Uncertainty quantification for deep learning-based schemes for solving
  high-dimensional backward stochastic differential equations
Lorenc Kapllani Long Teng Matthias Rottmann
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.