EGG-SR: Embedding Symbolic Equivalence into Symbolic Regression via Equality Graph

Abstract

Symbolic regression seeks to uncover physical laws from experimental data by searching for closed-form expressions, which is an important task in AI-driven scientific discovery. Yet the exponential growth of the search space of expression renders the task computationally challenging. A promising yet underexplored direction for reducing the effective search space and accelerating training lies in symbolic equivalence: many expressions, although syntactically different, define the same function -- for example, $\log(x_1^2x_2^3)$, $\log(x_1^2)+\log(x_2^3)$, and $2\log(x_1)+3\log(x_2)$. Existing algorithms treat such variants as distinct outputs, leading to redundant exploration and slow learning. We introduce EGG-SR, a unified framework that integrates equality graphs (e-graphs) into diverse symbolic regression algorithms, including Monte Carlo Tree Search (MCTS), deep reinforcement learning (DRL), and large language models (LLMs). EGG-SR compactly represents equivalent expressions through the proposed EGG module, enabling more efficient learning by: (1) pruning redundant subtree exploration in EGG-MCTS, (2) aggregating rewards across equivalence classes in EGG-DRL, and (3) enriching feedback prompts in EGG-LLM. Under mild assumptions, we show that embedding e-graphs tightens the regret bound of MCTS and reduces the variance of the DRL gradient estimator. Empirically, EGG-SR consistently enhances multiple baselines across challenging benchmarks, discovering equations with lower normalized mean squared error than state-of-the-art methods. Code implementation is available at: https://www.github.com/jiangnanhugo/egg-sr.