并非每个项都增添新结构：符号回归的Sobolev新颖性
Not Every Term Adds New Structure: Sobolev Novelty for Symbolic Regression
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中文总结 AI 辅助
针对符号回归中现有方法无法判断单个项是否冗余的问题，提出Sobolev新颖性度量，通过经验Sobolev签名和理论校准阈值识别结构新颖项，实验显示真值方程与SR方法结果存在显著差距，且作为插件可提升多种SR范式性能。
中文摘要 AI 辅助
符号回归（SR）旨在从数据中发现紧凑且有意义的数学方程，但搜索符号结构的巨大组合空间仍然具有挑战性。现有方法通常使用表达式级目标（如拟合误差）来指导这一过程，这些目标将候选方程作为一个整体进行评估，却很少提供关于单个项是否贡献了真正的新结构或与表达式其余部分基本冗余的信息。我们引入了Sobolev新颖性，一种用于符号方程的结构独立性的项级度量。对于每个项，我们根据其在观测输入上的函数值和精确导数构建经验Sobolev签名，并量化该行为中有多少无法由其余项重建。我们进一步推导出一个理论校准的阈值，从而得到一个有原则且无需调参的标准，用于识别结构新颖的项。使用该阈值，基准真值方程中92.6%的项表现出足够的结构新颖性，而15种SR方法生成的表达式中平均仅有38.3%，揭示了科学方程与当前SR解决方案之间的显著差距。作为一种轻量级插件，Sobolev新颖性可被纳入多种SR范式，以支持项剪枝、搜索引导、LLM反馈和数据选择，带来一致的性能提升，并展现出广泛的适用性。
英文摘要
Symbolic regression (SR) aims to discover compact and meaningful mathematical equations from data, but searching the vast combinatorial space of symbolic structures remains challenging. Existing methods typically guide this process using expression-level objectives, such as fitting error, which assess a candidate equation as a whole but provide little information about whether an individual term contributes genuinely new structure or is largely redundant with the rest of the expression. We introduce \textbf{Sobolev Novelty}, a term-level measure of structural independence for symbolic equations. For each term, we construct an empirical Sobolev signature from its function values and exact derivatives over the observed inputs, and quantify how much of this behavior cannot be reconstructed by the remaining terms. We further derive a theory-calibrated threshold, yielding a principled and tuning-free criterion for identifying structurally novel terms. Using this threshold, 92.6\% of terms in benchmark ground-truth equations exhibit sufficient structural novelty, compared with only 38.3\% on average for expressions produced by 15 SR methods, revealing a substantial gap between scientific equations and current SR solutions. As a lightweight plug-in, Sobolev Novelty can be incorporated into diverse SR paradigms to support term pruning, search guidance, LLM feedback, and data selection, yielding consistent performance gains and demonstrating broad applicability.
