FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula

Both the paper and the model agree on the pathwise C^k differentiability of the stochastic flow under locally Lipschitz spatial derivatives (Theorem 4.2) and on using a Taylor expansion with o(||h||^k) remainder for a fixed Brownian realization, as stated in the paper and used for the stochastic sampling trick . They also agree that when the SDE is linear in the initial condition (e.g., the GBM-derived FlowKac SDE), the first-order Taylor expansion is exact because the flow is linear in x . However, in the GBM example the paper gives a Jacobian (and hence flow multiplier) with exponent ((−μ+3σ^2)/2)t, whereas solving dX̃_t = (−μ+2σ^2)X̃_t dt + σ X̃_t dW_t yields X̃_t(x)=x exp((−μ+3σ^2/2)t + σ W_t), i.e., exponent ((−2μ+3σ^2)/2)t. The paper’s μ-term is off by a factor of 1/2 (compare eq. (19) and the subsequent lines giving J and X̃) . Finally, the claim that exact agreement pθ ≡ pFK implies zero training loss directly follows from the squared-error objective used in the paper (their discrete loss, eq. (14)) .