标题： 深度学习与迭代方法的混合用于加速粘性不可压缩流的求解 显示英文标题

标题： Hybrid deep learning and iterative methods for accelerated solutions of viscous incompressible flow

摘要： 压力泊松方程是不可压缩流动模拟中分数步法的核心，由于需要迭代求解大规模线性系统，因此计算成本较高。 为了解决这一挑战，我们引入了HyDEA，这是一种新颖的框架，将深度学习与经典迭代求解器相结合。 它利用了DeepONet（能够捕捉解的大尺度特征）和CG或PCG方法（能够高效解决细尺度误差）的互补优势。 具体来说，在线搜索方法的框架内，DeepONet预测搜索方向以加速稀疏、对称正定线性系统的收敛，而CG/PCG方法通过迭代细化确保鲁棒性。 该框架通过解耦的浸入边界投影方法无缝扩展到固体结构上的流场。 至关重要的是，DeepONet是在人造线性系统上训练的，而不是特定于流场数据，这使它能够在几何复杂性和雷诺数方面具有内在的泛化能力，而无需重新训练。 基准测试表明，HyDEA相对于CG/PCG方法在无障碍物、单个或多个固定障碍物以及一个移动障碍物的流动情况下表现出更高的效率和准确性——使用固定的网络权重。 值得注意的是，HyDEA还表现出超分辨率能力：尽管DeepONet是在Re=1000的128×128网格上训练的，但混合求解器通过插值在Re=10000的512×512网格上提供了准确的解决方案，尽管存在离散化不匹配。 相比之下，纯粹的数据驱动型DeepONet在复杂流场中失败，突显了将深度学习与迭代方法结合的必要性。 HyDEA在几何形状、分辨率和雷诺数方面的鲁棒性、效率和通用性展示了其作为现实世界流体力学问题变革性求解器的潜力。 显示英文摘要

摘要： The pressure Poisson equation, central to the fractional step method in incompressible flow simulations, incurs high computational costs due to the iterative solution of large-scale linear systems. To address this challenge, we introduce HyDEA, a novel framework that synergizes deep learning with classical iterative solvers. It leverages the complementary strengths of a DeepONet - capable of capturing large-scale features of the solution - and the CG or a PCG method, which efficiently resolves fine-scale errors. Specifically, within the framework of line-search methods, the DeepONet predicts search directions to accelerate convergence in solving sparse, symmetric-positive-definite linear systems, while the CG/ PCG method ensures robustness through iterative refinement. The framework seamlessly extends to flows over solid structures via the decoupled immersed boundary projection method. Crucially, the DeepONet is trained on fabricated linear systems rather than flow specific data, endowing it with inherent generalization across geometric complexities and Reynolds numbers without retraining. Benchmarks demonstrate superior efficiency and accuracy of HyDEA over the CG/PCG methods for flows with no obstacles, single or multiple stationary obstacles, and one moving obstacle - using fixed network weights. Remarkably, HyDEA also exhibits super-resolution capability: although the DeepONet is trained on a 128*128 grid for Re=1000, the hybrid solver delivers accurate solutions on a 512*512 grid for Re=10000 via interpolation, despite discretizations mismatch. In contrast, a purely data-driven DeepONet fails for complex flows, underscoring the necessity of hybridizing deep learning with iterative methods. Robustness, efficiency, and generalization across geometries, resolutions, and Reynolds numbers of HyDEA highlight its potential as a transformative solver for real world fluid dynamics problems.