The Analytical Artificial Neural Networks Method

This study introduces the Analytical Artificial Neural Networks Method (AANNM), a groundbreaking framework that systematically converts the discrete, black-box outputs of neural network solvers into closed-form analytical solutions. The efficacy of AANNM is demonstrated by solving the differential equation governing the Kelvin-Voigt viscoelastic model. First, a high-fidelity numerical solution is obtained using a Physics-Informed Neural Network (PINN). The core innovation of AANNM is then deployed: the discrete PINN data is used to construct a system of algebraic equations, the solution of which yields the coefficients for a precise polynomial analytical expression. The derived AANNM solution is directly validated against the known exact analytical solution, demonstrating exceptional agreement and providing a more rigorous benchmark than comparisons with purely numerical methods. Crucially, while demonstrated with PINNs, the AANNM framework is solver-agnostic, designed to convert discrete solutions from any artificial neural network into analytical form. This inherent flexibility ensures the method's applicability to future ANN advancements, making it both timeless and adaptable. The proposed framework establishes AANNM as a transformative pipeline that bridges data-driven numerical models with rigorous analytical mathematics, significantly enhancing the interpretability, utility, and trustworthiness of machine learning in computational science.