SMILE: Bridging Continuous Optimization and Discrete Symbolic Recovery

2026-09-03

AI Summary

SMILE框架通过结合连续优化和离散符号恢复解决符号回归中的慢收敛问题，利用数据结构分析、参数学习及符号化简化过程提高模型的准确性和鲁棒性。

Abstract

Symbolic regression (SR) discovers closed-form mathematical expressions from data, offering interpretability beyond black-box models. Existing methods suffer from slow convergence in combinatorial search spaces and lack mechanisms to exploit compositional structure in the data. We introduce SMILE (Sine, Multiplication, Identity, Logarithm, Exponential), a hybrid framework that unifies continuous gradient-based optimization with discrete symbolic recovery through three stages: structural analysis of the data to identify the compositional hierarchy of the target expression, continuous optimization to learn parameters of a network that encodes the target expression using interpretable activations, and symbolic recovery through structured pruning, coefficient optimization, and rounding. This final stage distills the learned network into a compact expression with exact symbolic constants. We evaluate SMILE on SRBench across ground-truth and black-box datasets, with ablation studies validating each component. SMILE achieves the highest symbolic solution rate at the largest noise levels, demonstrating strong robustness where competing methods degrade substantially. It consistently lies on the Pareto front of accuracy versus complexity, recovering significantly simpler expressions in a fraction of the time required by the competing methods.

Problem

• Symbolic Regression
• Combinatorial Search Spaces
• Compositional Structure

Innovation

• hybrid framework
• continuous optimization
• discrete symbolic recovery
• compositional hierarchy
• structured pruning