Experimental observation on a low-rank tensor model for eigenvalue problems

Neural networks-based machine learning methods are rapidly developed for various numerical problems, such as physics-informed neural networks (PINNs) [10, 11, 12], the deep Ritz method [2], and the deep Galerkin method [15]. One of the advantages of these approaches is that they show the possibility for solving high-dimensional problems. In [2], the deep learning techniques as well as the Monte-Carlo integration are used to solve eigenvalue problems, which provides a feasible strategy for high-dimensional cases. For the same eigenvalue problems, [16] applies a neural network-based low-rank tensor model, i.e. the tensor neural network (TNN), with a quadrature scheme to perform efficient numerical integration, and thus it achieves a much better result than [2]. Furthermore, [17] employs the TNN to solve the manybody Schrödinger equation, which emerges the practical value of such low-rank approximation method.