Neural networks suffer from spectral bias and have difficulty representing the high-frequency components of a function, whereas 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 behaviour 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; and it can also be used to precondition Krylov methods.

神经网络存在频谱偏差，难以表示函数的高频分量，而弛豫方法可以有效地解析高频，但在中低频时停滞不前。我们利用这两种方法的弱点，将它们协同组合，开发出一种大规模偏微分方程 （PDE） 的快速数值求解器。具体来说，我们提出了 HINTS，这是一种混合、迭代、数值和可转移的求解器，通过将深度算子网络 （DeepONet） 与标准松弛方法集成，从而为广泛的 PDE 实现并行效率和算法可扩展性，而现有的单片求解器则无法处理。HINTS 通过利用 DeepONet 的频谱偏差来平衡特征模态频谱的收敛行为，从而产生均匀的收敛速率，从而实现混合求解器的整体卓越性能。此外，HINTS 适用于大规模、多维系统;它在离散化、计算域和边界条件方面是灵活的;它也可以用于预处理 Krylov 方法。