A Greedy PDE Router for Blending Neural Operators and Classical Methods

2025-09-29

AI Summary

To address the slow convergence and poor numerical stability of single-method solvers for partial differential equations (PDEs), this paper proposes a hybrid iterative solving framework based on approximate greedy routing. At each iteration, the framework dynamically selects the optimal solver from a heterogeneous ensemble comprising neural operators and classical numerical solvers—without requiring ground-truth error feedback. By integrating an error estimation mechanism with a differentiable greedy selection strategy, it strikes a balance between combinatorial optimization complexity and theoretical interpretability. Theoretical analysis establishes a provable suboptimality bound for the routing policy. Experiments on Poisson and Helmholtz equations demonstrate that our method significantly outperforms both individual solvers and state-of-the-art hybrid approaches (e.g., HINTS) in terms of convergence rate and numerical stability.

Technology Category

Search and Optimization: Mixed Discrete/Continuous SearchPlanning, Routing, and Scheduling: Optimization of Spatio-temporal SystemsReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 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.

Problem

• Hybrid PDE solver selection poses combinatorial optimization challenges
• Greedy strategies require unavailable true error knowledge during solving
• Approximate router mimics greedy selection for improved convergence stability

Innovation

• Approximate greedy router for solver selection
• Blends neural operators with classical methods
• Mimics greedy approach without true error