The goal of hybrid adaptive modeling is to create a framework that combines Science-based Modeling (SBM) and Data-based Modeling (DBM) methodologies to achieve higher accuracy and wider adaptability. For SBM, perturbation methods are leveraged to solve the governing equation of the target system analytically and derive the asymptotic solutions. Based on that, frequency responses with varying parameters can be obtained by parameterizing the asymptotic solutions. Subsequently, various potential features are extracted from the frequency response curves to estimate desired parameters. The flowchart of the proposed method is demonstrated in the top-right figure.

In this study, we introduced the coupled Duffing oscillator system as an experimental setup. The frequency response obtained from MMS (methods of multiple scales) is demonstrated in the middle-right figure.

The asymptotic solutions are used to map changes in parameters onto geometrical changes in the frequency response through feature extraction. The bottom-right figure depicts the feature extraction from nonlinear frequency responses for oscillators X. The jump phenomenon observed in the nonlinear frequency response indicates the occurrence of bifurcation and the switch of stable states. These jump points are the most pronounced and sensitive characteristics of parameter changes. We label these points as numbers 1 and 2 for jump up and 3 and 4 for jump down in bottom-right. We select the amplitudes and frequencies at these points as our features, denoted by 'a' and 'f' respectively, where 'X' corresponds to oscillator X, and '1, 2, 3, 4' correspond to jump up and jump down points.

Furthermore, the bending of the frequency response curve, determined by the slopes at bifurcation points, is a quantitative measure of the nonlinearity of the system. As a result, the slopes 'a' are likewise chosen as features for our research, where 'X' and 'Y' correspond to oscillators X and Y, and '1st' and '2nd' refer to the bending at the first resonance and the second resonance.

These features are based on asymptotic solutions derived from MMS, which is more capable of capturing the dynamics and nonlinearity of the system. Subsequently, these features are ranked and selected by mutual information [1] and fed into neural networks. The neural network trained with physics-based features demonstrates higher accuracy and stronger robustness in estimating unknown parameters for model updates.