A Deep Learning Framework for Scalar-on-Function Models

We propose a deep learning framework for scalar-on-function regression with possibly multiple functional predictors observed on irregular grids. The method represents each functional covariate by finitely many basis coefficients computed from the observed trajectories and fits a deep neural network using these coefficients as features. We establish a general nonasymptotic excess risk bound for the neural network estimator that decomposes the error into statistical, truncation, discretization, and network approximation terms. We then specialize the theory to two important settings. (i) For continuous functional models, an equal-width neural network estimator achieves a nearly minimax optimal rate up to a log-logarithm factor. (ii) For functional multiple-index models, we design a bottleneck-equal-width neural network estimator that exploits the low-dimensional index-link structure and attains a nearly minimax optimal rate up to a logarithm factor. The results provide theoretical support for neural network-based scalar-on-function regression and highlight the importance of architecture design.