A Greedy PDE Router for Blending Neural Operators and Classical Methods

arXiv:2509.24814v3 Announce Type: replace-cross Abstract: 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 greedy selection is desirable for its constant-factor approximation guarantee to the optimal solution under Lipschitz assumptions, it requires knowledge of the true error at each step, which is 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, convection-diffusion, and anisotropic diffusion equations show that our method consistently reduces time-to-solution relative to single-solver baselines and existing hybrid approaches such as HINTS.

Abstract: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 greedy selection is desirable for its constant-factor approximation guarantee to the optimal solution under Lipschitz assumptions, it requires knowledge of the true error at each step, which is 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, convection-diffusion, and anisotropic diffusion equations show that our method consistently reduces time-to-solution relative to single-solver baselines and existing hybrid approaches such as HINTS.

Methodology (stat.ME); Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2509.24814 [stat.ME]

(or arXiv:2509.24814v3 [stat.ME] for this version)

https://doi.org/10.48550/arXiv.2509.24814 Focus to learn more arXiv-issued DOI via DataCite

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A Greedy PDE Router for Blending Neural Operators and Classical Methods

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