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.

翻译：偏微分方程的数值逼近在高维情形下面临巨大挑战，因为经典的网格方法饱受所谓的维度灾难之苦。近期的尝试依赖于蒙特卡洛方法与变分公式的结合，使用神经网络进行函数逼近。在前期工作（Richter 等人，2021）的基础上，我们认为张量列为抛物型偏微分方程提供了一个令人瞩目的框架：通过倒向随机微分方程的重述与回归类方法的结合，有望利用潜在的底层低秩结构，实现压缩与高效计算。强调连续时间视角，我们开发了在计算效率和鲁棒性方面各异的迭代方案。我们从理论和数值上证明，我们的方法能够在准确性与计算效率之间实现有利的权衡。尽管先前的方法要么精确但缓慢，要么快速但不够精确，但我们发现了一种新颖的数值策略，通常能够将这两方面优势结合起来。