Recent advances in deep learning have enabled us to address the curse of dimensionality (COD) by solving problems in higher dimensions. A subset of such approaches of addressing the COD has led us to solving high-dimensional PDEs. This has resulted in opening doors to solving a variety of real-world problems ranging from mathematical finance to stochastic control for industrial applications. Although feasible, these deep learning methods are still constrained by training time and memory. Tackling these shortcomings, Tensor Neural Networks (TNN) demonstrate that they can provide significant parameter savings while attaining the same accuracy as compared to the classical Dense Neural Network (DNN). In addition, we also show how TNN can be trained faster than DNN for the same accuracy. Besides TNN, we also introduce Tensor Network Initializer (TNN Init), a weight initialization scheme that leads to faster convergence with smaller variance for an equivalent parameter count as compared to a DNN. We benchmark TNN and TNN Init by applying them to solve the parabolic PDE associated with the Heston model, which is widely used in financial pricing theory.

翻译：深度学习的近期进展使我们能够通过求解高维问题来应对维数灾难。解决维数灾难的此类方法中的一个子集引导我们求解高维偏微分方程。这为求解从数理金融到工业应用中的随机控制等各类现实世界问题打开了大门。尽管这些深度学习方法可行，但仍受限于训练时间和内存。为克服这些缺陷，张量神经网络证明了在达到与经典密集神经网络相同精度的同时，能够显著节省参数。此外，我们还展示了在相同精度下，张量神经网络比密集神经网络训练得更快。除张量神经网络外，我们还引入了张量网络初始化器，这是一种权重初始化方案，在参数数量与密集神经网络相当的情况下，能实现更快收敛且方差更小。我们通过将张量神经网络和张量网络初始化器应用于求解与金融定价理论中广泛使用的赫斯顿模型相关的抛物型偏微分方程来进行基准测试。