Mapping-to-Parameter nonlinear functional regression with Iterative Local B-spline knot placement

Many real-world phenomena are inherently continuous, yet traditionally represented as finite-dimensional vectors or matrices of discrete data points. Functional data analysis offers a natural paradigm by modeling observations as continuous functions, preserving intrinsic continuity and structural dependencies, thereby better capturing real-world dynamics and their underlying truth. However, functional modeling within infinite-dimensional spaces presents significant challenges due to its infinite degrees of freedom and computational complexity. These difficulties have led most studies on functional regression to focus on linear models, with general nonlinear approaches remaining underdeveloped. This paper introduces the Mapping-to-Parameter model, a simple yet effective approach for nonlinear functional regression. The key idea is straightforward: transform nonlinear functional regression problems from infinite-dimensional function spaces to finite-dimensional parameter spaces, where standard machine learning techniques can be readily applied. This transformation is accomplished by uniformly approximating all input or output functions using a common set of B-spline basis functions of any chosen order and representing each function by its vector of basis coefficients. For optimal approximation, we develop a novel Iterative Local Placement algorithm that adaptively distributes knots according to localized function complexity while providing theoretical guarantees on approximation error bounds. The performance of the proposed knot placement algorithm is shown to be robust and efficient in both single-function approximation and multiple-function approximation contexts. Through several real-world applications, the effectiveness and superiority of the Mapping-to-Parameter model are demonstrated in handling both function-on-scalar regression and function-on-function regression problems, consistently outperforming state-of-the-art methods including statistical functional models, neural network models, and functional neural networks.