PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes

TL;DR

PDE-SHARP is a novel framework that reduces computational costs in PDE solving by combining analysis, generation, and iterative refinement using LLMs, achieving higher accuracy with fewer evaluations.

Contribution

It introduces a three-stage PDE solver generation framework that significantly decreases computational evaluations while improving solver accuracy across diverse PDEs.

Findings

Achieves 60-75% fewer evaluations than baseline methods.

Improves solver accuracy by 4x on average across tested PDEs.

Demonstrates robustness across various LLM architectures.

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: (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.