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.

Many random physical systems change in a fairly smooth way, even though each exact next step can be unpredictable. Stochastic Lifting is a method for learning those random transitions. During training, it gives each observed transition an extra random label like a hidden tag, and learns how the current state plus that tag leads to the next state. These tags help the model represent many possible futures instead of averaging them into one blurry prediction. When used later, the model draws new random tags to generate different realistic trajectories efficiently, one step at a time.