用概率方法解决符号回归中的不确定性问题，提升物理规律发现的可靠性。
VaSST: Variational Inference for Symbolic Regression using Soft Symbolic Trees
- 用软符号树替代传统符号表达式，实现连续优化与概率解释
- 在费曼数据集上优于现有方法，结构恢复准确率更高
- 适合需要可信符号模型的科学发现场景
符号回归（SR）在人工智能驱动的科学发现中用于学习闭合形式的物理定律。然而现有方法多依赖启发式搜索或数据密集型策略，常假设低噪声环境且缺乏严格的不确定性量化，而完全概率化的符号回归方法仍很少见。我们提出一种可扩展的概率框架VaSST，基于变分推断。该方法使用软符号树——将离散运算符和特征选择替换为允许组件上的概率分布——将组合式的符号搜索从天文数字的表达式空间转化为高效的梯度优化，同时保持清晰的概率解释。学习到的软表示可生成符号结构的后验分布，通过后验感知的选择实现对多种合理符号形式的不确定性量化。在模拟实验和费曼符号回归数据库上的结果表明，与当前最优方法相比，VaSST在结构恢复和预测精度方面表现更优。
原文摘要 · Abstract (English)
Symbolic regression (SR) has gained recent traction in AI-driven scientific discovery for learning closed-form physical laws. Yet existing methods are dominated by heuristic search or data-intensive approaches that often assume low-noise regimes and lack principled uncertainty quantification, while fully probabilistic SR formulations remain scarce. We introduce a scalable probabilistic framework for SR, VaSST, based on variational inference. VaSST uses soft symbolic trees, a continuous relaxation of symbolic expression trees in which discrete operator and feature assignments are replaced by probability distributions over allowable components. This transforms combinatorial symbolic search through an astronomically large expression space into efficient gradient-based optimization while preserving a coherent probabilistic interpretation. The learned soft representations induce posterior distributions over symbolic structures, enabling uncertainty quantification across plausible symbolic forms through posterior-aware symbolic model selection. On simulated experiments and the Feynman Symbolic Regression Database, VaSST achieves strong structural recovery and predictive accuracy compared to state-of-the-art competing SR methods.
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