The Frame Kernel Method for Multiscale Operator Learning
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.
Related
- Compressed Multiscale Kernel Collocation and Physics-Informed Kernel Learning for Elliptic BVPs
- Koopman operator learning for predictive control via Khatri-Rao kernel regression
- Neural means and kernel corrections for operator learning
- In-Context Learning for Multi-Fidelity Multiscale Model Closure and Cross-Operator Out-Of-Distribution Generalization
- Elucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs
BibTeX
@misc{frieden2026frame,
title = {{The Frame Kernel Method for Multiscale Operator Learning}},
author = {Branden Frieden and Ryan Whitehead and M. Keith Ballard and Robert M. Kirby and Varun Shankar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.25084},
eprint = {2608.25084},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25084},
}
RIS
TY - UNPB
TI - The Frame Kernel Method for Multiscale Operator Learning
AU - Branden Frieden
AU - Ryan Whitehead
AU - M. Keith Ballard
AU - Robert M. Kirby
AU - Varun Shankar
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.25084
UR - https://arxiv.org/abs/2608.25084
AB - 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.
ER -