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We propose a new adjoint dynamic neural network (ADNN) technique aimed at enhancing computer-aided design (CAD) of high-speed VLSI modules. A novel formulation for exact sensitivities is derived employing the Lagrange functions approach, and by defining an adjoint of a dynamic neural network (DNN), for the first time. The proposed ADNN is a dynamic model that we solve using integration backwards through time. One ADNN solution can be used to efficiently compute exact sensitivities of the corresponding DNN with respect to all its parameters. Using these sensitivities, we developed a training algorithm that facilitates DNN learning of nonlinear transients directly from continuous time-domain waveform data. Resulting accurate and fast DNN models can be straightaway used for carrying out high-speed VLSI CAD in SPICE-like time-domain environment. The technique can also speed-up physics-based nonlinear circuit CAD through faster sensitivity computations. Applications of the proposed ADNN technique in transient modeling and nonlinear design are demonstrated through high-speed interconnect driver examples.