Formal Uncertainty Propagation for Stochastic Dynamical Systems with Additive Noise

In this paper, we consider discrete-time non-linear stochastic dynamical systems with additive process noise in which both the initial state and noise distributions are uncertain. Our goal is to quantify how the uncertainty in these distributions is propagated by the system dynamics for possibly infinite time steps. In particular, we model the uncertainty over input and noise as ambiguity sets of probability distributions close in the $\rho$-Wasserstein distance and aim to quantify how these sets evolve over time. Our approach relies on results from quantization theory, optimal transport, and stochastic optimization to construct ambiguity sets of distributions centered at mixture of Gaussian distributions that are guaranteed to contain the true sets for both finite and infinite prediction time horizons. We empirically evaluate the effectiveness of our framework in various benchmarks from the control and machine learning literature, showing how our approach can efficiently and formally quantify the uncertainty in linear and non-linear stochastic dynamical systems.