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

TL;DR

HyDEA addresses the PPE bottleneck in fractional-step incompressible flow simulations by fusing a DeepONet-predicted line-search direction with CG-type iterative refinement, trained on fabricated linear systems to generalize across geometries and Reynolds numbers. The framework integrates seamlessly with the decoupled immersed boundary projection method for flows with solid structures and demonstrates robust super-resolution capabilities, solving higher-resolution problems from a lower-resolution training regime. Across lid-driven cavity benchmarks and multiple obstacle configurations (including moving cylinders), HyDEA significantly reduces iteration counts and wall-clock time while maintaining high-fidelity flow fields, outperforming purely data-driven approaches like DCDM. The results indicate HyDEA's potential as a general, scalable PDE solver for complex, real-world fluid dynamics problems, though implementation overhead and grid specificity motivate further optimization and extension to nonuniform grids and domain decomposition strategies.

Abstract

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.

Figures (44)

• Figure 1: The workflow of HyDEA, $n$ denotes the time step for an unsteady CFD simulation, $k$ represents the iteration index for solving the discrete PPE at the current time step, and $atol$ is the predefined absolute tolerance. The initial value for iteration is taken from last time step: $\delta p_k=\delta p^{n-1}$.
• Figure 2: The implementation details and two versions of HyDEA. (a) Detailed workflow. (b) For each round of the hybrid algorithm, a CG-type method is taken as the initial solver for maximum $Num_{CG-type}$ iterations followed by DLSM for maximum $Num_{DLSM}$ iterations. (c) For each round of the hybrid algorithm, DLSM is taken as the initial solver for maximum $Num_{DLSM}$ iterations followed by a CG-type method for maximum $Num_{CG-type}$ iterations.
• Figure 3: The DeepONet architecture.
• Figure 4: Iterative residuals of solving the 1D Poisson equation by CG and HyDEA (CG+DLSM).
• Figure 5: Comparison of the numerical solution obtained by HyDEA (CG+DLSM) and $u_{exact}$.