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This work advances the application of neural ordinary differential equations (ODEs) to circuit modeling. Prior works primarily utilized the recurrent neural network (RNN), which is a specific type of neural ODE. In this work, the capability of neural ODEs to represent different types of circuits is studied. Stability conditions are presented, both for neural ODEs in a standalone configuration and for neural ODEs with feedback connections, and practical techniques to impose the stability constraints during training are demonstrated. Based on the theoretical and experimental results, this work provides guidance as to when and how an accurate and stable neural ODE circuit model can be generated. [ABSTRACT FROM AUTHOR]