Symbolic Intermediaries as a Linguistic-Numerical Interface for LLM-Driven Geometric Reasoning

Researchers propose symbolic intermediaries—compact mathematical expressions derived from symbolic regression—to bridge the gap between Large Language Models and physics simulators by converting continuous numerical outputs into interpretable symbolic forms. LLM-based agents using this interface outperformed genetic algorithms by 19-53% on mechanism synthesis tasks, demonstrating that translating simulator behavior into symbolic language enables grounded geometric reasoning without model retraining.

This research addresses a fundamental architectural limitation in applying LLMs to spatially-grounded engineering problems. While LLMs excel at linguistic and discrete symbolic reasoning, they struggle with continuous numerical outputs from physics simulators—distances, curvatures, trajectories—that resist tokenization. The proposed solution uses symbolic regression to discover compact analytical expressions that serve as a bridge, allowing simulators' continuous outputs to be translated into a discrete symbolic vocabulary that language models can manipulate and reason over. The framework operates through an agentic loop where a design agent converts natural language specifications into simulation code, a critique agent reasons over shared symbolic expressions, and a revision step incorporates feedback into refinements. On the MSynth benchmark for planar mechanism synthesis, three different LLM architectures achieved 19-53% performance improvements over budget-matched genetic algorithms, with error reductions reaching 63% when incorporating critique feedback. Analysis shows the symbolic interface shifts LLM reasoning from generic commentary toward concrete geometric verification, indicating meaningful engagement with domain semantics rather than surface-level pattern matching. This approach generalizes beyond mechanism design to any domain requiring linguistic interpretation of simulator behavior, potentially unlocking LLM applications across physics-based engineering, robotics, and computational design. The inference-time generalization without parameter updates suggests practical deployment advantages over fine-tuning approaches, though the scalability to complex, multi-constraint problems remains to be demonstrated.

• →Symbolic intermediaries translate continuous simulator outputs into discrete symbolic forms interpretable by LLMs, bridging a critical gap in geometric reasoning.
• →LLM agents using this interface outperform genetic algorithms by up to 53% on mechanism synthesis tasks, with 63% error reduction when incorporating critique feedback.
• →The agentic coordination loop—design, critique, revision—enables inference-time generalization without retraining, suggesting practical deployment efficiency.
• →Analysis confirms the symbolic interface drives grounded geometric reasoning rather than generic structural commentary across multiple model architectures.
• →The approach generalizes to any domain where continuous simulation behavior must be interpreted through language, expanding LLM applications in engineering.