Learning solution operators of PDEs with sparse approximation methods

2026-06-04

AI Summary

This work addresses the heavy reliance on large numbers of PDE solution samples in operator learning by proposing an efficient approximation framework based on incremental dimensionality expansion. By integrating tensor-product basis expansions with sparse recovery techniques—such as orthogonal matching pursuit—the method identifies low-dimensional structures and critical variable interactions, substantially reducing the required number of PDE solves. In multiple numerical experiments, the approach achieves accuracy comparable to or better than conventional quadrature methods and Fourier neural operators, while demanding significantly lower sample complexity and computational cost. Furthermore, the recovered sparse index sets offer interpretable insights into the underlying solution structure.

Abstract

We investigate the approximation of solution operators for partial differential equations (PDEs) using sparse high-dimensional techniques. Building on a dimension-incremental framework, we combine product basis expansions with sparse recovery methods, specifically orthogonal matching pursuit (OMP), to substantially reduce the required sample size compared with a previously considered cubature-based approach. We evaluate the resulting method numerically on several examples, comparing it against both cubature-based sparse approximation and Fourier neural operators in terms of accuracy, runtime, and sample size. The experiments show that our approach considerably reduces the number of required PDE solves relative to its predecessor while maintaining competitive accuracy, particularly when the solution admits a sparse representation in the chosen basis. Furthermore, the recovered sparse index sets yield interpretable insights into the relevant variables and parameter interactions.

Problem

• solution operators
• partial differential equations
• sparse approximation
• sample complexity
• high-dimensional approximation

Innovation

• sparse approximation
• solution operators
• orthogonal matching pursuit
• dimension-incremental framework
• PDE learning