提出新型多尺度核框架方法，高效建模复杂微分方程的代理模型。
The Frame Kernel Method for Multiscale Operator Learning
- 用多尺度核框架将算子学习转化为系数映射问题。
- 在挑战性问题上精度显著优于主流神经算子，且收敛率可验证。
- 支持事后多尺度分解，适合科学计算与物理模拟场景。
我们提出一种原生多尺度算子学习方法，用于（数值求解）多尺度偏微分方程的代理建模。该方法的核心创新在于一种新颖的多尺度核框架函数逼近技术。利用这一新框架，我们将算子学习问题转化为输出函数的框架系数作为输入函数框架系数的函数来学习。泛化过程自动实现输出函数的多尺度分解。本方法适用于张量积网格和点云。我们给出了插值证明、误差估计及数值收敛率。实验展示了该方法在本质上多尺度的偏微分方程代理建模中的适用性。新提出的多尺度框架核方法在文献中的挑战性问题上显著优于主流神经算子，同时在泛化后仍能提供事后多尺度分解。
原文摘要 · Abstract (English)
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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