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

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 (Hybrid Deep lEarning line-search directions and iterative methods for Accelerated solutions), a novel framework that synergizes deep learning with classical iterative solvers. It leverages the complementary strengths of a deep operator network (DeepONet) – capable of capturing large-scale features of the solution – and the conjugate gradient (CG) or a preconditioned conjugate gradient (PCG) (with Incomplete Cholesky, Jacobi, or Multigrid preconditioner) 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, preserving the linear system’s structure. 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 HyDEA’s superior efficiency and accuracy over the CG/PCG methods for flows with no obstacles, single/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 Reynolds number $Re=1000$, the hybrid solver delivers accurate solutions on a 512 × 512 grid for $Re=10,000$ 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. HyDEA’s robustness, efficiency, and generalization across geometries, resolutions, and Reynolds numbers highlight its potential as a transformative solver for real-world fluid dynamics problems.