🤖 AI Summary
Existing symbolic regression methods treat explicit (y = f(x)) and implicit (F(x, y) = 0) relationships separately and neglect structural reuse across expressions, resulting in low search efficiency and poor generalization. To address this, we propose the first unified, structure-aware evolutionary framework supporting both relationship types. Our method introduces a biologically inspired, reusable motif library; employs multi-island genetic programming; incorporates implicit derivative-guided fitness evaluation; and enforces subtree semantic consistency during crossover and mutation. This enables the first joint discovery of explicit and implicit relations. On the Nguyen benchmark, our approach achieves a 5.1% accuracy improvement and uniquely solves the highly challenging Nguyen-12 problem. It attains state-of-the-art performance on the Feynman equation suite and accelerates symbolic search by up to 100× over baseline methods on the Eureqa dataset—significantly enhancing both efficiency and robustness in scientific law discovery.
📝 Abstract
Symbolic regression (SR) seeks to recover closed-form mathematical expressions that describe observed data. While existing methods have advanced the discovery of either explicit mappings (i.e., $y = f(mathbf{x})$) or discovering implicit relations (i.e., $F(mathbf{x}, y)=0$), few modern and accessible frameworks support both. Moreover, most approaches treat each expression candidate in isolation, without reusing recurring structural patterns that could accelerate search. We introduce SymMatika, a hybrid SR algorithm that combines multi-island genetic programming (GP) with a reusable motif library inspired by biological sequence analysis. SymMatika identifies high-impact substructures in top-performing candidates and reintroduces them to guide future generations. Additionally, it incorporates a feedback-driven evolutionary engine and supports both explicit and implicit relation discovery using implicit-derivative metrics. Across benchmarks, SymMatika achieves state-of-the-art recovery rates, achieving 5.1% higher performance than the previous best results on Nguyen, the first recovery of Nguyen-12, and competitive performance on the Feynman equations. It also recovers implicit physical laws from Eureqa datasets up to $100 imes$ faster. Our results demonstrate the power of structure-aware evolutionary search for scientific discovery. To support broader research in interpretable modeling and symbolic discovery, we have open-sourced the full SymMatika framework.