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 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 Green’s function. In this article, a novel approximation method for Green’s function and Green’s function gradient is presented, which leverages group-equivariant convolutional neural networks (G-CNNs). This can accompany existing multitier Green’s function data precomputation schemes. Additionally, it enables performing the crucial step of sampling according to Green’s function without explicit calculation of 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 through numerical experiments. The proposed scheme enjoys massive gains in terms of speed for advanced CMOS nm node setups with nonplanar dielectric media, compared to a straight finite differences method (FDM) solution, while maintaining good accuracy levels.