Hybrid Adaptive Modeling using Neural Networks Trained with Nonlinear Dynamics Based Features

TL;DR

The paper tackles parameter drift in nonlinear dynamic systems by proposing a hybrid adaptive framework that integrates physics-based modeling with data-driven updates. It leverages perturbation-based nonlinear dynamics, via the method of multiple scales, to extract informative frequency-response features from a pair of coupled Duffing oscillators and uses mutual information to select a compact feature set. An artificial neural network maps these dynamics-based features to estimates of coupling and damping, enabling rapid adaptation under changing conditions. Compared with gray-box and purely numerical models, the approach delivers higher accuracy and much faster predictions, highlighting the practical value of embedding analytical dynamical insights into data-driven estimators.

Abstract

Accurate models are essential for design, performance prediction, control, and diagnostics in complex engineering systems. Physics-based models excel during the design phase but often become outdated during system deployment due to changing operational conditions, unknown interactions, excitations, and parametric drift. While data-based models can capture the current state of complex systems, they face significant challenges, including excessive data dependence, limited generalizability to changing conditions, and inability to predict parametric dependence. This has led to combining physics and data in modeling, termed physics-infused machine learning, often using numerical simulations from physics-based models. This paper introduces a novel approach that departs from standard techniques by uncovering information from nonlinear dynamical modeling and embedding it in data-based models. The goal is to create a hybrid adaptive modeling framework that integrates data-based modeling with newly measured data and analytical nonlinear dynamical models for enhanced accuracy, parametric dependence, and improved generalizability. By explicitly incorporating nonlinear dynamic phenomena through perturbation methods, the predictive capabilities are more realistic and insightful compared to knowledge obtained from brute-force numerical simulations. In particular, perturbation methods are utilized to derive asymptotic solutions which are parameterized to generate frequency responses. Frequency responses provide comprehensive insights into dynamics and nonlinearity which are quantified and extracted as high-quality features. A machine-learning model, trained by these features, tracks parameter variations and updates the mismatched model. The results demonstrate that this adaptive modeling method outperforms numerical gray box models in prediction accuracy and computational efficiency.

Figures (14)

• Figure 1: The overview of the adaptive modeling approach
• Figure 2: Frequency response of (a) oscillator $X$ and (b) oscillator $Y$ versus coupling coefficient $\delta$ and detuning parameter $\sigma_{1}$. For this case, $f = 1.0$, $d = 1.0$, $\epsilon = 0.1$, $\sigma_2 = 7.3$, $\beta = 10$ and $\delta$ ranging from $0.2$ to $2.0$.
• Figure 3: Frequency response of (a) oscillator $X$ and (b) oscillator $Y$ versus nonlinear coefficient $\beta$ and detuning parameter $\sigma_{1}$. In this case, $f = 1.0$, $d = 1.0$, $\epsilon = 0.1$, $\sigma_2 = 7.3$, $\delta = 1.0$, and $\beta$ ranges from $10$ to $70$.
• Figure 4: Frequency response of (a) oscillator $X$ and (b) oscillator $Y$ versus damping $d$ and detuning parameter $\sigma_{1}$. In this case, $d$ changes from $0.8$ to $2.0$, $f = 1.0$, $\epsilon = 0.1$, $\sigma_2 = 7.3$, $\delta = 1.0$, and $\beta=50$.
• Figure 5: Frequency response of (a) oscillator $X$ and (b) oscillator $Y$ versus excitation amplitude $f$ and detuning parameter $\sigma_{1}$. For these plots, $f$ ranges from $0.5$ to $1.5$, $d=1.0$, $\epsilon = 0.1$, $\sigma_2 = 7.3$, $\delta = 1.0$, and $\beta=50$.