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：在现代应用如心电图监测、神经影像学、可穿戴传感及工业设备诊断中，复杂且连续的结构化数据普遍存在，为函数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：传统函数模型受限于线性关系，而深度学习方法则缺乏针对稀疏效应的可解释区域选择。为解决这些不足，我们提出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，以捕捉复杂非线性关系，同时利用结构化先验实现对关键影响域的可解释区域选择，并提供量化不确定性。理论上，我们建立了严格的逼近误差界、后验一致性及区域选择一致性。这些结果为贝叶斯深度函数模型提供了首项理论保证，确保了其可靠性与统计严谨性。实证方面，全面的仿真实验和真实案例研究证实了sBayFDNN的有效性与优越性。关键在于，sBayFDNN擅长识别复杂依赖性以实现精准预测，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。