GENSR: Symbolic Regression Based in Equation Generative Space

TL;DR

GenSR introduces a generative latent space approach for symbolic regression, enabling more effective exploration of equations by leveraging a structured, smooth space guided by a Bayesian framework, improving accuracy and robustness.

Contribution

The paper proposes GenSR, a novel SR framework using a CVAE to create a continuous, smooth latent space for better equation search and a Bayesian perspective for theoretical guarantees.

Findings

Outperforms existing methods in accuracy and simplicity.

Robust to noisy data and computationally efficient.

Provides a theoretical foundation for the generative approach.

Abstract

Symbolic Regression (SR) tries to reveal the hidden equations behind observed data. However, most methods search within a discrete equation space, where the structural modifications of equations rarely align with their numerical behavior, leaving fitting error feedback too noisy to guide exploration. To address this challenge, we propose GenSR, a generative latent space-based SR framework following the `map construction -> coarse localization -> fine search'' paradigm. Specifically, GenSR first pretrains a dual-branch Conditional Variational Autoencoder (CVAE) to reparameterize symbolic equations into a generative latent space with symbolic continuity and local numerical smoothness. This space can be regarded as a well-structured `map'' of the equation space, providing directional signals for search. At inference, the CVAE coarsely localizes the input data to promising regions in the latent space. Then, a modified CMA-ES refines the candidate region, leveraging smooth latent gradients. From a Bayesian perspective, GenSR reframes the SR task as maximizing the conditional distribution $p(\mathrm{Equ.} \mid \mathrm{Num.})$, with CVAE training achieving this objective through the Evidence Lower Bound (ELBO). This new perspective provides a theoretical guarantee for the effectiveness of GenSR. Extensive experiments show that GenSR jointly optimizes predictive accuracy, expression simplicity, and computational efficiency, while remaining robust under noise.