Spectral modal operator learning framework with decoupled spatial and temporal representations for unsteady flows

Unsteady physical phenomena are governed by time-dependent partial differential equations, for which repeated numerical simulations under varying parameters remain computationally expensive. Although reduced-basis methods and operator learning frameworks, such as Proper Orthogonal Decomposition DeepONet (POD-DeepONet), have shown promise for time-dependent problems, they often struggle to capture high-frequency dynamics and nonlinear modal interactions due to the coupling between spatial and temporal representations inherent in modal decomposition-based approaches. To overcome these limitations, we propose Spectral Proper Orthogonal Decomposition DeepONet (SPOD-DeepONet), a novel operator learning framework that incorporates frequency-resolved modal representations into the DeepONet architecture. The proposed method extracts spectrally coherent spatial bases across varying parameter conditions and embeds them into the trunk network, while the branch network, composed of separate temporal and parameter sub-networks, predicts the corresponding time- and parameter-dependent coefficients. This architectural design explicitly decouples spatial representations from temporal and parametric variations, thereby mitigating spatiotemporal entanglement and spectral bias. Comparative evaluations on two benchmark unsteady-flow problems indicate that SPOD-DeepONet accurately captures both large- and fine-scale dynamics over all time intervals. In comparison with Naïve-DeepONet, Fourier Neural Operator, and POD-DeepONet, the proposed model converged more rapidly, improved reconstruction of high-frequency vortical structures, and produced physically consistent power spectral density distributions. This work presents a generalized operator learning framework that combines spectral-domain decomposition with neural operator architecture, enabling efficient and interpretable modeling for complex unsteady systems.