Solving the Heat Conduction Equation Using Butterfly Algorithm Guided by Machine Learning

Abstract

In this paper, we propose a hybrid computational framework based on a machine-learning model for the one-dimensional transient heat conduction equation, where the hybrid model is optimized using the Butterfly Optimization Algorithm (BOA). Rather than using classical numerical discretization techniques, the temperature distribution  is modeled by a feed-forward neural network capable of learning the spaio-temporal relation of space and time coordinates. Inspired by its potential as a derivative-free metaheuristic optimizer, BOA guides the training process to approximate the best parameters of the network, within physical constraints. We use a hybrid loss formulation that satisfies boundary conditions and initial conditions, interior supervised temperature samples, and the governing heat-equation residual to maintain numerical accuracy and consistency with physical principles. This BOA–ML solver is then assessed by representative simulations of fundamental heat conduction problems, with results provided in the form of convergence curves, temperature-field heatmaps, and absolute errors distributions. The results from the various simulation cases validate that on the one hand the proposed combination of data-driven and physics-based learning approach leverages the stable temperature prediction capability of the hybrid BOA-guided learning approach, which consistently yields smaller approximation errors than purely data-driven and purely physics-based training variants. The new proposed method may serve as a flexible alternative for heat-transfer modeling, and extensions to higher-dimensional conduction problems and non-simplified thermal boundary conditions are possible directions for future work.