用AI分析+迭代优化，让解偏微分方程更省计算、更准。
PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes
- 先分析方程类型和稳定性，再生成解法，最后通过AI裁判不断优化
- 平均只需13次评估，准确率比传统方法高4倍，节省60%-75%算力
- 适合想高效求解复杂方程的科研人员，兼容各类大模型
现有基于大模型的测试时计算方法需大量求解器采样以找到高精度解法，对复杂偏微分方程（PDE）而言计算开销巨大。我们提出PDE-SHARP框架，通过将昂贵的科学计算替换为低成本的大模型推理，实现更优解法精度，同时减少60%-75%的计算评估次数。该框架包含三个阶段：(1) 分析阶段，进行方程分类、解型检测与稳定性分析；(2) 创生阶段，基于数学洞察生成求解器；(3) 综合阶段，通过协同选择-混合竞赛，由大模型裁判迭代反馈以优化实现。PDE-SHARP平均仅需少于13次求解器评估，显著优于基线方法（超30次），在各类测试方程上平均准确率提升4倍，并在不同大模型架构（通用与专用推理模型）间表现出强鲁棒性。
原文摘要 · Abstract (English)
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.
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