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.

中文翻译：

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使用张量列格式求解高维抛物线形PDE

高维偏微分方程（PDE）在经济学，科学和工程学中无处不在。但是，由于传统的基于网格的方法往往会因维数的诅咒而受挫，因此它们的数值处理提出了巨大的挑战。在本文中，我们认为张量列为抛物线偏微分方程提供了一个吸引人的近似框架：以反向随机微分方程和张量格式中的回归类型方法进行的重新组合的组合，有望利用潜在的低阶结构实现两者压缩和高效计算。遵循这种范例，我们开发了新颖的迭代方案，涉及显式和快速或隐式和准确的更新。