Title: PDE-SHARP: PDE Solver Hybrids Through Analysis & Refinement Passes
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Large Language Models, Partial Differential Equations, Solver Code Generation

Abstract: 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: $\textbf{(1) Analysis}$: mathematical chain-of-thought analysis including PDE classification, solution type detection, and stability analysis; $\textbf{(2) Genesis}$: solver generation based on mathematical insights from the previous stage; and $\textbf{(3) Synthesis}$: 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.