Data-driven identification of nonlinear normal modes via physics-integrated deep learning

Identifying the characteristic coordinates or modes of nonlinear dynamical systems is critical for understanding, analysis, and reduced-order modeling of the underlying complex dynamics. While normal modal transformation exactly characterizes any linear systems, there exists no such a general mathematical framework for nonlinear dynamical systems. Nonlinear normal modes (NNMs) are natural generalization of the normal modal transformation for nonlinear systems; however, existing research for identifying NNMs has relied on theoretical derivation or numerical computation from the closed-form equation of the system, which is usually unknown. In this work, we present a new data-driven framework based on physics-integrated deep learning for nonlinear modal identification of unknown nonlinear dynamical systems from the system response data only. Leveraging the universal modeling capacity and learning flexibility of deep neural networks, we first represent the forward and inverse nonlinear modal transformations through the physically interpretable deep encoder–decoder architecture, generalizing the modal superposition to nonlinear dynamics. Furthermore, to guarantee correct nonlinear modal identification, the proposed deep learning architecture integrates prior physics knowledge of the defined NNMs by embedding a unique dynamics-coder with physics-based constraints, including generalized modal properties, dynamics evolution, and future-state prediction. We test the proposed method by a series of study on the conservative and non-conservative Duffing systems with cubic nonlinearity and observe that the proposed data-driven framework is able to identify NNMs with invariant manifolds, energy-dependent nonlinear modal spectrum, and future-state prediction for unknown nonlinear dynamical systems from response data only; these identification results are found consistent with those from theoretically derived or numerically computed from closed-form equations. We also discuss its implementations and limitations for nonlinear modal identification of dynamical systems.