Abstract: To address the limitations of Physics-informed neural networks (PINN) and Physics-informed extreme learning machines (PIELM) in solving heat conduction problems, such as low computational efficiency and large boundary singular errors, an Energy-based physics-informed extreme learning machine (EPIELM) method is proposed. Based on the principle of minimum potential energy, this method transforms the governing partial differential equations of heat conduction into an energy functional extremum problem, which reduces the required order of derivatives and improves boundary solution accuracy. Furthermore, distance functions and the Karush-Kuhn-Tucker (KKT) Lagrange multiplier method are introduced to impose hard constraints on both regular and complex geometric domains. By utilizing the extreme learning machine (ELM) architecture, the constrained functional extremum problem is converted into a system of linear equations to solve for the output weights, thereby achieving a fast, mesh-free, and efficient solution. Numerical experiments demonstrate that the EPIELM method combines the high efficiency of ELM with the global stability of the energy variational principle, achieving a solution time that is only 2.45% of that required by PINN, and the maximum absolute error over the entire domain is 0.044 ℃. By reducing computational costs while ensuring high accuracy, this method provides a mesh-free numerical approach for the temperature field analysis of complex engineering structures.