用对称性约束和语言模型，从噪声数据中自动发现物理方程。
Symmetry-Constrained Language-Guided Program Synthesis for Discovering Governing Equations from Noisy and Partial Observations
- 基于对称性规则生成候选方程，提前排除71.3%无效表达式。
- 在10%噪声下实现83.7%方程结构准确恢复，优于前人22.4个百分点。
- 能识别方程歧义，避免错误确定单一公式，适合物理建模研究者。
从实验观测中发现简洁的支配方程是定量科学的核心目标，但实际中常因测量噪声、状态变量缺失或多个符号结构在统计不确定性下等效而失败。本文提出SymLang（对称性约束的语言引导方程发现）框架，融合三项关键技术：(i) 类型化对称性约束语法，通过量纲分析、群论不变性和奇偶性约束作为硬规则，平均提前剔除71.3%候选表达式；(ii) 基于微调70亿参数语言模型的程序合成，结合可解释数据描述符高效搜索受限空间；(iii) MDL正则化贝叶斯模型选择与块自举稳定性分析，量化结构不确定性而非锁定单一最优方程。在涵盖经典力学、电磁学、热力学、种群动力学及非线性振子的133个动力系统上，该框架在10%观测噪声下实现83.7%的精确结构恢复率，较次优基线提升22.4个百分点，外推误差降低61%，守恒律违反近乎消除（物理漂移3.1×10⁻³对比对手187.3×10⁻³）。所有测试场景均正确识别结构退化，明确报告而非输出错误单一方程。框架完全开源可复现，为从原始数据到可解释、可审计物理定律提供可靠路径。
原文摘要 · Abstract (English)
Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are unobserved, or multiple symbolic structures explain the data equally well within statistical uncertainty. Here we introduce SymLang (Symmetry-constrained Language-guided equation discovery), a unified framework that brings together three previously separate ideas: (i) typed symmetry-constrained grammars that encode dimensional analysis, group-theoretic invariance, and parity constraints as hard production rules, eliminating on average 71.3% of candidate expression trees before any fitting; (ii) language-model-guided program synthesis in which a fine-tuned 7B-parameter proposer, conditioned on interpretable data descriptors, efficiently navigates the constrained search space; and (iii) MDL-regularized Bayesian model selection coupled with block-bootstrap stability analysis that quantifies structural uncertainty rather than committing to a single best equation. Across 133 dynamical systems spanning classical mechanics, electrodynamics, thermodynamics, population dynamics, and nonlinear oscillators, SymLang achieves an exact structural recovery rate of 83.7% under 10% observational noise - a 22.4 percentage-point improvement over the next-best baseline - while reducing out-of-distribution extrapolation error by 61% and near-eliminating conservation-law violations (3.1 x 10-3 vs. 187.3 x 10-3 physical drift for the closest competitor). In all tested regimes the framework correctly identifies structural degeneracy, reporting it explicitly rather than returning a confidently wrong single equation. The framework is fully open-source and reproducible, providing a principled pathway from raw data to interpretable, physically auditable symbolic laws.
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