We focus on nonlinear Function-on-Scalar regression, where the predictors are scalar variables, and the responses are functional data. Most existing studies approximate the hidden nonlinear relationships using linear combinations of basis functions, such as splines. However, in classical nonparametric regression, it is known that these approaches lack adaptivity, particularly when the true function exhibits high spatial inhomogeneity or anisotropic smoothness. To capture the complex structure behind data adaptively, we propose a simple adaptive estimator based on a deep neural network model. The proposed estimator is straightforward to implement using existing deep learning libraries, making it accessible for practical applications. Moreover, we derive the convergence rates of the proposed estimator for the anisotropic Besov spaces, which consist of functions with varying smoothness across dimensions. Our theoretical analysis shows that the proposed estimator mitigates the curse of dimensionality when the true function has high anisotropic smoothness, as shown in the classical nonparametric regression. Numerical experiments demonstrate the superior adaptivity of the proposed estimator, outperforming existing methods across various challenging settings. Moreover, the proposed method is applied to analyze ground reaction force data in the field of sports medicine, demonstrating more efficient estimation compared to existing approaches.

翻译：本文研究非线性函数对标量回归问题，其中预测变量为标量，响应变量为函数型数据。现有研究大多通过基函数（如样条函数）的线性组合来近似隐藏的非线性关系。然而，在经典的非参数回归中，已知这类方法缺乏自适应性，尤其当真实函数表现出高度空间非均匀性或各向异性光滑性时。为自适应地捕捉数据背后的复杂结构，我们提出一种基于深度神经网络模型的简易自适应估计器。该估计器可利用现有深度学习库轻松实现，便于实际应用。此外，我们推导了该估计器在各向异性Besov空间中的收敛速率，该空间包含具有跨维度变光滑性特征的函数。理论分析表明，当真实函数具有高度各向异性光滑性时，所提估计器能够缓解维度灾难问题，这与经典非参数回归中的结论一致。数值实验证明了所提估计器卓越的自适应能力，在多种挑战性设定下均优于现有方法。此外，我们将所提方法应用于运动医学领域的地面反作用力数据分析，结果表明其估计效率较现有方法更高。