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Existing research mainly focuses on a single function. This study generalizes the scope to analyse a collection of functions observed at discretized times, with sampling frequencies varying from sparse to dense designs. The range of ODE models studied caters to diverse dynamic systems, and includes the complex nonlinear and non-Lipschitz scenarios. We introduce a new concept named functional moment method, a novel approach for parameter estimation within these ODE models and facilitating the recovery of curves for the discretely observed functions. 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