Deep learning-based numerical schemes for solving high-dimensional backward stochastic differential equations (BSDEs) have recently raised plenty of scientific interest. While they enable numerical methods to approximate very high-dimensional BSDEs, their reliability has not been studied and is thus not understood. In this work, we study uncertainty quantification (UQ) for a class of deep learning-based BSDE schemes. More precisely, we review the sources of uncertainty involved in the schemes and numerically study the impact of different sources. Usually, the standard deviation (STD) of the approximate solutions obtained from multiple runs of the algorithm with different datasets is calculated to address the uncertainty. This approach is computationally quite expensive, especially for high-dimensional problems. Hence, we develop a UQ model that efficiently estimates the STD of the approximate solution using only a single run of the algorithm. The model also estimates the mean of the approximate solution, which can be leveraged to initialize the algorithm and improve the optimization process. Our numerical experiments show that the UQ model produces reliable estimates of the mean and STD of the approximate solution for the considered class of deep learning-based BSDE schemes. The estimated STD captures multiple sources of uncertainty, demonstrating its effectiveness in quantifying the uncertainty. Additionally, the model illustrates the improved performance when comparing different schemes based on the estimated STD values. Furthermore, it can identify hyperparameter values for which the scheme achieves good approximations.

翻译：近年来，基于深度学习的数值方案在求解高维倒向随机微分方程方面引发了大量科学兴趣。尽管这类方法能近似计算极高维度的倒向随机微分方程，但其可靠性尚未得到研究，因此仍不明确。本文针对一类基于深度学习的倒向随机微分方程求解方案，研究其不确定性量化问题。具体而言，我们梳理了这类方案中涉及的不确定性来源，并通过数值实验分析了不同来源的影响。传统上，为处理不确定性，需通过多次运行算法并采用不同数据集求解，再计算近似解的标准差，该方法计算成本极高，尤其在高维问题中更为突出。为此，我们提出一种不确定性量化模型，该模型仅需单次算法运行即可高效估计近似解的标准差。该模型同时能估计近似解的均值，可将其用于初始化算法并优化求解过程。数值实验表明，针对所考虑的这类基于深度学习的倒向随机微分方程求解方案，该不确定性量化模型能够可靠地估计近似解的均值与标准差。估计的标准差能综合反映多种不确定性来源，验证了其量化不确定性的有效性。此外，该模型通过比较不同方案估计的标准差值，展示了方案性能的提升；同时还可识别使方案获得良好近似效果的超参数取值。