  The stochastic finite volume method offers an efficient one-pass approach for
assessing uncertainty in hyperbolic conservation laws. Still, it struggles with
the curse of dimensionality when dealing with multiple stochastic variables. We
introduce the stochastic finite volume method within the tensor-train framework
to counteract this limitation. This integration, however, comes with its own
set of difficulties, mainly due to the propensity for shock formation in
hyperbolic systems. To overcome these issues, we have developed a
tensor-train-adapted stochastic finite volume method that employs a global WENO
reconstruction, making it suitable for such complex systems. This approach
represents the first step in designing tensor-train techniques for hyperbolic
systems and conservation laws involving shocks.
