  High-dimensional partial differential equations (PDEs) are ubiquitous in
economics, science and engineering. However, their numerical treatment poses
formidable challenges since traditional grid-based methods tend to be
frustrated by the curse of dimensionality. In this paper, we argue that tensor
trains provide an appealing approximation framework for parabolic PDEs: the
combination of reformulations in terms of backward stochastic differential
equations and regression-type methods in the tensor format holds the promise of
leveraging latent low-rank structures enabling both compression and efficient
computation. Following this paradigm, we develop novel iterative schemes,
involving either explicit and fast or implicit and accurate updates. We
demonstrate in a number of examples that our methods achieve a favorable
trade-off between accuracy and computational efficiency in comparison with
state-of-the-art neural network based approaches.
