  Generating eye diagrams by using a circuit simulator can be very
computationally intensive, especially in the presence of nonlinearities. It
often involves multiple Newton-like iterations at every time step when a
SPICE-like circuit simulator handles a nonlinear system in the transient
regime. In this paper, we leverage machine learning methods, to be specific,
the recurrent neural network (RNN), to generate black-box macromodels and
achieve significant reduction of computation time. Through the proposed
approach, an RNN model is first trained and then validated on a relatively
short sequence generated from a circuit simulator. Once the training completes,
the RNN can be used to make predictions on the remaining sequence in order to
generate an eye diagram. The training cost can also be amortized when the
trained RNN starts making predictions. Besides, the proposed approach requires
no complex circuit simulations nor substantial domain knowledge. We use two
high-speed link examples to demonstrate that the proposed approach provides
adequate accuracy while the computation time can be dramatically reduced. In
the high-speed link example with a PAM4 driver, the eye diagram generated by
RNN models shows good agreement with that obtained from a commercial circuit
simulator. This paper also investigates the impacts of various RNN topologies,
training schemes, and tunable parameters on both the accuracy and the
generalization capability of an RNN model. It is found out that the long
short-term memory (LSTM) network outperforms the vanilla RNN in terms of the
accuracy in predicting transient waveforms.
