mlsanityData to 14 Oct 2026

The Frame Kernel Method for Multiscale Operator Learning

arXiv 2608.25084 ↗First posted 25 Aug 2026cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

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.

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References · 17

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2021Machine learning accelerated computational fluid dynamicscited by 471
  4. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 413
  5. 2019Machine Learning for Fluid Mechanicscited by 385
  6. 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 326
  7. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 234
  8. 2018Survey of multifidelity methods in uncertainty propagation, inference, and optimizationcited by 142
  9. 2025Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometriescited by 97
  10. 2015Operator-valued Kernels for Learning from Functional Response Datacited by 60
  11. 2023Kernel Methods are Competitive for Operator Learningcited by 50
  12. 2022Mitigating spectral bias for the multiscale operator learningcited by 44
5 more
  1. 2013Generalized Multiscale Finite Element Methods (GMsFEM)cited by 21
  2. 2024Basis-to-Basis Operator Learning Using Function Encoderscited by 17
  3. 2024Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learningcited by 11
  4. 2024Multiscale scattered data analysis in samplet coordinatescited by 3
  5. 2026Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flowscited by 3

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