d}+r)}$; $D_6$ is uncompressed (all entries retained).

Theorems & Definitions (5)

  • Proposition 3.2: Properties of $\mathbf{A},\mathbf{U},\mathbf{W}$
  • Definition 4.1: Matrix truncation
  • Definition 4.2: Matrix compression
  • Lemma 4.3
  • Theorem 5.1