Can Large Language Models Solve Engineering Equations? A Systematic Comparison of Direct Prediction and Solver-Assisted Approaches

TL;DR

This study systematically compares large language models' ability to solve engineering equations directly versus using hybrid approaches with classical solvers, finding hybrid methods significantly improve accuracy.

Contribution

It provides a comprehensive evaluation of LLMs for engineering problem-solving, demonstrating their strengths in symbolic manipulation and limitations in iterative numerical precision.

Findings

Solver-assisted approaches reduce errors by up to 81.8%.

LLMs excel at symbolic manipulation but struggle with precise iterative calculations.

Domain-specific improvements vary, with electronics showing the most significant gains.

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.