Physics-informed neural network for spectrum design in delayed vibration control

Time-delay coupling complicates spectrum design and stability-constrained optimization in active vibration control (AVC) systems, owing to infinite-dimensional dynamics and costly eigenvalue analysis. To address this problem, a physics-informed neural network is developed for surrogate-assisted leading spectrum design of delayed AVC systems. At its core, a physics-constrained eigenvalue regression network (PCER-Net) is constructed by embedding the residual of the delayed characteristic equation into training, thereby improving physical consistency, convergence stability, and generalization relative to a purely data-driven network. Based on the trained PCER-Net, a unified framework is established for stability classification, stability maximization, spectral-constrained inverse design, and stability-margin-constrained anti-resonance optimization. Experimental validation is performed on an active quasi-zero-stiffness (QZS) isolator with multiple delayed control structures, including single-delay feedback, dual-delay feedback, and hybrid feedforward-feedback configurations. In repeated optimization, the PCER-Net reduces the average task time from 14.53 s to 0.51 s for local optimization and from 351.22 s to 5.53 s for hybrid search, with break-even points reached after 116 and 6 tasks, respectively. Experimentally, the optimized hybrid controller achieves target transmissibility values of 0.2079 at 5 Hz and 0.1884 at 8 Hz, corresponding to 85.98% and 92.14% reductions relative to the passive QZS case.