Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations

The paper states the LL–ODE, the score PDE, and the probability-flow ODE for a general Itô SDE using A(x,t)=f−(1/2)∇·(GG^T) and L[s] exactly as in the candidate solution. The model’s derivation expands the Fokker–Planck equation, defines q_t=log p_t and s_t=∇q_t, and shows that ∂_t q_t=L[s_t], ∂_t s_t=∇L[s_t], and that the deterministic ODE ẋ=A−(1/2)GG^T s_t yields the same marginals as the SDE. These are the same identities asserted in the paper (eqs. (3), (5), (6), (7)), and the proof technique is the natural algebra behind those statements. The paper’s exposition is terse and cites prior work for derivations, while the model fills in the algebraic details; there is no substantive disagreement.