A Greedy PDE Router for Blending Neural Operators and Classical Methods
This addresses the computational expense of classical solvers and spectral bias in machine learning methods for PDEs, offering an incremental improvement in hybrid solver design.
The paper tackles the problem of designing an optimal hybrid iterative solver for PDEs by proposing an approximate greedy router that selects solvers from an ensemble to leverage complementary strengths, resulting in faster and more stable convergence compared to single-solver baselines and existing hybrid approaches like HINTS.
When solving PDEs, classical numerical solvers are often computationally expensive, while machine learning methods can suffer from spectral bias, failing to capture high-frequency components. Designing an optimal hybrid iterative solver--where, at each iteration, a solver is selected from an ensemble of solvers to leverage their complementary strengths--poses a challenging combinatorial problem. While the greedy selection strategy is desirable for its constant-factor approximation guarantee to the optimal solution, it requires knowledge of the true error at each step, which is generally unavailable in practice. We address this by proposing an approximate greedy router that efficiently mimics a greedy approach to solver selection. Empirical results on the Poisson and Helmholtz equations demonstrate that our method outperforms single-solver baselines and existing hybrid solver approaches, such as HINTS, achieving faster and more stable convergence.