Construction of quasi-potentials for stochastic dynamical systems: an optimization approach

TL;DR

This paper introduces a novel polynomial optimization method based on control theory to construct quasi-potentials for stochastic systems, providing analytical landscapes with bounds that improve understanding of system dynamics.

Contribution

It extends the Sum-of-Squares technique to generate explicit polynomial quasi-potentials for polynomial stochastic systems, offering bounds and decompositions of the dynamics.

Findings

Successfully computes quasi-potentials for high-dimensional linear systems.

Accurately models nonlinear stochastic systems with the proposed method.

Provides bounds that become tight under orthogonal decomposition.

Abstract

The construction of effective and informative landscapes for stochastic dynamical systems has proven a long-standing and complex problem. In many situations, the dynamics may be described by a Langevin equation while constructing a landscape comes down to obtaining the quasi-potential, a scalar function that quantifies the likelihood of reaching each point in the state-space. In this work we provide a novel method for constructing such landscapes by extending a tool from control theory: the Sum-of-Squares method for generating Lyapunov functions. Applicable to any system described by polynomials, this method provides an analytical polynomial expression for the potential landscape, in which the coefficients of the polynomial are obtained via a convex optimization problem. The resulting landscapes are based upon a decomposition of the deterministic dynamics of the original system, formed in terms of the gradient of the potential and a remaining "curl" component. By satisfying the condition that the inner product of the gradient of the potential and the remaining dynamics is everywhere negative, our derived landscapes provide both upper and lower bounds on the true quasi-potential; these bounds becoming tight if the decomposition is orthogonal. The method is demonstrated to correctly compute the quasi-potential for high-dimensional linear systems and also for a number of nonlinear examples.