Deep-Learning-Driven Random Walk Method for Capacitance Extraction

Abstract

Random-walk based methods for solving Partial Differential Equations (PDEs) exhibit significant advantages and applicability to numerous domains. One of these is parasitic capacitance extraction in modern silicon technologies, where they are essentially called upon to solve a boundary-value problem. In this regime, fast and accurate calculation of the Green’s function within the transition domains of the random walk is critical for competitive performance. Silicon processes in advanced nodes introduce complex or special dielectric structures that challenge currently established methods for characterizing the Green’s function. In this paper, a novel approximation method for the Green’s function and the Green’s function gradient is presented, which leverages Group-Equivariant Convolutional Neural Networks (G-CNNs). This can accompany existing multi-tier Green’s function data pre-computation schemes. Additionally, it enables performing the crucial step of sampling according to the Green’s function without explicit calculation of the Green’s function itself. A novel end-to-end random-walk architecture is discussed which incorporates such deep neural-networks, and its utility is demonstrated though numerical experiments. The proposed scheme enjoys massive gains in terms of speed for advanced CMOS nm node setups with non-planar dielectric media, compared to a straight Finite Differences Method (FDM) solution, while maintaining good accuracy levels.