Extrapolation With Range Determination of 2-D Spectral Transposed Convolutional Neural Network for Advanced Packaging Problems

In this article, we propose the application of a 2-D spectral transposed convolutional neural network (S-TCNN) with extrapolation to reduce the number of trainable parameters during the upsampling process, leading to reduced training time and a decrease in computational resources. Our proposed model consists of three stages, namely the prediction stage that uses high-accuracy 2-D S-TCNN to predict partial frequency responses; the extrapolation stage that takes the output of the prediction stage and uses the Gaussian process (GP) to construct the mean and covariance matrix; and the extrapolation range determining stage responses to numerically compute the optimized output length. For extrapolation, we construct new kernel combinations and use maximum likelihood estimation (MLE) to address the problem of losing correlation between priors and extrapolated points as distances increase. We also form a quantitative method to determine the extrapolation range, which takes the distance and confidence interval (CI) into consideration. We apply our method to three application examples: 1) a staggered via; 2) an on-chip microstrip line; and 3) an 1× 8 slot antenna array. Results show that our model can reduce about 20% of the trainable parameters, 30% of memory storage, and 20% of the training time compared with the 2-D S-TCNN. The proposed method can also achieve a similar normalized mean-squared error (NMSE) level with a small tradeoff in the final loss.