UNCERTAINTY QUANTIFICATION FOR DEEP LEARNING-BASED SCHEMES FOR SOLVING HIGH-DIMENSIONAL BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

Deep learning-based numerical schemes for solving high-dimensional backward stochastic differential equations (BSDEs) have recently received a lot of scientific interest. Although 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 experiments demonstrate the UQ model's reliability in estimating mean and STD for deep learning-based BSDE schemes. The estimated STD effectively captures various uncertainties, showcasing its effectiveness. 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.