A Data-Driven Approach for Discovering the Most Probable Transition Pathway for a Stochastic Carbon Cycle System

The paper correctly formulates the Onsager–Machlup (OM) action for the stochastic carbonate system with state-dependent diagonal diffusion, derives the Euler–Lagrange (EL) equations, computes that the scalar curvature term vanishes for the induced metric, and presents data-driven computations that match the numerical claims in the candidate solution (notably the sharp drop in c_end at ν=0.1 to 9.8782 μmol·kg−1, the peak of the OM action at ν=0.1, the rightward endpoint drift on the limit cycle as T increases at ν=0, and the optimal transition time T*≈2.9×10^4 y) . The candidate solution reaches the same quantitative conclusions and adds standard calculus-of-variations structure (existence, regularity, and the natural transversality condition for a free terminal point on the limit cycle manifold). The paper does not state this transversality condition explicitly, but its two-step discretize-then-minimize procedure is consistent with it. Hence both are correct; the model supplies additional theoretical details not spelled out in the paper.