Matrix-Free Transfer Function Prediction Using Model Reduction and Machine Learning

We propose a novel technique for fast predicting the transfer function of linear systems without information on system matrices by combining machine learning (ML) with model-order reduction. The transfer function of a linear time-invariant system can be written as <inline-formula> <tex-math notation="LaTeX">$H(s)=C G(s)^{-1} B+D$ </tex-math></inline-formula>. In some situations, it is difficult to obtain the individual systems matrices that are implicitly included in <inline-formula> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula>. The only information available is the data of <inline-formula> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> and a number of frequency samples <inline-formula> <tex-math notation="LaTeX">$s_{i}$ </tex-math></inline-formula>. The proposed method derives a reduced-order model (ROM) of the transfer function in the form of neural networks using limited data of <inline-formula> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula>. Discrete structure-preserving reduced transfer functions at training samples <inline-formula> <tex-math notation="LaTeX">$s_{i}, i=1,\ldots,\ell $ </tex-math></inline-formula>, of the frequency are first generated based on the function values of <inline-formula> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> at those training samples, i.e., <inline-formula> <tex-math notation="LaTeX">$\hat H(s_{i})= \hat C \hat G(s_{i})^{-1} \hat B+D$ </tex-math></inline-formula>. An ROM of the original transfer function as a continuous function of the frequency is then learned using the data of the discrete reduced transfer functions. The original transfer function at any testing frequency can then be quickly predicted using either a compact ML model or a deep learning (DL) model with a few layers. The discrete reduced transfer functions are guaranteed to be accurate by a cheap and sharp error estimator, which guarantees the accuracy of the training data for ML. If only the data of the original transfer function are available, then the proposed ML method and DL method can be directly applied without generating the data of the discrete reduced transfer function. The proposed methods are tested on three partial element equivalent circuit (PEEC) models with many delays. The high efficiency and accuracy of the ROM are demonstrated.