Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and engineering. In recent years, a great number of computational approaches have been developed, most of them relying on a combination of Monte Carlo sampling and deep learning based approximation. For elliptic and parabolic problems, existing methods can broadly be classified into those resting on reformulations in terms of $\textit{backward stochastic differential equations}$ (BSDEs) and those aiming to minimize a regression-type $L^2$-error ($\textit{physics-informed neural networks}$, PINNs). In this paper, we review the literature and suggest a methodology based on the novel $\textit{diffusion loss}$ that interpolates between BSDEs and PINNs. Our contribution opens the door towards a unified understanding of numerical approaches for high-dimensional PDEs, as well as for implementations that combine the strengths of BSDEs and PINNs. The diffusion loss furthermore bears close similarities to $\textit{(least squares) temporal difference}$ objectives found in reinforcement learning. We also discuss eigenvalue problems and perform extensive numerical studies, including calculations of the ground state for nonlinear Schr\"odinger operators and committor functions relevant in molecular dynamics.

翻译：求解高维偏微分方程是经济学、科学与工程领域中的一个反复出现的挑战。近年来，大量计算方法被提出，其中大部分依赖于蒙特卡洛采样与基于深度学习的近似相结合。对于椭圆型和抛物型问题，现有方法大致可分为两类：一类基于$\textit{倒向随机微分方程}$（BSDEs）的重新表述，另一类旨在最小化回归型$L^2$误差（$\textit{物理信息神经网络}$，PINNs）。本文综述了相关文献，并提出了一种基于新颖的$\textit{扩散损失}$的方法，该方法在BSDE与PINN之间进行插值。我们的贡献为统一理解高维PDE的数值方法以及结合BSDE与PINN优势的实现打开了大门。此外，扩散损失与强化学习中的$\textit{（最小二乘）时序差分}$目标函数具有密切的相似性。我们还讨论了特征值问题，并进行了广泛的数值研究，包括非线性薛定谔算子的基态计算以及分子动力学中相关的committor函数。