Complex Equation Learner: Rational Symbolic Regression with Gradient Descent in Complex Domain

TL;DR

This paper introduces a complex-valued extension of the Equation Learner model that enables stable gradient-based symbolic regression involving operators with singularities, such as division and logarithms.

Contribution

It proposes a complex weight approach to overcome real-domain optimization issues, allowing unconstrained use of operators with domain restrictions.

Findings

The method converges stably with target expressions containing poles.

It can recover singular behaviors from frequency response data.

Validated on symbolic regression benchmarks.

Abstract

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.