The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination of Monte Carlo methods and variational formulations, using neural networks for function approximation. Extending previous work (Richter et al., 2021), we argue that tensor trains provide an appealing framework for parabolic PDEs: The combination of reformulations in terms of backward stochastic differential equations and regression-type methods holds the promise of leveraging latent low-rank structures, enabling both compression and efficient computation. Emphasizing a continuous-time viewpoint, we develop iterative schemes, which differ in terms of computational efficiency and robustness. We demonstrate both theoretically and numerically that our methods can achieve a favorable trade-off between accuracy and computational efficiency. While previous methods have been either accurate or fast, we have identified a novel numerical strategy that can often combine both of these aspects.

数值逼近偏微分方程在高维度上面临巨大挑战，因为传统的基于网格的方法受到维度灾难的困扰。最近的尝试依赖于蒙特卡罗方法和变分公式的结合，使用神经网络进行函数逼近。本文在前人工作的基础上，认为张量网络为抛物型偏微分方程提供了一个吸引人的框架：在反向随机微分方程和回归型方法的重构相结合的方法中，利用潜在低秩结构实现了压缩和高效计算的优势。强调连续时间观点，我们开发了迭代方案，其在计算效率和稳健性方面有所不同。我们在理论上和数值上都证明了我们的方法能在精度和计算效率之间取得有利的权衡。而以往的方法要么准确要么快速，我们已经找到了一种新的数值策略，经常能够同时兼顾这两个方面。

从连续时间公式到离散化方案：张量列车和强健回归用于BSDEs和抛物型PDEs