Research·arXiv cs.LG·May 5

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

Symbolic regression, a core technique for discovering interpretable equations from data, has long struggled with operators that create mathematical singularities or domain constraints like division and logarithms. Researchers propose extending gradient-based equation learners into the complex number domain, allowing optimization to sidestep real-axis degeneracies and converge reliably even when target expressions contain poles. This removes artificial constraints that previously narrowed the search space, potentially expanding the class of discoverable models and improving interpretability in scientific machine learning workflows where symbolic equations drive downstream analysis.

Modelwire context

Explainer

The key insight is that moving optimization into the complex plane doesn't just add mathematical generality. It actively converts a hard constraint (avoiding division, logarithms) into a solvable problem by letting gradients flow through poles rather than hitting dead ends on the real axis.

This connects directly to the broader push toward interpretable scientific ML we've covered. The HyCOP paper from May 1st tackled opacity in neural operators by introducing modularity and symbolic reasoning. This work takes the opposite angle: it expands what symbolic regression can discover by removing artificial search-space boundaries. Both are attacking the same core tension in scientific ML (interpretability vs. expressiveness), but from different directions. Where HyCOP adds structure to neural surrogates, this work removes constraints from symbolic learners. Together they suggest the field is converging on hybrid approaches that blend symbolic and learned components.

If follow-up work demonstrates that complex-domain symbolic regression recovers known physics equations with poles (e.g., Coulomb potentials, resonance phenomena) faster than real-domain baselines on standard benchmarks within the next 6 months, the method has moved beyond theoretical interest. If adoption remains confined to toy problems or synthetic data, the practical bottleneck likely isn't the optimization domain but the search algorithm itself.

Coverage we drew on

HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs · arXiv cs.LG

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MentionsEquation Learner · Symbolic Regression

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