A compiler-based framework for automated numerical method selection via deterministic finite automata

Abstract

This study introduces a compiler-integrated framework for the automated selection of numerical methods through deterministic finite automata (DFA)-based structural classification of mathematical expressions. Traditional numerical method selection approaches typically rely on expert knowledge or data-driven models, which often lack interpretability and fail to integrate seamlessly with compiler systems. In contrast, the proposed framework employs a domain-specific language, lexical analysis, canonicalization, and DFA-based pattern recognition to classify mathematical expressions into categories such as linear, polynomial, nonlinear, recurrence relations, and ordinary differential equations (ODEs). Subsequently, a rule-based selection engine maps each expression to the most appropriate numerical solver, including Newton’s method, Bisection, Secant, Euler, RK2, and RK4 methods. Evaluation of the framework on a dataset of 150 expressions yielded 96.67% classification accuracy and 97% correctness in method selection. The solvers demonstrate convergence behavior consistent with theoretical expectations, while the computational overhead remains under 2 milliseconds. This approach offers an interpretable and efficient alternative to machine-learning-based methods and demonstrates the feasibility of integrating numerical reasoning directly into compiler systems and mathematical software.