  Stochastic vegetation-water dynamical systems play a pivotal role in
ecological stability, biodiversity, water resource management, and adaptation
to climate change. This research proposes a machine learning-based method for
analyzing rare events in stochastic vegetation-water dynamical systems with
multiplicative Gaussian noise. Utilizing the Freidlin-Wentzell large deviation
theory, we derive the asymptotic expressions for the quasipotential and the
mean first exit time. Based on the decomposition of vector field, we design a
neural network architecture to compute the most probable transition paths and
the mean first exit time for both non-characteristic and characteristic
boundary scenarios. The results indicate that this method can effectively
predict early warnings of vegetation degradation, providing new theoretical
foundations and mathematical tools for ecological management and conservation.
Moreover, the method offers new possibilities for exploring more complex and
higher-dimensional stochastic dynamical systems.
