High-dimensional functional time series (HDFTS) are often characterized by nonlinear trends and high spatial dimensions. Such data poses unique challenges for modeling and forecasting due to the nonlinearity, nonstationarity, and high dimensionality. We propose a novel probabilistic functional neural network (ProFnet) to address these challenges. ProFnet integrates the strengths of feedforward and deep neural networks with probabilistic modeling. The model generates probabilistic forecasts using Monte Carlo sampling and also enables the quantification of uncertainty in predictions. While capturing both temporal and spatial dependencies across multiple regions, ProFnet offers a scalable and unified solution for large datasets. Applications to Japan's mortality rates demonstrate superior performance. This approach enhances predictive accuracy and provides interpretable uncertainty estimates, making it a valuable tool for forecasting complex high-dimensional functional data and HDFTS.

翻译：高维函数时间序列（HDFTS）通常具有非线性趋势和高空间维度的特征。由于非线性、非平稳性和高维度，此类数据对建模和预测提出了独特的挑战。我们提出了一种新颖的概率函数神经网络（ProFnet）来应对这些挑战。ProFnet 融合了前馈神经网络、深度神经网络与概率建模的优势。该模型通过蒙特卡洛采样生成概率预测，并能够量化预测中的不确定性。在捕捉多个区域间时空依赖性的同时，ProFnet 为大型数据集提供了一个可扩展的统一解决方案。在日本死亡率数据上的应用证明了其优越性能。该方法提升了预测准确性，并提供了可解释的不确定性估计，使其成为预测复杂高维函数数据和 HDFTS 的有价值工具。