Accurate calculations of the heat transfer and the resulting maximum wall temperature are essential for the optimal design of reliable and efficient regenerative cooling systems. However, predicting the heat transfer of supercritical methane flowing in cooling channels of a regeneratively cooled rocket combustor presents a significant challenge. High-fidelity CFD calculations provide sufficient accuracy but are computationally too expensive to be used within elaborate design optimization routines. In a previous work it has been shown that a surrogate model based on neural networks is able to predict the maximum wall temperature along straight cooling channels with convincing precision when trained with data from CFD simulations for simple cooling channel segments. In this paper, the methodology is extended to cooling channels with curvature. The predictions of the extended model are tested against CFD simulations with different boundary conditions for the representative LUMEN combustor contour with varying geometries and heat flux densities. The high accuracy of the extended model’s predictions, suggests that it will be a valuable tool for designing and analyzing regenerative cooling systems with greater efficiency and effectiveness.