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Using deep learning to learn physics of conduction heat transfer


In the present study, an advanced type of artificial intelligence, a deep neural network, is employed to learn the physic of conduction heat transfer in 2D geometries. A dataset containing 44,160 samples is produced by using the conventional finite volume method in a uniform grid of 64 × 64. The dataset includes four geometries of the square, triangular, regular hexagonal, and regular octagonal with random sizes and random Dirichlet boundary conditions. Then, the dataset of the solved problems was introduced to a convolutional Deep Neural Network (DNN) to learn the physics of 2D heat transfer without knowing the partial differential equation underlying the conduction heat transfer. Two loss functions based on the Mean Square Errors (MSE) and Mean of Maximum Square Errors (MMaSE) are introduced. The MMaSE is a new loss function, tailored for the physic of heat transfer. The 70%, 15%, and 15% of images are used for training DNN, testing DNN, and validation of the DNN during the training process, respectively. In the validation stage, the 2D domain with random boundary conditions, in which DNN has never seen them before, is introduced to DNN. Then, DNN is asked to estimate the temperature distribution. The results show that the DNNs are capable of learning physical problems without knowing the underlying fundamental governing equation. The error analysis for various training methods is reported and discussed. The outcomes reveal that DNNs are capable of learning physics, but using MMaSE as a tailored loss function could improve the training quality. A DNN trained by MMaSE provides a better temperature distribution compared to a DNN trained by MSE. As the 2D heat equation is a Laplace equation, which is practical in multiple physics, the results of the present study indicate a new direction for future computational methods and advanced modeling of physical phenomena, using a big dataset of observations.

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T :

Temperature (K)

T i :

Isothermal boundary conditions at the boundaries of the domain (K)

x :

x-Cartesian coordinate (m)

y :

y-Cartesian coordinate (m)

T :

Temperature field, actual temperature distribution image (target images)

P :

Predicted temperature distribution image (K)

N :

Number of images in a collection of images

R :

Number of rows in an image

C :

Number of columns in an image

β :

Parameters of Adam optimizer


First parameter of Adam optimizer


Second parameter of Adam optimizer


The segment of the boundary condition


Sigma index for summation on a collection of images


Sigma index for summation on all rows


Sigma index for summation on all columns


Square Error


Mean Square Errors


Mean of Maximum Square Errors


Maximum of Square Errors


Absolute Error


Mean Absolute Errors


Mean of Maximum Absolute Errors


Maximum of Absolute Errors


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Correspondence to Mohammad Bagher Tavakoli.

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Edalatifar, M., Tavakoli, M.B., Ghalambaz, M. et al. Using deep learning to learn physics of conduction heat transfer. J Therm Anal Calorim 146, 1435–1452 (2021).

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  • Conduction heat transfer
  • Deep convolutional neural networks
  • Deep learning
  • Laplace equation
  • Large dataset