PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes
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| Format: | Preprint |
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2025
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| _version_ | 1866917062270517248 |
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| author | Fazliani, Shaghayegh Udell, Madeleine |
| author_facet | Fazliani, Shaghayegh Udell, Madeleine |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00183 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes Fazliani, Shaghayegh Udell, Madeleine Machine Learning 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. |
| title | PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2511.00183 |