Prediction multimodal optical responses for ultrafast plasmonic based functional universal approximation theorem with exponential data efficiency

Machine learning offers efficient alternatives to traditional solvers for modeling nonlinear relationships between nanophotonic structures and their optical responses. However, neural network (NN)-based methods typically represent functional data as high-dimensional pointwise vectors without incorporating structural priorities, which limits their ability to capture the smoothness and continuity of such data and results in a strong dependence on large training datasets. To address this challenge, we propose a modeling framework based on the functional universal approximation (FUA) theorem, which leverages functional data analysis (FDA) to efficiently learn nonlinear function-on-scalar mappings by explicitly modeling functional structure. We validate the method on an aluminum nano ring-disk dimer, accurately predicting absorption, scattering, near-field spectra, and time-resolved electric fields using only 300 training samples, achieving R² values of 0.86, 0.84, 0.96, and 0.98, respectively. Compared to NN models, FUA shows superior data efficiency, robustness, and generalization, offering a promising approach for small-sample inverse design in nanophotonics.