Modeling Electrically Long Interconnects Using Physics-Informed Delayed Gaussian Processes

This work presents a machine learning technique to model wide-band scattering parameters (S-parameters) of interconnects in the frequency domain using a new Gaussian processes (GP) model. Standard GPs with a general-purpose kernel typically assume high smoothness and therefore are not suitable to model S-parameters that are highly dynamic and oscillating due to propagation delays. The new delayed Gaussian process ( $\tau$ GP) model employs a physics-informed kernel consisting of periodic components, whose fundamental frequencies are interpreted as tunable propagation delays. Then, the model hyperparameters are tuned using a combination of maximum marginal likelihood estimation (MMLE) and delay estimation using Gabor transform. The delay estimation allows one to automatically identify the optimal fundamental frequencies for the kernel, thus increasing the numerical stability of the hyperparameters tuning process. The resulting delayed Gaussian process model accurately predicts the S-parameter values at desired frequency points in the training interval. Two application examples demonstrate the increased accuracy of the new technique, compared to standard Gaussian processes, vector fitting (VF), and delayed vector fitting (DVF) rational models.