Deep Learning Solves High-Dimensional Reflected Brownian Motion

A novel deep learning method has been introduced to tackle the challenging problem of determining the stationary distribution of high-dimensional Reflected Brownian Motion (RBM). RBMs are crucial for analyzing complex stochastic systems, but their closed-form solutions are rare, and computing performance metrics like tail probabilities has been largely intractable. The new framework leverages the basic adjoint relationship (BAR) and combines a carefully designed loss function, an optimized training data sampling procedure, and a specialized neural network architecture. This integrated approach allows the model to accurately and efficiently learn the Laplace transform of RBMs. Evaluations against RBM instances with known ground-truth tail probabilities demonstrated the method's effectiveness, achieving near-perfect predictions in high-dimensional settings. This breakthrough positions the deep learning approach as a powerful and generalizable tool for analyzing stochastic systems that extend beyond the capabilities of traditional analytical methods.