PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes
This addresses computational efficiency for researchers and practitioners in scientific computing by reducing resource needs for PDE solver development, though it is incremental as it builds on existing LLM-based methods.
The paper tackles the high computational cost of LLM-driven PDE solver generation by introducing PDE-SHARP, a framework that uses cheaper LLM inference to reduce evaluations by 60-75%, achieving a 4x average accuracy improvement with fewer than 13 solver evaluations compared to over 30 for baselines.
Current LLM-driven approaches using test-time computing to generate PDE solvers execute a large number of solver samples to identify high-accuracy solvers. These paradigms are especially costly for complex PDEs requiring substantial computational resources for numerical evaluation. We introduce PDE-SHARP, a framework to reduce computational costs by replacing expensive scientific computation by cheaper LLM inference that achieves superior solver accuracy with 60-75% fewer computational evaluations. PDE-SHARP employs three stages: (1) Analysis: mathematical chain-of-thought analysis including PDE classification, solution type detection, and stability analysis; (2) Genesis: solver generation based on mathematical insights from the previous stage; and (3) Synthesis: collaborative selection-hybridization tournaments in which LLM judges iteratively refine implementations through flexible performance feedback. To generate high-quality solvers, PDE-SHARP requires fewer than 13 solver evaluations on average compared to 30+ for baseline methods, improving accuracy uniformly across tested PDEs by $4\times$ on average, and demonstrates robust performance across LLM architectures, from general-purpose to specialized reasoning models.