Convergence, design and training of continuous-time dropout as a random batch method

We study continuous-time dropout in controlled differential equations. We introduce a random-batch approximation of additive vector fields. On each time interval of length h, a random subset of components is activated and rescaled by its inclusion probabilities, yielding an unbiased approximation of the full field. We prove a uniform-in-time mean-square trajectory error of order mathcal O(h). At the distribution level, we derive Wasserstein and pointwise density estimates, together with global L1 bounds under moment assumptions. For supervised training, we prove uniform mathcal O(√ h) root-mean-square fluctuations of the randomized objective and an mathcal O(h) weak error for its expectation, leading to consistency of optimal values and near-minimizers and, under quadratic growth, convergence in control space. We further characterize how the sampling law affects the error constant through inclusion probabilities and field variance, compare fixed-size sampling, fixed partitions, and Bernoulli dropout, and derive a work--accuracy model for selecting the switching scale. Numerical experiments with neural ODEs illustrate trajectory and transport errors, sampling effects, and objective consistency at a fixed control.