Partial Differential Equations (PDEs) are used to model a variety of
dynamical systems in science and engineering. Recent advances in deep learning
have enabled us to solve them in a higher dimension by addressing the curse of
dimensionality in new ways. However, deep learning methods are constrained by
training time and memory. To tackle these shortcomings, we implement Tensor
Neural Networks (TNN), a quantum-inspired neural network architecture that
leverages Tensor Network ideas to improve upon deep learning approaches. We
demonstrate that TNN 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. We benchmark TNN by applying them to solve parabolic PDEs,
specifically the Black-Scholes-Barenblatt equation, widely used in financial
pricing theory, empirically showing the advantages of TNN over DNN. Further
examples, such as the Hamilton-Jacobi-Bellman equation, are also discussed.

本文介绍了如何通过 Tensor Neural Networks 来解决 Partial Differential Equations 的问题，实现了与 Deep Neural Network 同样精度的情况下更小的参数及更快的训练速度，并以 Black-Scholes-Barenblatt equation 模型为例进行了测试验证。