让符号回归具备不确定性建模能力，可同时推断公式与参数分布。
Deep Variational Inference Symbolic Regression
- 用变分推断扩展深度符号回归，实现模型与参数的后验分布推断。
- 在简单场景下能准确恢复真实后验，表达式空间增大时仍保持稳定性能。
- 适合需要解释性且关注不确定性的科学建模任务，如物理规律发现。
符号回归可在不预先假设函数形式的情况下发现显式的可解释方程。贝叶斯方法通过在候选表达式上定义概率分布，增强了对噪声和有限数据下的不确定性量化能力。深度符号回归（DSR）利用神经网络生成符号表达式，但仅聚焦于寻找最优表达式，而非推断模型的后验分布。本文提出深度变分推断符号回归（DVISR），作为DSR的贝叶斯扩展。DVISR将原始奖励替换为证据下界（ELBO）的被积函数，并扩展网络架构以输出表达式中常数的分布，从而实现对表达式树及其对应常数的联合后验推断。我们验证了DVISR在简单设定下可准确恢复真实后验，无论是否包含常数项，并研究了其在表达式空间规模增大时的表现变化。结果表明，DVISR是迈向可扩展贝叶斯符号回归、实现完整符号模型不确定性建模的重要一步。
原文摘要 · Abstract (English)
Symbolic regression discovers explicit, interpretable equations without assuming a functional form in advance. A Bayesian approach strengthens this through probability distributions over candidate expressions, thus quantifying uncertainty in the presence of noisy and limited data. Deep Symbolic Regression (DSR) uses a neural network to generate symbolic expressions, but it is designed to identify a single best-fitting expression rather than infer a posterior distribution over models. We introduce Deep Variational Inference Symbolic Regression (DVISR), a variational Bayesian extension of DSR. DVISR replaces the original reward with the integrand of the evidence lower bound. It also extends the network architecture to output distributions over constants within expressions, enabling posterior inference over both expression trees and their associated constants. We show that DVISR can recover the true posterior in simple settings, both with and without constant tokens, and we examine how its performance changes as the size of the expression space increases. These results position DVISR as a step toward scalable Bayesian symbolic regression with uncertainty over full symbolic models.
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