The miniaturization of transistors down to 5nm and beyond, plus the increasing complexity of integrated circuits, significantly aggravate short channel effects, and demand analysis and optimization of more design corners and modes. Simulators need to model output variables related to circuit timing, power, noise, etc., which exhibit nonlinear behavior. The existing simulation and sign-off tools, based on a combination of closed-form expressions and lookup tables are either inaccurate or slow, when dealing with circuits with more than billions of transistors. In this work, we present CSM-NN, a scalable simulation framework with optimized neural network structures and processing algorithms. CSM-NN is aimed at optimizing the simulation time by accounting for the latency of the required memory query and computation, given the underlying CPU and GPU parallel processing capabilities. Experimental results show that CSM-NN reduces the simulation time by up to $6\times$ compared to a state-of-the-art current source model based simulator running on a CPU. This speedup improves by up to $15\times$ when running on a GPU. CSM-NN also provides high accuracy levels, with less than $2\%$ error, compared to HSPICE.