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

2026-05-05

AI Summary

This work addresses a key limitation in existing gradient-based symbolic regression methods, which struggle to incorporate operations such as division and logarithms due to singularities or domain restrictions on the real line, thereby constraining the search space. To overcome this, the paper introduces complex-domain optimization into symbolic regression for the first time. By performing gradient descent in the complex plane, the optimization trajectory can circumvent singularities on the real axis, enabling stable and unconstrained use of a broad class of nonlinear operations. This approach effectively avoids degeneracies inherent in real-valued optimization, substantially expanding the set of learnable symbolic expressions. The method successfully reconstructs target functions with singularities on standard benchmarks and accurately recovers singular behaviors from experimental frequency response data.

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.

Problem

• symbolic regression
• singularities
• domain constraints
• gradient-based methods
• interpretable equations

Innovation

• symbolic regression
• complex domain optimization
• gradient descent
• singularities
• interpretable equations