Deep Divide-and-Reduce in Symbolic Regression

2026-07-26

AI Summary

Existing symbolic regression methods are limited in applicability due to their insufficient integration of mathematical and physical principles and reliance on inefficient brute-force search for expression simplification. This work proposes a novel paradigm grounded in rigorous mathematical derivation, eliminating brute-force substructure search in favor of a theoretically guaranteed deep divide-and-conquer reduction mechanism to enable efficient and reliable expression discovery. By synergistically combining formal mathematical reasoning, expression decomposition theory, and symbolic regression algorithms, the proposed method substantially outperforms current approaches in both expression decomposition and numerical regression tasks, demonstrating its generality, efficacy, and theoretical soundness.

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. 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. 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. 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.

Problem

• Symbolic Regression
• Expression Simplification
• Brute-force Search
• Mathematical Principles
• Complex Equations

Innovation

• Symbolic Regression
• Expression Decomposition
• Mathematical Deduction
• Brute-force Search Avoidance
• Theoretical Correctness