CFIR-NR: A causal gray-box modeling architecture for low-frequency vibration signal prediction

Vibration transfer response prediction of isolation systems is critical for transfer path analysis, isolation design, and online monitoring in noise and vibration control. Under complex multi-level and mixed tonal-broadband excitations typical of engineering practice, conventional methods based on frequency response function (FRF) inversion suffer from noise sensitivity and ill-conditioning, making high-precision time-domain prediction difficult—particularly when isolation elements exhibit amplitude-dependent nonlinear characteristics. This paper proposes a gray-box modeling architecture (CFIR-NR) that combines a causal FIR backbone with a short-window nonlinear residual, establishing a strictly causal, single-point supervision time-domain streaming prediction framework. The long-memory causal FIR network captures the dominant linear transfer characteristics of the isolation system, while a lightweight MLP residual network compensates for amplitude-dependent nonlinear deviations. Experiments across 20 methods show that CFIR-NR achieves low-frequency prediction accuracy comparable to or better than the best baseline models. With approximately 12 discrete sweep conditions, it stabilizes time-domain prediction accuracy to R² ≥ 0.98 and enables extrapolation to complex excitation responses. Without requiring FRF measurement, CFIR-NR provides an accurate, interpretable, and practical solution for low-frequency vibration transfer problems.