The Frame Kernel Method for Multiscale Operator Learning

2026-08-25

AI Summary

本文提出了一种基于新型多尺度核框架函数逼近技术的多尺度算子学习方法，用于求解多尺度偏微分方程，并在文献中的难题上展示了其优越性。

Abstract

We present a natively multiscale operator learning method for the surrogate modeling of (numerical solvers for) multiscale partial differential equations (PDEs). The primary novelty of our method lies in a novel multiscale kernel frame function approximation technique. Leveraging this new kernel frame technique, we cast the operator learning problem as one of learning frame coefficients of output functions as a function of frame coefficients of input functions. The generalization step then automatically allows for a multiscale decomposition of the output functions. Our method is applicable to both tensor-product grids and point clouds. We present interpolation proofs, error estimates, and numerical convergence rates for our frame approximation. We the demonstrate the applicability of our method for the surrogate modeling of inherently multiscale PDEs. The new multiscale frame kernel method is significantly more accurate than popular neural operators on challenging problems from the literature, while simultaneously admitting an a posteriori multiscale decomposition upon generalization.

Problem

• multiscale
• partial differential equations
• operator learning
• surrogate modeling

Innovation

• multiscale kernel frame
• operator learning
• PDEs surrogate modeling
• frame coefficients