Some Thoughts on Compositional Tensor Networks

In these notes we present some first ideas on the composition of tensor trains for the use in scientific computing. We discuss the relation to deep neural networks and the potential role compositional tensor trains can have in efficiently representing the solutions to PDEs. We illustrate the potential role compositional tensor trains might have for efficiently representing the solutions to high-dimensional PDEs circumventing the curse of dimensionality. Lastly, we embed the task of function regression on the set of compositional tensor trains into the context of semiglobal optimal control and mean field games.