Masterarbeit (laufend)

Fan Yang

Parallel Deep Operator Network as a Surrogate Component for Large-Scale Multigrid PDE Solvers

Multigrid methods are among the most efficient iterative solvers for large-scale discretized partial differential equations (PDEs), achieving optimal computational complexity by combining smoothing iterations with coarse-grid corrections. However, the efficiency and robustness of classical multigrid methods strongly depend on the design of algorithmic components, including the choice of smoother, restriction and prolongation operators. These choices become particularly challenging for problems involving complex geometries, variable coefficients, and complicated boundary conditions.
This work investigates the integration of Deep Operator Networks (DeepONet) as a learned component within a multigrid framework. DeepONet approximates nonlinear operators by learning mappings between function spaces, providing a data-driven approach for constructing adaptive corrections. Motivated by the spectral interpretation of multigrid methods, where smoother reduces high-frequency error components and coarse-grid correction eliminates remaining low-frequency components, DeepONet is explored as an adaptive mechanism to improve the error correction process. The goal of this work is to develop a hybrid solver that combines the scalability and mathematical structure of classical multigrid methods with the learning capability of neural operators.