Sparse identification of quasipotentials via a combined data-driven method

The paper states the orthogonal decomposition f = −∇V + g with V solving the stationary Hamilton–Jacobi equation ∇V·∇V + f·∇V = 0 and g·∇V = 0, and notes U = 2V gives the small-noise invariant density pε ∝ exp(−U/ε). For the archetypal model (system (17)), it lists the exact U, ∇V, and g matching the candidate’s computations; the paper also uses the same asymptotic density formula pε(x) = Z−1 exp(−U(x)/ε). The candidate solution verifies these identities by direct calculation and derives the WKB/Hamilton–Jacobi relation from the stationary Fokker–Planck equation. Hence both are correct and aligned in substance, with the model giving explicit checks and the paper providing the general framework and the same exact example. See the paper’s Eqs. (3)–(5) for decomposition and orthogonality, the archetypal model with its exact U, ∇V, g, and the small-noise density formula p(x) = Z−1 exp(−U(x)/ε) as used in the applications .