This paper introduces a tensor neural network (TNN) to address nonparametric regression problems. Characterized by its distinct sub-network structure, the TNN effectively facilitates variable separation, thereby enhancing the approximation of complex, unknown functions. Our comparative analysis reveals that the TNN outperforms conventional Feed-Forward Networks (FFN) and Radial Basis Function Networks (RBN) in terms of both approximation accuracy and generalization potential, despite a similar scale of parameters. A key innovation of our approach is the integration of statistical regression and numerical integration within the TNN framework. This integration allows for the efficient computation of high-dimensional integrals associated with the regression function. The implications of this advancement extend to a broader range of applications, particularly in scenarios demanding precise high-dimensional data analysis and prediction.

翻译：本文提出一种张量神经网络（TNN）以解决非参数回归问题。该网络以其独特的子网络结构为特征，有效促进了变量分离，从而增强了对复杂未知函数的逼近能力。我们的对比分析表明，在参数规模相近的情况下，TNN在逼近精度与泛化潜力方面均优于传统前馈网络（FFN）和径向基函数网络（RBN）。本方法的核心创新在于将统计回归与数值积分整合到TNN框架中。这种整合使得与回归函数相关的高维积分得以高效计算。这一进展的意义延伸至更广泛的应用领域，尤其在对高维数据分析与预测精度要求较高的场景中。