  This paper investigates the application of mini-batch gradient descent to
semiflows. Given a loss function, we introduce a continuous version of
mini-batch gradient descent by randomly selecting sub-loss functions over time,
defining a piecewise flow. We prove that, under suitable assumptions on the
gradient flow, the mini-batch descent flow trajectory closely approximates the
original gradient flow trajectory on average. Additionally, we propose a
randomized minimizing movement scheme that also approximates the gradient flow
of the loss function. We illustrate the versatility of this approach across
various problems, including constrained optimization, sparse inversion, and
domain decomposition. Finally, we validate our results with several numerical
examples.
