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.

The quasipotential function allows for comprehension and prediction of the escape mechanisms from metastable states in nonlinear dynamical systems. This function acts as a natural extension of the potential function for non-gradient systems and it unveils important properties such as the maximum likelihood transition paths, transition rates and expected exit times of the system. Here, we leverage on machine learning via the combination of two data-driven techniques, namely a neural network and a sparse regression algorithm, to obtain symbolic expressions of quasipotential functions. The key idea is first to determine an orthogonal decomposition of the vector field that governs the underlying dynamics using neural networks, then to interpret symbolically the downhill and circulatory components of the decomposition. These functions are regressed simultaneously with the addition of mathematical constraints. We show that our approach discovers a parsimonious quasipotential equation for an archetypal model with a known exact quasipotential and for the dynamics of a nanomechanical resonator. The analytical forms deliver direct access to the stability of the metastable states and predict rare events with significant computational advantages. Our data-driven approach is of interest for a wide range of applications in which to assess the fluctuating dynamics.

In this paper, we present large deviation theory that characterizes the exponential estimate for rare events of stochastic dynamical systems in the limit of weak noise. We aim to consider next-to-leading-order approximation for more accurate calculation of mean exit time via computing large deviation prefactors with the research efforts of machine learning. More specifically, we design a neural network framework to compute quasipotential, most probable paths and prefactors based on the orthogonal decomposition of vector field. We corroborate the higher effectiveness and accuracy of our algorithm with a practical example. Numerical experiments demonstrate its powerful function in exploring internal mechanism of rare events triggered by weak random fluctuations.

The mean exit time escaping basin of attraction in the presence of white noise is of practical importance in various scientific fields. In this work, we propose a strategy to control mean exit time of general stochastic dynamical systems to achieve a desired value based on the quasipotential concept and machine learning. Specifically, we develop a neural network architecture to compute the global quasipotential function. Then we design a systematic iterated numerical algorithm to calculate the controller for a given mean exit time. Moreover, we identify the most probable path between metastable attractors with help of the effective Hamilton-Jacobi scheme and the trained neural network. Numerical experiments demonstrate that our control strategy is effective and sufficiently accurate.

We apply two independent data analysis methodologies to locate stable climate states in an intermediate complexity climate model. First, drawing from the theory of quasipotentials, and viewing the state space as an energy landscape with valleys and mountain ridges, we infer the relative likelihood of the identified multistable climate states, and investigate the most likely transition trajectories as well as the expected transition times between them. Second, harnessing techniques from data science, specifically manifold learning, we characterize the data landscape of the simulation data to find climate states and basin boundaries within a fully agnostic and unsupervised framework. Both approaches show remarkable agreement, and reveal, apart from the well known warm and snowball earth states, a third intermediate stable state in one of the two climate models we consider. The combination of our approaches allows to identify how the negative feedback of ocean heat transport and entropy production via the hydrological cycle drastically change the topography of the dynamical landscape of Earth's climate.