🤖 AI Summary
This study addresses the widespread challenge in engineering of solving transcendental equations that typically require iterative numerical methods. It systematically evaluates the performance of large language models (LLMs) in both direct numerical prediction and hybrid symbolic–numerical approaches. The authors propose a novel hybrid solving paradigm that integrates LLM-based symbolic manipulation, initial value estimation, and the Newton–Raphson method, evaluated across 100 problems spanning seven engineering domains. Results demonstrate that LLMs are better suited as intelligent interfaces for symbolic operations rather than high-precision numerical computation. The hybrid approach reduces average relative error by 67.9%–81.8%, with improvements reaching up to 93.1% in electronics, significantly outperforming pure LLM-based prediction and confirming its effectiveness and broad applicability in engineering computation.
📝 Abstract
Transcendental equations requiring iterative numerical solution pervade engineering practice, from fluid mechanics friction factor calculations to orbital position determination. We systematically evaluate whether Large Language Models can solve these equations through direct numerical prediction or whether a hybrid architecture combining LLM symbolic manipulation with classical iterative solvers proves more effective. Testing six state-of-the-art models (GPT-5.1, GPT-5.2, Gemini-3-Flash, Gemini-2.5-Lite, Claude-Sonnet-4.5, Claude-Opus-4.5) on 100 problems spanning seven engineering domains, we compare direct prediction against solver-assisted computation where LLMs formulate governing equations and provide initial conditions while Newton-Raphson iteration performs numerical solution. Direct prediction yields mean relative errors of 0.765 to 1.262 across models, while solver-assisted computation achieves 0.225 to 0.301, representing error reductions of 67.9% to 81.8%. Domain-specific analysis reveals dramatic improvements in Electronics (93.1%) due to exponential equation sensitivity, contrasted with modest gains in Fluid Mechanics (7.2%) where LLMs exhibit effective pattern recognition. These findings establish that contemporary LLMs excel at symbolic manipulation and domain knowledge retrieval but struggle with precision-critical iterative arithmetic, suggesting their optimal deployment as intelligent interfaces to classical numerical solvers rather than standalone computational engines.