论文标题
混合神经操作员和PDE数值求解器中的放松方法
Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers
论文作者
论文摘要
神经网络患有光谱偏置在表示功能的高频成分时难以有效，而松弛方法可以有效地解决高频，但以中度至低频的停滞。我们通过协同结合两种方法来开发偏微分方程（PDE）的快速数值求解器来利用两种方法的弱点。具体而言，我们通过将深层运算符网络（DeepOnet）与标准弛豫方法集成，从而提出提示，杂种，迭代，数值和可转移的求解器，从而导致广泛的PDES的平行效率和算法可伸缩性，而不是与现有单片求解器的算法。提示通过利用deponet的频谱偏置来平衡跨本征频谱的收敛行为，从而导致均匀的收敛速率，从而使混合求解器总体上的出色表现。此外，提示适用于大规模的多维系统，就离散化，计算域和边界条件而言，它具有灵活性。
Neural networks suffer from spectral bias having difficulty in representing the high frequency components of a function while relaxation methods can resolve high frequencies efficiently but stall at moderate to low frequencies. We exploit the weaknesses of the two approaches by combining them synergistically to develop a fast numerical solver of partial differential equations (PDEs) at scale. Specifically, we propose HINTS, a hybrid, iterative, numerical, and transferable solver by integrating a Deep Operator Network (DeepONet) with standard relaxation methods, leading to parallel efficiency and algorithmic scalability for a wide class of PDEs, not tractable with existing monolithic solvers. HINTS balances the convergence behavior across the spectrum of eigenmodes by utilizing the spectral bias of DeepONet, resulting in a uniform convergence rate and hence exceptional performance of the hybrid solver overall. Moreover, HINTS applies to large-scale, multidimensional systems, it is flexible with regards to discretizations, computational domain, and boundary conditions.
