发表机构
Kahlert School of Computing, University of Utah; U.S. Air Force Research Laboratory（犹他大学卡勒特计算机学院; 美国空军研究实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出原生多尺度算子学习的框架核方法，用于多尺度PDE代理建模，其精度优于流行神经算子，可实现后验多尺度分解。
AI 中文摘要
我们提出了一种原生多尺度算子学习方法，用于多尺度偏微分方程（PDE，Partial Differential Equations）的（数值求解器的）代理建模。该方法的主要创新在于一种新型多尺度核框架函数逼近技术，借助这一新技术，我们将算子学习问题转化为学习输出函数的框架系数（以输入函数的框架系数为函数）的问题。泛化步骤会自动实现输出函数的多尺度分解。该方法适用于张量积网格和点云。我们为该框架逼近提供了插值证明、误差估计和数值收敛速率，并展示了其在固有多尺度PDE代理建模中的适用性。在文献中具有挑战性的问题上，新的多尺度框架核方法比流行的神经算子（neural operators）精度显著更高，同时在泛化后可实现后验多尺度分解。
英文摘要
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.
