  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.
