Uncertainty quantification for deep learning-based schemes for solving high-dimensional backward stochastic differential equations

TL;DR

This paper develops an efficient uncertainty quantification model for deep learning schemes solving high-dimensional BSDEs, enabling reliable estimation of solution variability with a single run, thus improving scheme evaluation and hyperparameter tuning.

Contribution

The authors introduce a novel UQ model that accurately estimates mean and standard deviation of solutions from a single run, reducing computational cost and enhancing reliability in high-dimensional BSDE solutions.

Findings

The UQ model reliably estimates mean and STD of solutions.

Estimated STD captures multiple sources of uncertainty.

Model improves scheme comparison and hyperparameter selection.

Abstract

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.