Approximate Stochastic Reachability for High Dimensional Systems

TL;DR

This paper introduces a data-driven method using kernel embeddings and random Fourier features to efficiently compute stochastic reachability safety probabilities in high-dimensional systems, avoiding the curse of dimensionality.

Contribution

It presents a novel approach combining kernel methods and Fourier features to handle high-dimensional stochastic reachability without prior system structure knowledge.

Findings

Successfully applied to a double integrator system.

Demonstrated capability on a million-dimensional nonlinear system.

Achieved computational efficiency in high-dimensional stochastic safety analysis.

Abstract

We present a method to compute the stochastic reachability safety probabilities for high-dimensional stochastic dynamical systems. Our approach takes advantage of a nonparametric learning technique known as conditional distribution embeddings to model the stochastic kernel using a data-driven approach. By embedding the dynamics and uncertainty within a reproducing kernel Hilbert space, it becomes possible to compute the safety probabilities for stochastic reachability problems as simple matrix operations and inner products. We employ a convergent approximation technique, random Fourier features, in order to alleviate the increased computational requirements for high-dimensional systems. This technique avoids the curse of dimensionality, and enables the computation of safety probabilities for high-dimensional systems without prior knowledge of the structure of the dynamics or uncertainty. We validate this approach on a double integrator system, and demonstrate its capabilities on a million-dimensional, nonlinear, non-Gaussian, repeated planar quadrotor system.