An augmented physics-informed neural network approach with trainable scaling for nonlinear dynamic analysis

Abstract

Nonlinear dynamic problems are ubiquitous in engineering applications, and accurately solving their governing equations is essential for understanding system behavior. Physics-informed neural networks (PINNs) have emerged as a new computing paradigm for solving partial differential equations. However, conventional PINNs struggle to predict accurate solutions for dynamic systems affected by strong nonlinearity, damping, and spatiotemporal coupling. To address this challenge, this work proposes a nonlinear vibration stepping PINN (NVS-PINN) approach for analyzing the complex nonlinear behavior of dynamic systems. This approach introduces a trainable scaling parameter within each time segment to adaptively adjust the network output. Furthermore, the hyperbolic tangent function is adopted as hard constraints to ensure that the network output consistently satisfies the initial and/or Dirichlet boundary conditions of nonlinear dynamic systems. Three illustrative examples, including a single-degree-of-freedom parametric Duffing oscillator, a two-degree-of-freedom nonlinear damped vibratory system, and a nonlinear elastic circular arch under wind load, are considered for validation. Numerical results demonstrate that the NVS-PINN approach can accurately predict long-duration nonlinear vibration responses, including complex multi-stable limit cycles, escape motions, and wind-induced vibrations. For the Duffing oscillator, the NVS-PINN approach achieves highly accurate results, with average relative errors of 5.3444×10−3 for the strongly nonlinear system and 4.9573×10−4 for the strongly nonlinear system with damping. For the elastic arch under wind load, incorporating a trainable scaling parameter in NVS-PINN reduces the maximum relative error by about 82.6 %. Moreover, using smaller time intervals can improve network accuracy.