  Many natural systems exhibit tipping points where changing environmental
conditions spark a sudden shift to a new and sometimes quite different state.
Global climate change is often associated with the stability of marine carbon
stocks. We consider a stochastic carbonate system of the upper ocean to capture
such transition phenomena. Based on the Onsager-Machlup action functional
theory, we calculate the most probable transition pathway between the
metastable and oscillatory states via a neural shooting method, and further
explore the effects of external random carbon input rates on the most probable
transition pathway, which provides a basis to recognize naturally occurring
tipping points. Particularly, we investigate the effect of the transition time
on the transition pathway and further compute the optimal transition time using
physics informed neural network, towards the maximum carbonate concentration
state in the oscillatory regimes. This work offers some insights on the effects
of random carbon input on climate transition in a simple model. Key words:
Onsager-Machlup action functional, the most probable transition pathway, neural
shooting method, stochastic carbon cycle system.
