Exploring the Learning Potential of ELM from Finite Difference Solutions for Heat Equation

Abstract

The study compares two methods, the finite difference and extreme learning machine (ELM), for solving the one-dimensional heat equation. The finite difference is a classical numerical method, while ELM is a machine learning-based approach that does not use a trial function to represent the solution. The results show that ELM can learns the finite difference. However, it should be noted that ELM faces challenges when directly solve the one-dimensional heat equation itself, as shown in the results. Despite the limitations observed in directly solving the one-dimensional heat equation with ELM, the study suggests that ELM still holds promise as a potentially viable alternative to classical numerical methods for solving partial differential equations (PDEs). Further research could explore incorporating optimization methods or employing a two-phase neural network, as proposed in our future work, to improve the accuracy of ELM's predictions for PDEs.