Neural-based Transient Behavioral Modeling of IC Buffers for High-speed Interconnect Design

Artificial neural networks (ANN) have gained attention as fast and flexible vehicles to microwave modeling and design. This paper reviews a recent advance of neural network modeling, i.e., state-space dynamic neural network (SSDNN) for transient behavioral modeling of high-speed nonlinear circuits. The SSDNN model can be directly trained from the input and output waveforms without relying on the circuit internal details. A training algorithm exploiting adjoint sensitivities is summarized for training the model in an efficient manner. An example of the SSDNN technique for IC buffer modeling and its use with transmission line elements in high-speed interconnect design are included. DOI: 10.2529/PIERS060907175229 With the continuous increase of signal speed and frequency, signal integrity (SI) in VLSI packages becomes more and more prominent. Fast and accurate representations of the nonlinear analog behaviors of driver/receiver buffers are the key to the success of SI-based design of high-speed interconnects with nonlinear terminations [1, 2]. As such, developing efficient buffer models for transient applications has become an important topic [3–5]. To ensure model reliability in circuit simulations, the model stability remains one of the most critical aspects of nonlinear transient modeling. In the neural network community, global asymptotical stability and global exponential stability have been studied for some special classes of dynamic networks, e.g., Hopfield neural networks [6], recurrent neural networks [7], and discrete-time state-space neural networks [8]. Recently stability for ANN-based analog microwave modeling has also been addressed [9]. This paper summarizes a state-space dynamic neural network (SSDNN) technique for modeling nonlinear transient behaviors of IC drivers and receivers [5]. We describe the detailed structure of SSDNN and how to train the model based on the transient waveforms from the original circuits. An example is provided to demonstrate the application of the SSDNN model in coupled transmission line environments. Let u ∈ <M be transient input signals of a nonlinear circuit, e.g., input voltages and currents, and y ∈ <K be transient output signals of a nonlinear circuit, e.g., output voltages and currents where M and K are the numbers of circuit inputs and outputs respectively. Based on combining state-space concept and continuous recurrent neural network method [10], the SSDNN nonlinear model is formulated as [5] { ẋ(t) = −x(t) + τgANN (u(t),x(t),w) y(t) = Cx(t) (1) where x = [x1, . . . , xN ]T ∈ <N is the vector of the state variables and N denotes the dimension of the state-space, or order of the model. Here gANN = [gANN 1, . . . , gANN N ]T represents a feedforward Multilayer Perceptron (MLP) neural network [11], which has M + N input neurons and N output neurons, with weight parameters w and a constant scaling parameter τ . C = [cij ] ∈ <K×N is the output matrix, which maps state-space into output-space. Training is an essential step to ensure the SSDNN model can accurately represent the original nonlinear circuit. The training data for SSDNN is generated from the detailed nonlinear circuit in the form of transient waveforms. Let L denote total number of transient waveforms used for training. Let ud(t) and y i d(t) represent ith input and output transient waveforms sampled in the time-interval [0 , T1]. Let yi(t) represent SSDNN prediction of ith input waveform ud(t). The goal of training process is to adjust parameters in w and C such that the objective function