Quasi-potential and drift decomposition in stochastic systems by sparse identification

TL;DR

This paper introduces a machine learning-based method combining sparse identification and action minimization to efficiently compute the quasi-potential landscape and drift decomposition in stochastic systems, applicable in physics, biology, and economics.

Contribution

It presents a novel approach that uses sparse learning to decompose the drift and determine the quasi-potential from a single trajectory, improving computational efficiency.

Findings

Successfully applied to 2D and 3D systems with various landscapes.

Accurately decomposes drift into gradient and orthogonal components.

Provides complete quasi-potential landscapes from minimal data.

Abstract

The quasi-potential is a key concept in stochastic systems as it accounts for the long-term behavior of the dynamics of such systems. It also allows us to estimate mean exit times from the attractors of the system, and transition rates between states. This is of significance in many applications across various areas such as physics, biology, ecology, and economy. Computation of the quasi-potential is often obtained via a functional minimization problem that can be challenging. This paper combines a sparse learning technique with action minimization methods in order to: (i) Identify the orthogonal decomposition of the deterministic vector field (drift) driving the stochastic dynamics; (ii) Determine the quasi-potential from this decomposition. This decomposition of the drift vector field into its gradient and orthogonal parts is accomplished with the help of a machine learning-based sparse identification technique. Specifically, the so-called sparse identification of non-linear dynamics (SINDy) [1] is applied to the most likely trajectory in a stochastic system (instanton) to learn the orthogonal decomposition of the drift. Consequently, the quasi-potential can be evaluated even at points outside the instanton path, allowing our method to provide the complete quasi-potential landscape from this single trajectory. Additionally, the orthogonal drift component obtained within our framework is important as a correction to the exponential decay of transition rates and exit times. We implemented the proposed approach in 2- and 3-D systems, covering various types of potential landscapes and attractors.