This work presents the application of the inverse Physics-Informed Neural Networks (PINNs) algorithm for the estimation of parameters of the nonlinear dynamic Duffing system. To achieve fast convergence and accuracy during the neural network training with the smallest possible set of training data, a sine activation function was used. The research provides a comprehensive comparison of the capabilities of inverse PINNs in reproducing the parameters of the studied system using classical activation functions such as sigmoid, tanh, GELU, and the proposed sine function. The issue of training data resources (dense vs. sparse data) is also discussed. This research demonstrates the advantage of the model with the sine activation function when analysing very sparse data across a wide time domain of the Duffing system's solution. However, these studies also indicate the difficulty of estimating the frequency of the periodic driving force in the studied system using a sparse dataset for training. As an example, for a time domain of a chaotic solution of the Duffing equation t=100s, with a training set containing only 50 measurement points, PINNs with sine activation can easily reproduce the damping parameter and the oscillator potential, as well as the attractor in phase space, while models with the sigmoid, tanh, or GELU activation function are not even able to converge. Additionally, an attempt was made to reproduce the system's parameters, as well as the time series and the attractor, for noisy data.