Spectrum and Physics-Informed Neural Networks (SaPINNs) for Input-State-Parameter Estimation in Dynamic Systems Subjected to Natural Hazards-Induced Excitation

System identification under unknown external excitation is an inherently ill-posed problem, typically requiring additional knowledge or simplifying assumptions to enable reliable state and parameter estimation. The difficulty of the problem is further amplified in structural systems subjected to natural hazards such as earthquakes or windstorms, where responses are often highly transient, nonlinear, and spatially distributed. To address this challenge, we introduce Spectrum and Physics-Informed Neural Networks (SaPINNs) for efficient input--state--parameter estimation in systems under complex excitations characteristic of natural hazards. The proposed model enhances the neural network with governing physics of the system dynamics and incorporates spectral information of natural hazards by using empirically derived spectra as priors on the unknown excitations. This integration improves inference of unmeasured inputs, system states, and parameters without imposing restrictive assumptions on their dynamics. The performance of the proposed framework is demonstrated through comparative studies on both linear and nonlinear systems under various types of excitation, including the El Centro earthquake, where the seismic spectrum is assumed to be not precisely known. To account for predictive uncertainty, the proposed architecture is embedded within a Deep Ensemble (DEns) networks architecture, providing distributions over possible solutions. The results demonstrate that the proposed approach outperforms conventional PINNs, as the incorporation of spectral information introduces an inductive bias that guides the network more effectively through the solution space and enhances its ability to recover physically consistent state and parameter estimates with realistic uncertainty levels.