The Koopman Expectation: An Operator Theoretic Method for Efficient Analysis and Optimization of Uncertain Hybrid Dynamical Systems

Abstract

For dynamical systems involving decision making, the success of the system greatly depends on its ability to make good decisions with incomplete and uncertain information. By leveraging the Koopman operator and its adjoint property, we introduce the Koopman Expectation, an efficient method for computing expectations as propagated through a dynamical system. Unlike other Koopman operator-based approaches in the literature, this is possible without an explicit representation of the Koopman operator. Furthermore, the efficiencies enabled by the Koopman Expectation are leveraged for optimization under uncertainty when expected losses and constraints are considered. We show how the Koopman Expectation is applicable to discrete, continuous, and hybrid non-linear systems driven by process noise with non-Gaussian initial condition and parametric uncertainties. We finish by demonstrating a 1700x acceleration for calculating probabilistic quantities of a hybrid dynamical system over the naive Monte Carlo approach with many orders of magnitudes improvement in accuracy. * For example, systems without explicit differential equations, i.e., data-driven models

Control of Stochastic Quantum Dynamics by Differentiable Programming

“…based on NNs also require a (potentially costly) training procedure [60]. Second, the expected values of the loss functions could be optimized by leveraging the Koopman expectation for direct computation of expected values from stochastic and uncertain models [61]. Additionally, one could approach this control problem by using an SDE moment expansion to generate ordinary differential equations for the moments and apply a closure relationship [62].…”

j-Wave: An open-source differentiable wave simulator

“…j-Wave as a differentiable forward model can also be exploited for uncertainty quantification. Besides Monte Carlo methods that can be accelerated in j-Wave using single-device and multipledevice parallel transformations, there is a growing body of techniques that are being developed to exploit simulation gradients for simulation-based inference [8,45]. For example, in [46], the use of linear uncertainty propagation (LUP) was proposed as a meta-programming method to endow arbitrary (differential) simulations with uncertainty propagation in the Julia language [47].…”