用复数优化解决符号回归中的奇点问题，让模型能自然处理除法、对数等复杂运算。
Complex Equation Learner: Rational Symbolic Regression with Gradient Descent in Complex Domain
- 将符号回归的权重扩展到复数域，避开实数轴上的奇异点
- 在含极点的目标函数上仍能稳定收敛，无需人为约束分母
- 适用于需要对数、平方根等运算的科学建模任务
符号回归旨在从数据中发现可解释的方程，但现代基于梯度的方法对引入奇点或定义域限制的运算（如除法、对数和平方根）失效。因此，现有方程学习模型通常回避这些运算或施加约束（如限制分母以避免极点），从而缩小了假设空间。本文提出一种复数权重扩展的方程学习方法，通过允许优化路径绕过实轴上的退化点，缓解实值优化的病态问题。该方法即使在目标表达式存在实域极点时也能稳定收敛，并支持对数、平方根等运算的无约束使用。我们在符号回归基准上验证了该方法的有效性，结果表明其能从实验频率响应数据中恢复出奇异行为。
原文摘要 · Abstract (English)
Symbolic regression aims to discover interpretable equations from data, yet modern gradient-based methods fail for operators that introduce singularities or domain constraints, including division, logarithms, and square roots. As a result, Equation Learner-type models typically avoid these operators or impose restrictions, e.g. constraining denominators to prevent poles, which narrows the hypothesis class. We propose a complex weight extension of the Equation Learner that mitigates real-valued optimization pathologies by allowing optimization trajectories to bypass real-axis degeneracies. The proposed approach converges stably even when the target expression has real-domain poles, and it enables unconstrained use of operations such as logarithm and square root. We Validate the method on symbolic regression benchmarks and show it can recover singular behavior from experimental frequency response data.
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