Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Lévy Equations

Using the paper’s Fourier-transform definition of the fractional Laplacian and the definition of the fractional score, one directly verifies (−Δ)^{α/2}p_t = −∇·(p_t S_t^{(α)}) (Eq. (6)), and substituting this into the FPL equation (Eq. (3)) yields the conservation form with a plus sign in front of the second-order term; this aligns with the LL-PDE (Eq. (8)) and A_α (Eq. (9)) stated in the paper. The candidate solution carries out exactly this computation (same method), notes the sign, and derives the LL-PDE correctly. The paper’s displayed Eq. (7) contains a sign typo on the second-order term and an extraneous α on σ(t), but Eqs. (6), (8), and (9) are correct and mutually consistent with the corrected sign, so the substance matches the candidate’s proof. See Eqs. (3)–(9) in the PDF for the definitions and statements used here .