Neuro-Symbolic Hypothesis Engine: Autonomous Scientific Discovery via Hybrid Large Language Models and Symbolic Regression

Extracting governing mathematical laws directly from raw empirical datasets is a fundamental challenge in scientific computing and artificial intelligence. Classical symbolic regression methods suffer from combinatorial search spaces, while pure deep learning approaches lack mathematical interpretability. In this paper, we introduce the Neuro-Symbolic Hypothesis Engine, an autonomous quantitative discovery agent that combines the high-level semantic reasoning of Large Language Models (LLMs) with the speed of deterministic numerical solvers and localized evolutionary search. By leveraging a 6-stage hybrid pipeline—incorporating automated domain literature retrieval, whitelisted Abstract Syntax Tree (AST) safety gating, SciPy non-linear optimization, and local expression tree mutations—the engine discovers robust mathematical formulations across astrophysics, digital signals, and computer graphics. We validate the engine on four real-world scientific datasets, achieving R² > 0.977 in all cases while maintaining computational safety and efficiency.