Comparative Analysis of Finite Difference Method and Neural Network Models for Solving the One-Dimensional Heat Equal
DOI:
https://doi.org/10.5281/zenodo.17411006Keywords:
Heat Equation, Finite Difference Method (FDM), Neural Network (NN), Data-Driven ModelingAbstract
This study explores the capability of a simple feedforward neural network (NN) to approximate the one-dimensional heat equation traditionally solved by the Finite Difference Method (FDM). Using data generated from a stable FTCS scheme, the NN was trained to map spatial–temporal inputs (x, t) to temperature outputs u(x, t). The model achieved a mean squared error of 4.6×10⁻⁵, accurately reproducing the FDM temperature distribution with minor deviations near the boundary at x = 1. While the NN’s inference time (0.827 s) exceeded that of FDM (0.018 s), it demonstrated strong generalization and reusability across finer grids, suggesting potential scalability for high-dimensional and real-time applications. The findings indicate that even a basic data-driven NN can closely emulate classical numerical solvers, bridging conventional and machine-learning-based approaches to partial differential equations. Future work will extend the model to higher-dimensional PDEs and incorporate physics-informed neural networks (PINNs) for improved boundary precision and physical consistency.
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