用神经网络先提取特征,再高效搜数学公式,提升复杂表达式恢复率。
FePySR: A Neural Feature Extraction Framework for Efficient and Scalable Symbolic Regression
- 先用异构神经网络筛选候选特征,缩小公式搜索空间
- 在75个复杂方程中成功恢复36个,误差更小且耗时更短
- 对含噪数据和生物微分方程均表现稳定,适合科学建模
符号回归(SR)的核心挑战是从观测数据中高效恢复复杂数学表达式。尽管该问题为NP难,但许多实际关注的表达式可自然分解为非线性特征模块的组合,使结构复杂度集中于少量可复用组件。本文提出两阶段框架FePySR,通过先提取有效特征来压缩符号回归的搜索空间。第一阶段使用异构神经网络将观测数据约束为一组候选表达式;第二阶段在该优化后的表达式空间内,利用PySR进行结构优化。在五个标准基准上,FePySR比现有方法获得更高方程恢复率。在75个高度复杂的合成方程测试中,成功恢复36个,未恢复项的均方误差显著降低,且计算时间少于PySR。FePySR第一阶段在不同特征数和噪声水平下均保持一致性能。应用于描述生物系统的常微分方程时,在100次测试中成功识别24个真实方程,而PySR全未恢复。综上,FePySR是一种通用性强的框架,可显著提升符号回归求解器效率与可靠性,适用于多科学领域。
原文摘要 · Abstract (English)
A fundamental challenge in symbolic regression (SR) is efficiently recovering complex mathematical expressions from observational data. Although this problem is NP-hard, many expressions of practical interest decompose naturally into combinations of nonlinear feature modules, concentrating structural complexity into a small number of reusable components. Here, we introduce FePySR, a two-stage framework that reduces the SR search space by extracting valid features prior to equation search. FePySR first employs a heterogeneous neural network to constrain observational data to a set of candidate expressions, then performs structural optimization within this refined expression space using PySR. Across five standard benchmarks, FePySR outperforms state-of-the-art methods by achieving higher equation recovery rates. On a set of 75 highly complex synthesized equations, FePySR recovers 36 equations, while producing substantially smaller mean squared errors on the remaining unrecovered cases, with reduced computation time compared to PySR. FePySR's first stage also maintains consistent performance under varying numbers of selected top features and increasing levels of noise in the observational data. Applied to ordinary differential equations governing biological systems, FePySR successfully identifies governing equations in 24 out of 100 tests where PySR recovers none. Taken together, FePySR is a generalizable framework that can enhance the SR solvers, enabling the efficient and reliable recovery of symbolic expressions across scientific domains.
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