  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.
