In modern applications such as ECG monitoring, neuroimaging, wearable sensing, and industrial equipment diagnostics, complex and continuously structured data are ubiquitous, presenting both challenges and opportunities for functional data analysis. However, existing methods face a critical trade-off: conventional functional models are limited by linearity, whereas deep learning approaches lack interpretable region selection for sparse effects. To bridge these gaps, we propose a sparse Bayesian functional deep neural network (sBayFDNN). It learns adaptive functional embeddings through a deep Bayesian architecture to capture complex nonlinear relationships, while a structured prior enables interpretable, region-wise selection of influential domains with quantified uncertainty. Theoretically, we establish rigorous approximation error bounds, posterior consistency, and region selection consistency. These results provide the first theoretical guarantees for a Bayesian deep functional model, ensuring its reliability and statistical rigor. Empirically, comprehensive simulations and real-world studies confirm the effectiveness and superiority of sBayFDNN. Crucially, sBayFDNN excels in recognizing intricate dependencies for accurate predictions and more precisely identifies functionally meaningful regions, capabilities fundamentally beyond existing approaches.

翻译：在心电图监测、神经影像、可穿戴传感和工业设备诊断等现代应用中，复杂且连续结构的数据无处不在，这为函数型数据分析带来了挑战与机遇。然而，现有方法面临一个关键权衡：传统的函数模型受限于线性假设，而深度学习方法则缺乏对稀疏效应进行可解释区域选择的能力。为弥补这些不足，我们提出了一种稀疏贝叶斯函数深度神经网络（sBayFDNN）。该模型通过深度贝叶斯架构学习自适应函数嵌入，以捕捉复杂的非线性关系；同时，一种结构化先验使得能够对具有量化不确定性的影响域进行可解释的、区域层面的选择。在理论上，我们建立了严格的近似误差界、后验一致性以及区域选择一致性。这些结果为贝叶斯深度函数模型提供了首个理论保证，确保了其可靠性与统计严谨性。在实证方面，全面的模拟实验和真实世界研究证实了sBayFDNN的有效性和优越性。至关重要的是，sBayFDNN在识别复杂依赖关系以实现精准预测方面表现卓越，并能更精确地识别具有功能意义的区域，这些能力从根本上超越了现有方法。