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In this paper, artificial neural-network approaches to electromagnetic (EM)-based modeling in both frequency and time domains and their applications to nonlinear circuit optimization are presented. Through accurate and fast EM-based neural models of passive components, we enable consideration of EM effects in high-frequency and high-speed computer-aided design, including component's geometrical/physical parameters as optimization variables. Formulations for standard frequency-domain neural modeling approach, and recent time-domain neural modeling approach based on state-space concept, are described. A new EM-based time-domain neural modeling approach combining existing knowledge in the form of equivalent circuits (ECs), with state-space equations (SSEs) and neural networks (NNs), called the EC-SSE-NN, is proposed. The EC-SSE-NN models allow EM behaviors of passive components in the circuit to interact with nonlinear behaviors of active devices, and facilitate nonlinear circuit optimization in the time domain. An automatic mechanism for EM data generation, which can lead to efficient training of neural models for EM components, is presented. Demonstration examples including EM-based frequency-domain optimization of a three-stage amplifier, time-domain circuit optimization in a multilayer printed circuit board, including geometrical/physical-oriented neural models of power-plane effects, and EM-based optimization of a high-speed interconnect circuit with embedded passive terminations and nonlinear buffers in the time domain are presented.