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This paper introduces a deep learning framework to learn the Laplace transform of high-dimensional reflected Brownian motion (RBM), addressing the challenge of computing stationary distributions and tail probabilities in complex stochastic systems. By leveraging the basic adjoint relationship (BAR) and optimizing the loss function alongside a tailored neural network architecture, the authors achieve near-perfect predictions in scenarios where traditional analytical methods fall short. The results underscore the method's potential as a versatile tool for performance analysis in high-dimensional stochastic environments, expanding the capabilities of existing analytical approaches.
Achieving near-perfect predictions of tail probabilities in high-dimensional reflected Brownian motion using deep learning could revolutionize performance analysis in complex stochastic systems.
The stationary distribution of reflected Brownian motion (RBM) plays an important role in the analysis of high-dimensional stochastic systems, yet closed-form solutions are known only for a few special cases. Computing important performance metrics, such as tail probabilities, is even more intractable, despite their practical relevance. In this paper, we develop a deep learning approach that accurately and efficiently learns the Laplace transform of high-dimensional RBMs based on the basic adjoint relationship (BAR). Our framework combines a careful design of the loss function, training data sampling procedure, and neural network architecture. We evaluate the proposed method on RBM instances with known ground-truth tail probabilities and demonstrate near-perfect prediction in high-dimensional settings, highlighting its potential as a general tool for analyzing stochastic systems beyond analytically tractable regimes. Our code can be found at https://github.com/zhangz73/NN4MGF.
