Physics-informed convolutional neural networks for temperature field of heat source without labeled data

Recently, surrogate models based on deep learning have attracted much attention for engineering analysis and optimization. Since constructing data pairs in most engineering problems is time-consuming, data acquisition is becoming the predictive capability bottleneck of most deep surrogate models, which also exist in surrogate for thermal analysis and design. In contrast with data-driven learning, enforcing the physical laws in building surrogates has emerged as a promising alternative to reduce the dependence on annotated data. This paper develops a physics-informed convolutional neural network (CNN) for the thermal simulation surrogate without labeled data. Firstly, we leverage the finite difference method to integrate heat conduction equation and loss function construction, guiding surrogate model training to minimize the violation of physical laws. Since the solution is sensitive to boundary conditions, we properly impose hard constraints by padding in the Dirichlet and Neumann boundaries. The proposed network can learn a mapping from heat source layout to the steady-state temperature field without labeled data, which equals solving an entire family of partial difference equations (PDEs). Moreover, the neural network architecture is well-designed to improve the prediction accuracy of the problem at hand, and pixel-level online hard example mining is proposed to overcome the imbalance of optimization difficulty in the computation domain, which is beneficial to the network training of physics-informed learning. The experiments demonstrate that the proposed method can provide comparable predictions with numerical methods and data-driven deep learning models. We also conduct various ablation studies to investigate the effectiveness of the proposed network components and training methods in this paper. Furthermore, the developed methods can be applied to other design and optimization applications which need to solve parameterized PDEs.