HilbertNet: A Probabilistic Machine Learning Framework for Frequency Response Extrapolation of Electromagnetic Structures

In this article, we propose a probabilistic machine learning framework for extrapolating the frequency response of distributed physical circuits. For the structures where there is hidden dependency between higher and lower frequency features, we propose a method to extrapolate the response while providing confidence intervals harnessing Bayesian recurrent neural networks (RNN) thereby avoiding extensive simulations and saving computational time. To address complex-valued impedance, Hilbert transform is used to relate the real and imaginary parts where a Hilbert-based RNN architecture is proposed called Hilbert Net to extrapolate the frequency response. We apply the technique to four applications: 1) A simple microstrip transmission line circuit for proof of concept, 2) coupled waveguide filter operating in D-band comparing with measured results, 3) fifth-order interdigital bandpass filter for 28 GHz band, and 4) complex stack-up power delivery network (PDN) having a sharply changing response to test the framework limits. Results show that our architecture performs accurate extrapolation with a normalized mean square error of 0.008 squared with 95% confidence for a typical PDN. Using probabilistic networks, we achieve a tight confidence bound on our results. Furthermore, the reliability of Hilbert Net is assessed as to how far the response can be extrapolated.