Stochastic Lifting for Generating Trajectories of Stochastic Physical Systems

Many stochastic physical systems evolve smoothly over time in the sense that the distribution of states changes regularly with time. The precise transition from current to next state is often modeled as the interplay of a smooth map and an explicit source of randomness. Stochastic Lifting leverages this premise by attaching an independent, high-dimensional random label to each state transition in the training data and fitting a transition map from the current state and label to the next state using a standard regression loss. The labels act as auxiliary coordinates that let the model represent multiple plausible outcomes for similar current states, avoiding collapse to a mean prediction in the finite-sample size regime. At inference, drawing fresh labels and rolling the map forward generates diverse trajectories with a single network evaluation per time step, with the smoothness bias of the learned map supporting accurate sampling in practice.