Thermoelectric Cooler Modeling and Optimization via Surrogate Modeling Using Implicit Physics-Constrained Neural Networks

Thermoelectric cooler (TEC) is a promising active cooling device to remove the localized hot spots precisely in VLSI chips. In this article, we use a novel implicit physics-constrained neural networks (called IPCNNs) to build a surrogate model for the single TEC device with the reduction from 3-D to 1-D. First, the surrogate model represented by the deep neural networks (DNNs) allows parameterization of key design and running parameters, such as current density, length, and thermal boundary conditions of the TEC. Second, the proposed method tries to partition the physics laws into two different groups, which then are enforced by supervised learning and physics-informed neural networks (PINNs) framework sequentially. Such implicit PCNN scheme can lead to much faster training speed and better convergent accuracy for the unsupervised training. The existing plain PINN enforces all the physics laws via the loss functions and the network tends to have very slow training speed and a large convergent error for large problems. An extreme learning machine (ELM) is used for the networks in the first stage. Compared with fully connected network (FCN) trained by the traditional back-propagation algorithm, ELM can be easily trained and converges much faster. Furthermore, by leveraging the differential nature of the DNN model, we can directly estimate the derivative of the cooling heat flux with respect to current density instead of using a finite difference approximation. The calculated derivatives are used to find the optimal current density to achieve maximum cooling heat flux via Newton’s method. Last but not least, we propose a novel hybrid finite element neural network (FENN) method to perform thermal analysis of the VLSI chip system with the TEC device. The DNN model is embedded into COMSOL through the heat flux boundary conditions. Experimental results show that the machine learning-based method can achieve about $8.5\times $ speedup with good accuracy than the COMSOL-based finite element method. Furthermore, the proposed IPCNN is more stable and accurate than the existing PINN. The proposed FENN can have a $5.1\times $ speedup and $5.4\times $ memory reduction over the traditional numerical method.