Deep Divide-and-Reduce in Symbolic Regression

符号回归中的深度分治与约简 (Deep Divide-and-Reduce)

Abstract: Symbolic regression (SR) is the task of discovering underlying patterns from data and representing them using mathematical expressions. Current machine learning approaches to SR often lack a profound understanding of the intrinsic mathematical and physical principles governing these expressions.

摘要： 符号回归（SR）是一项从数据中发现潜在模式并使用数学表达式对其进行表征的任务。当前用于符号回归的机器学习方法往往缺乏对支配这些表达式的内在数学和物理原理的深刻理解。

While the pioneering AI Feynman method leverages the mathematical properties underlying the data, its expression simplification mechanism suffers from a narrow scope of applicability and is prone to failure on complex equations. Furthermore, its underlying mechanisms rely heavily on brute-force searches for sub-expressions, severely limiting its practical utility.

尽管开创性的 AI Feynman 方法利用了数据背后的数学属性，但其表达式简化机制存在适用范围狭窄的问题，且在处理复杂方程时容易失效。此外，其底层机制严重依赖于对子表达式的暴力搜索，这极大地限制了其实际应用价值。

Through rigorous mathematical deduction and proofs, we propose our method, Deep Divide and Reduce in Symbolic Regression (DDRSR). DDRSR fundamentally broadens the applicability of expression decomposition and reduction, circumvents the need for brute-force sub-structure searches, and ensures both wider versatility and strict theoretical correctness.

通过严谨的数学推导和证明，我们提出了“符号回归中的深度分治与约简”（DDRSR）方法。DDRSR 从根本上拓宽了表达式分解与约简的适用性，规避了对子结构进行暴力搜索的需求，并确保了更广泛的通用性和严格的理论正确性。

Empirical evaluations demonstrate that these theoretical principles yield significant advantages in both expression decomposition and numerical regression tasks. Finally, we discuss the applicable scenarios and inherent limitations of this paradigm, alongside promising directions for future research.

实证评估表明，这些理论原则在表达式分解和数值回归任务中均表现出显著优势。最后，我们讨论了该范式的适用场景和固有局限性，并提出了未来研究的有益方向。