AT-PINN-HC: A refined time-sequential method incorporating hard-constraint strategies for predicting structural behavior under dynamic loads

Physics-informed neural networks (PINNs) have been rapidly developed and offer a new computational paradigm for solving partial differential equations (PDEs) in various engineering fields. Hard constraints on boundary and initial conditions represent a significant advancement in PINNs. Given that existing hard-constraint strategies are unsuitable for structural vibration problems, this work addresses this challenge by proposing three effective hard-constraint strategies specifically for vibrational issues. Notably, the relationship between solution accuracy and the derivatives of auxiliary functions for hard constraints is identified. Based on this, various types of auxiliary functions, including polynomial, power, trigonometric, exponential, and logarithmic functions, are proposed for each hard-constraint strategy. Integrating these hard-constraint strategies and auxiliary functions into PINNs, the advanced time-marching physics-informed neural networks with hard constraints (AT-PINN-HC) are introduced. A series of numerical experiments, involving a classical Euler−Bernoulli beam, a supersonic vehicle skin panel under multi-physics loads, and a vertical standing glass plate under wind load, demonstrate that the AT-PINN-HC methods can accurately solve vibration problems in long-duration simulations. Compared to existing PINNs, AT-PINN-HC can reduce solution errors by one to four orders of magnitude and enhance training efficiency by reducing the number of iterations by up to 78 %. Additionally, the present results indicate that appropriate hard-constraint strategies and auxiliary functions must be selected on a case-by-case basis: trigonometric auxiliary functions are most effective for imposing hard constraints on boundary displacement, while exponential auxiliary functions are optimal for implementing hard constraints on initial displacement and velocity. This study not only provides effective hard-constraint strategies for vibrational problems but also provides insights into constructing hard constraints and auxiliary functions for solving other time-dependent PDEs.