A Causality- and Frequency-Aware Deep Learning Framework for Wave Elevation Prediction Behind Floating Breakwaters

Predicting the elevations of nonlinear wave fields behind floating breakwaters (FBs) is crucial for optimizing coastal engineering structures, enhancing safety, and improving design efficiency. Existing deep learning approaches exhibit limited generalization capability under unseen operating conditions. To address this challenge, this study proposes the Exogenous-to-Endogenous Frequency-Aware Network (E2E-FANet), a novel end-to-end neural network designed to model relationships between waves and structures. First, the Dual-Basis Frequency Mapping (DBFM) module leverages orthogonal cosine and sine bases to generate an adaptive time-frequency representation, enabling the model to effectively disentangle the evolving spectral components of wave signals. Second, the Exogenous-to-Endogenous Cross-Attention (E2ECA) module employs cross attention to explicitly model the unidirectional causal influence of floating breakwater motion on wave elevations. Additionally, a Temporal-wise Attention (TA) mechanism is incorporated that adaptively captures complex dependencies in endogenous variables. Extensive experiments, including generalization tests across diverse wave conditions and adaptability tests under varying relative water density (RW) conditions, demonstrate that E2E-FANet achieves superior predictive accuracy and robust generalization compared to mainstream models. This work emphasizes the importance of integrating causality and frequency-aware modeling in deep learning architectures for modeling nonlinear dynamics systems.